# lfunc_search downloaded from the LMFDB on 26 July 2026. # Search link: https://www.lmfdb.org/L/2/3800/152.123/c0-0 # Query "{'degree': 2, 'conductor': 3800, 'spectral_label': 'c0-0'}" returned 306 lfunc_searchs, sorted by root analytic conductor. # Each entry in the following data list has the form: # [Label, $\alpha$, $A$, $d$, $N$, $\chi$, $\mu$, $\nu$, $w$, prim, arith, $\mathbb{Q}$, self-dual, $\operatorname{Arg}(\epsilon)$, $r$, First zero, Origin] # For more details, see the definitions at the bottom of the file. "2-3800-152.11-c0-0-0" 1.3771154809983834 1.8964470480054092 2 3800 "152.11" [] [[0.0, 0.0]] 0 true true false false -0.36720997736447175 0 0.68423752526969071470027603493 ["ModularForm/GL2/Q/holomorphic/3800/1/bd/a/3051/1", "ArtinRepresentation/2.3800.6t5.d.a"] "2-3800-152.11-c0-0-1" 1.3771154809983834 1.8964470480054092 2 3800 "152.11" [] [[0.0, 0.0]] 0 true true false false -0.033876644031138395 0 0.799358572018411046409269381166 ["ModularForm/GL2/Q/holomorphic/3800/1/bd/d/3051/1", "ArtinRepresentation/2.3800.12t18.e.a"] "2-3800-152.11-c0-0-2" 1.3771154809983834 1.8964470480054092 2 3800 "152.11" [] [[0.0, 0.0]] 0 true true false false -0.28387664403113844 0 1.01708103101501137684679887016 ["ModularForm/GL2/Q/holomorphic/3800/1/bd/f/3051/2"] "2-3800-152.11-c0-0-3" 1.3771154809983834 1.8964470480054092 2 3800 "152.11" [] [[0.0, 0.0]] 0 true true false false 0.21612335596886162 0 1.06910423054643075536548621834 ["ModularForm/GL2/Q/holomorphic/3800/1/bd/f/3051/1"] "2-3800-152.11-c0-0-4" 1.3771154809983834 1.8964470480054092 2 3800 "152.11" [] [[0.0, 0.0]] 0 true true false false -0.033876644031138395 0 1.14076181499277350436845037254 ["ModularForm/GL2/Q/holomorphic/3800/1/bd/b/3051/1", "ArtinRepresentation/2.3800.6t5.c.b"] "2-3800-152.11-c0-0-5" 1.3771154809983834 1.8964470480054092 2 3800 "152.11" [] [[0.0, 0.0]] 0 true true false false 0.29945668930219493 0 1.30942933133815676285840403259 ["ModularForm/GL2/Q/holomorphic/3800/1/bd/c/3051/1", "ArtinRepresentation/2.3800.12t18.d.a"] "2-3800-152.11-c0-0-6" 1.3771154809983834 1.8964470480054092 2 3800 "152.11" [] [[0.0, 0.0]] 0 true true false false -0.36720997736447175 0 2.72319435301908874115882307727 ["ModularForm/GL2/Q/holomorphic/3800/1/bd/e/3051/1", "ArtinRepresentation/2.3800.12t18.f.b"] "2-3800-152.123-c0-0-0" 1.3771154809983834 1.8964470480054092 2 3800 "152.123" [] [[0.0, 0.0]] 0 true true false false -0.28029332098734944 0 0.14961762086333696360935269126 ["ModularForm/GL2/Q/holomorphic/3800/1/cv/c/2251/1"] "2-3800-152.123-c0-0-1" 1.3771154809983834 1.8964470480054092 2 3800 "152.123" [] [[0.0, 0.0]] 0 true true false false 0.3863733456793173 0 0.63099925734261402372765677228 ["ModularForm/GL2/Q/holomorphic/3800/1/cv/b/2251/1"] "2-3800-152.123-c0-0-2" 1.3771154809983834 1.8964470480054092 2 3800 "152.123" [] [[0.0, 0.0]] 0 true true false false -0.25251554320957165 0 0.71757907700913506663827882447 ["ModularForm/GL2/Q/holomorphic/3800/1/cv/f/2251/1"] "2-3800-152.123-c0-0-3" 1.3771154809983834 1.8964470480054092 2 3800 "152.123" [] [[0.0, 0.0]] 0 true true false false 0.05304001234598393 0 0.75136841781040699102854412938 ["ModularForm/GL2/Q/holomorphic/3800/1/cv/a/2251/1"] "2-3800-152.123-c0-0-4" 1.3771154809983834 1.8964470480054092 2 3800 "152.123" [] [[0.0, 0.0]] 0 true true false false 0.3863733456793173 0 1.54505486513936447597751823243 ["ModularForm/GL2/Q/holomorphic/3800/1/cv/d/2251/1"] "2-3800-152.123-c0-0-5" 1.3771154809983834 1.8964470480054092 2 3800 "152.123" [] [[0.0, 0.0]] 0 true true false false 0.05304001234598393 0 1.56323275383286706708476674064 ["ModularForm/GL2/Q/holomorphic/3800/1/cv/e/2251/1"] "2-3800-152.123-c0-0-6" 1.3771154809983834 1.8964470480054092 2 3800 "152.123" [] [[0.0, 0.0]] 0 true true false false 0.24748445679042838 0 1.71637813427509189513980870520 ["ModularForm/GL2/Q/holomorphic/3800/1/cv/f/2251/2"] "2-3800-152.131-c0-0-0" 1.3771154809983834 1.8964470480054092 2 3800 "152.131" [] [[0.0, 0.0]] 0 true true false false -0.05304001234598393 0 0.73053505432368097083646628390 ["ModularForm/GL2/Q/holomorphic/3800/1/cv/a/1651/1"] "2-3800-152.131-c0-0-1" 1.3771154809983834 1.8964470480054092 2 3800 "152.131" [] [[0.0, 0.0]] 0 true true false false -0.24748445679042838 0 0.74751150436094506008877153765 ["ModularForm/GL2/Q/holomorphic/3800/1/cv/f/1651/2"] "2-3800-152.131-c0-0-2" 1.3771154809983834 1.8964470480054092 2 3800 "152.131" [] [[0.0, 0.0]] 0 true true false false -0.05304001234598393 0 0.893618540191142883717432693184 ["ModularForm/GL2/Q/holomorphic/3800/1/cv/e/1651/1"] "2-3800-152.131-c0-0-3" 1.3771154809983834 1.8964470480054092 2 3800 "152.131" [] [[0.0, 0.0]] 0 true true false false -0.3863733456793173 0 1.10532613996399056735028032042 ["ModularForm/GL2/Q/holomorphic/3800/1/cv/d/1651/1"] "2-3800-152.131-c0-0-4" 1.3771154809983834 1.8964470480054092 2 3800 "152.131" [] [[0.0, 0.0]] 0 true true false false -0.3863733456793173 0 1.38416345661263996789630470374 ["ModularForm/GL2/Q/holomorphic/3800/1/cv/b/1651/1"] "2-3800-152.131-c0-0-5" 1.3771154809983834 1.8964470480054092 2 3800 "152.131" [] [[0.0, 0.0]] 0 true true false false 0.28029332098734944 0 1.51538650696186285296774844298 ["ModularForm/GL2/Q/holomorphic/3800/1/cv/c/1651/1"] "2-3800-152.131-c0-0-6" 1.3771154809983834 1.8964470480054092 2 3800 "152.131" [] [[0.0, 0.0]] 0 true true false false 0.25251554320957165 0 1.52863128661068222437468926906 ["ModularForm/GL2/Q/holomorphic/3800/1/cv/f/1651/1"] "2-3800-152.139-c0-0-0" 1.3771154809983834 1.8964470480054092 2 3800 "152.139" [] [[0.0, 0.0]] 0 true true false false -0.358699448860543 0 0.000832598882436861477407193692 ["ModularForm/GL2/Q/holomorphic/3800/1/cv/f/1051/1"] "2-3800-152.139-c0-0-1" 1.3771154809983834 1.8964470480054092 2 3800 "152.139" [] [[0.0, 0.0]] 0 true true false false 0.4468561066950126 0 0.06633626023977120830009807562 ["ModularForm/GL2/Q/holomorphic/3800/1/cv/d/1051/1"] "2-3800-152.139-c0-0-2" 1.3771154809983834 1.8964470480054092 2 3800 "152.139" [] [[0.0, 0.0]] 0 true true false false 0.11352277336167924 0 0.856843530204907589373333334017 ["ModularForm/GL2/Q/holomorphic/3800/1/cv/a/1051/1"] "2-3800-152.139-c0-0-3" 1.3771154809983834 1.8964470480054092 2 3800 "152.139" [] [[0.0, 0.0]] 0 true true false false -0.2198105599716541 0 1.16414841980705346392993377780 ["ModularForm/GL2/Q/holomorphic/3800/1/cv/c/1051/1"] "2-3800-152.139-c0-0-4" 1.3771154809983834 1.8964470480054092 2 3800 "152.139" [] [[0.0, 0.0]] 0 true true false false 0.14130055113945703 0 1.60945670089102455026612486123 ["ModularForm/GL2/Q/holomorphic/3800/1/cv/f/1051/2"] "2-3800-152.139-c0-0-5" 1.3771154809983834 1.8964470480054092 2 3800 "152.139" [] [[0.0, 0.0]] 0 true true false false 0.11352277336167924 0 1.77345951636550670047352189157 ["ModularForm/GL2/Q/holomorphic/3800/1/cv/e/1051/1"] "2-3800-152.139-c0-0-6" 1.3771154809983834 1.8964470480054092 2 3800 "152.139" [] [[0.0, 0.0]] 0 true true false false 0.4468561066950126 0 1.84220947526851715491844553185 ["ModularForm/GL2/Q/holomorphic/3800/1/cv/b/1051/1"] "2-3800-152.35-c0-0-0" 1.3771154809983834 1.8964470480054092 2 3800 "152.35" [] [[0.0, 0.0]] 0 true true false false -0.14130055113945703 0 0.40174486317415070186052478663 ["ModularForm/GL2/Q/holomorphic/3800/1/cv/f/1251/2"] "2-3800-152.35-c0-0-1" 1.3771154809983834 1.8964470480054092 2 3800 "152.35" [] [[0.0, 0.0]] 0 true true false false -0.4468561066950126 0 0.58391819050418152457507952929 ["ModularForm/GL2/Q/holomorphic/3800/1/cv/b/1251/1"] "2-3800-152.35-c0-0-2" 1.3771154809983834 1.8964470480054092 2 3800 "152.35" [] [[0.0, 0.0]] 0 true true false false -0.11352277336167924 0 0.61065719934470144284918960642 ["ModularForm/GL2/Q/holomorphic/3800/1/cv/a/1251/1"] "2-3800-152.35-c0-0-3" 1.3771154809983834 1.8964470480054092 2 3800 "152.35" [] [[0.0, 0.0]] 0 true true false false 0.2198105599716541 0 1.45223696666061835205671100235 ["ModularForm/GL2/Q/holomorphic/3800/1/cv/c/1251/1"] "2-3800-152.35-c0-0-4" 1.3771154809983834 1.8964470480054092 2 3800 "152.35" [] [[0.0, 0.0]] 0 true true false false -0.11352277336167924 0 1.50786051014669123445812855798 ["ModularForm/GL2/Q/holomorphic/3800/1/cv/e/1251/1"] "2-3800-152.35-c0-0-5" 1.3771154809983834 1.8964470480054092 2 3800 "152.35" [] [[0.0, 0.0]] 0 true true false false 0.358699448860543 0 1.54056080972528817997209847102 ["ModularForm/GL2/Q/holomorphic/3800/1/cv/f/1251/1"] "2-3800-152.35-c0-0-6" 1.3771154809983834 1.8964470480054092 2 3800 "152.35" [] [[0.0, 0.0]] 0 true true false false -0.4468561066950126 0 1.64588176062350009232667802072 ["ModularForm/GL2/Q/holomorphic/3800/1/cv/d/1251/1"] "2-3800-152.37-c0-0-0" 1.3771154809983834 1.8964470480054092 2 3800 "152.37" [] [[0.0, 0.0]] 0 true true false false -0.4375 0 0.27668080574447311944771080826 ["ModularForm/GL2/Q/holomorphic/3800/1/o/g/1101/2"] "2-3800-152.37-c0-0-1" 1.3771154809983834 1.8964470480054092 2 3800 "152.37" [] [[0.0, 0.0]] 0 true true false false -0.3125 0 0.51044433845691587461890953830 ["ModularForm/GL2/Q/holomorphic/3800/1/o/g/1101/4"] "2-3800-152.37-c0-0-10" 1.3771154809983834 1.8964470480054092 2 3800 "152.37" [] [[0.0, 0.0]] 0 true true true true 0.0 0 1.17694312738159186309599882069 ["ModularForm/GL2/Q/holomorphic/3800/1/o/b/1101/1", "ModularForm/GL2/Q/holomorphic/3800/1/o/b", "ArtinRepresentation/2.3800.6t3.c.a", "ArtinRepresentation/2.3800.6t3.c"] "2-3800-152.37-c0-0-11" 1.3771154809983834 1.8964470480054092 2 3800 "152.37" [] [[0.0, 0.0]] 0 true true false false -0.06250000000000001 0 1.21349285112509968177632613427 ["ModularForm/GL2/Q/holomorphic/3800/1/o/g/1101/8"] "2-3800-152.37-c0-0-12" 1.3771154809983834 1.8964470480054092 2 3800 "152.37" [] [[0.0, 0.0]] 0 true true false true 0.0 0 1.22480060681873618762671818129 ["ModularForm/GL2/Q/holomorphic/3800/1/o/f/1101/1", "ArtinRepresentation/2.3800.9t3.a.c"] "2-3800-152.37-c0-0-13" 1.3771154809983834 1.8964470480054092 2 3800 "152.37" [] [[0.0, 0.0]] 0 true true false true 0.0 0 1.24394717335128879486726389519 ["ModularForm/GL2/Q/holomorphic/3800/1/o/f/1101/2", "ArtinRepresentation/2.3800.9t3.a.b"] "2-3800-152.37-c0-0-14" 1.3771154809983834 1.8964470480054092 2 3800 "152.37" [] [[0.0, 0.0]] 0 true true true true 0.0 0 1.26627674389081139921670943886 ["ModularForm/GL2/Q/holomorphic/3800/1/o/a/1101/1", "ModularForm/GL2/Q/holomorphic/3800/1/o/a", "ArtinRepresentation/2.3800.6t3.b.a", "ArtinRepresentation/2.3800.6t3.b"] "2-3800-152.37-c0-0-15" 1.3771154809983834 1.8964470480054092 2 3800 "152.37" [] [[0.0, 0.0]] 0 true true false false 0.1875 0 1.50711970670807164268213917037 ["ModularForm/GL2/Q/holomorphic/3800/1/o/g/1101/5"] "2-3800-152.37-c0-0-16" 1.3771154809983834 1.8964470480054092 2 3800 "152.37" [] [[0.0, 0.0]] 0 true true false false -0.1875 0 1.54463113061535235245841511351 ["ModularForm/GL2/Q/holomorphic/3800/1/o/g/1101/6"] "2-3800-152.37-c0-0-17" 1.3771154809983834 1.8964470480054092 2 3800 "152.37" [] [[0.0, 0.0]] 0 true true false true 0.0 0 1.55059409854462001210166804268 ["ModularForm/GL2/Q/holomorphic/3800/1/o/c/1101/3"] "2-3800-152.37-c0-0-18" 1.3771154809983834 1.8964470480054092 2 3800 "152.37" [] [[0.0, 0.0]] 0 true true false true 0.0 0 1.68694758213333139295167510214 ["ModularForm/GL2/Q/holomorphic/3800/1/o/e/1101/2"] "2-3800-152.37-c0-0-19" 1.3771154809983834 1.8964470480054092 2 3800 "152.37" [] [[0.0, 0.0]] 0 true true false true 0.0 0 1.72408719718483107940422132223 ["ModularForm/GL2/Q/holomorphic/3800/1/o/e/1101/3"] "2-3800-152.37-c0-0-2" 1.3771154809983834 1.8964470480054092 2 3800 "152.37" [] [[0.0, 0.0]] 0 true true false true 0.0 0 0.53475195151724166465600046100 ["ModularForm/GL2/Q/holomorphic/3800/1/o/c/1101/2"] "2-3800-152.37-c0-0-20" 1.3771154809983834 1.8964470480054092 2 3800 "152.37" [] [[0.0, 0.0]] 0 true true false false 0.4375 0 1.79224391649443448737962314976 ["ModularForm/GL2/Q/holomorphic/3800/1/o/g/1101/1"] "2-3800-152.37-c0-0-21" 1.3771154809983834 1.8964470480054092 2 3800 "152.37" [] [[0.0, 0.0]] 0 true true false true 0.0 0 1.83520021970847513060548334122 ["ModularForm/GL2/Q/holomorphic/3800/1/o/f/1101/3", "ArtinRepresentation/2.3800.9t3.a.a"] "2-3800-152.37-c0-0-3" 1.3771154809983834 1.8964470480054092 2 3800 "152.37" [] [[0.0, 0.0]] 0 true true false true 0.0 0 0.790026702321987899437065594714 ["ModularForm/GL2/Q/holomorphic/3800/1/o/c/1101/1"] "2-3800-152.37-c0-0-4" 1.3771154809983834 1.8964470480054092 2 3800 "152.37" [] [[0.0, 0.0]] 0 true true false true 0.0 0 0.821521948736724844879885488022 ["ModularForm/GL2/Q/holomorphic/3800/1/o/d/1101/1"] "2-3800-152.37-c0-0-5" 1.3771154809983834 1.8964470480054092 2 3800 "152.37" [] [[0.0, 0.0]] 0 true true false true 0.0 0 0.861330360567961029172910553528 ["ModularForm/GL2/Q/holomorphic/3800/1/o/e/1101/1"] "2-3800-152.37-c0-0-6" 1.3771154809983834 1.8964470480054092 2 3800 "152.37" [] [[0.0, 0.0]] 0 true true false false 0.3125 0 0.869058986565000773675567928839 ["ModularForm/GL2/Q/holomorphic/3800/1/o/g/1101/3"] "2-3800-152.37-c0-0-7" 1.3771154809983834 1.8964470480054092 2 3800 "152.37" [] [[0.0, 0.0]] 0 true true false true 0.0 0 0.888642821641465817593507004954 ["ModularForm/GL2/Q/holomorphic/3800/1/o/d/1101/2"] "2-3800-152.37-c0-0-8" 1.3771154809983834 1.8964470480054092 2 3800 "152.37" [] [[0.0, 0.0]] 0 true true false false 0.06250000000000001 0 1.05240540401779602092582256845 ["ModularForm/GL2/Q/holomorphic/3800/1/o/g/1101/7"] "2-3800-152.37-c0-0-9" 1.3771154809983834 1.8964470480054092 2 3800 "152.37" [] [[0.0, 0.0]] 0 true true false true 0.0 0 1.16998028326487072405259761940 ["ModularForm/GL2/Q/holomorphic/3800/1/o/d/1101/3"] "2-3800-152.43-c0-0-0" 1.3771154809983834 1.8964470480054092 2 3800 "152.43" [] [[0.0, 0.0]] 0 true true false false -0.20064719165680855 0 0.29971038553498067229088233058 ["ModularForm/GL2/Q/holomorphic/3800/1/cv/b/651/1"] "2-3800-152.43-c0-0-1" 1.3771154809983834 1.8964470480054092 2 3800 "152.43" [] [[0.0, 0.0]] 0 true true false false -0.20064719165680855 0 0.956308949984380190610881682794 ["ModularForm/GL2/Q/holomorphic/3800/1/cv/d/651/1"] "2-3800-152.43-c0-0-2" 1.3771154809983834 1.8964470480054092 2 3800 "152.43" [] [[0.0, 0.0]] 0 true true false false -0.22842496943458634 0 1.02515826655293376180756107969 ["ModularForm/GL2/Q/holomorphic/3800/1/cv/f/651/2"] "2-3800-152.43-c0-0-3" 1.3771154809983834 1.8964470480054092 2 3800 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Label -- # Each L-function $L$ has a label of the form d-N-q.k-x-y-i, where # * $d$ is the degree of $L$. # * $N$ is the conductor of $L$. When $N$ is a perfect power $m^n$ we write $N$ as $m$e$n$, since $N$ can be very large for some imprimitive L-functions. # * q.k is the label of the primitive Dirichlet character from which the central character is induced. # * x-y is the spectral label encoding the $\mu_j$ and $\nu_j$ in the analytically normalized functional equation. # * i is a non-negative integer disambiguating between L-functions that would otherwise have the same label. #$\alpha$ (root_analytic_conductor) -- # If $d$ is the degree of the L-function $L(s)$, the **root analytic conductor** $\alpha$ of $L$ is the $d$th root of the analytic conductor of $L$. It plays a role analogous to the root discriminant for number fields. #$A$ (analytic_conductor) -- # The **analytic conductor** of an L-function $L(s)$ with infinity factor $L_{\infty}(s)$ and conductor $N$ is the real number # \[ # A := \mathrm{exp}\left(2\mathrm{Re}\left(\frac{L_{\infty}'(1/2)}{L_{\infty}(1/2)}\right)\right)N. # \] #$d$ (degree) -- # The **degree** of an L-function is the number $J + 2K$ of Gamma factors occurring in its functional equation # \[ # \Lambda(s) := N^{s/2} # \prod_{j=1}^J \Gamma_{\mathbb R}(s+\mu_j) \prod_{k=1}^K \Gamma_{\mathbb C}(s+\nu_k) # \cdot L(s) = \varepsilon \overline{\Lambda}(1-s). # \] # The degree appears as the first component of the Selberg data of $L(s).$ In all known cases it is the degree of the polynomial of the inverse of the Euler factor at any prime not dividing the conductor. #$N$ (conductor) -- # The **conductor** of an L-function is the integer $N$ occurring in its functional equation # \[ # \Lambda(s) := N^{s/2} # \prod_{j=1}^J \Gamma_{\mathbb R}(s+\mu_j) \prod_{k=1}^K \Gamma_{\mathbb C}(s+\nu_k) # \cdot L(s) = \varepsilon \overline{\Lambda}(1-s). # \] # The conductor of an analytic L-function is the second component in the Selberg data. For a Dirichlet L-function # associated with a primitive Dirichlet character, the conductor of the L-function is the same as the conductor of the character. For a primitive L-function associated with a cusp form $\phi$ on $GL(2)/\mathbb Q$, the conductor of the L-function is the same as the level of $\phi$. # In the literature, the word _level_ is sometimes used instead of _conductor_. #$\chi$ (central_character) -- # An L-function has an Euler product of the form # $L(s) = \prod_p L_p(p^{-s})^{-1}$ # where $L_p(x) = 1 + a_p x + \ldots + (-1)^d \chi(p) x^d$. The character $\chi$ is a Dirichlet character mod $N$ and is called **central character** of the L-function. # Here, $N$ is the conductor of $L$. #$\mu$ (mus) -- # All known analytic L-functions have a **functional equation** that can be written in the form # \[ # \Lambda(s) := N^{s/2} # \prod_{j=1}^J \Gamma_{\mathbb R}(s+\mu_j) \prod_{k=1}^K \Gamma_{\mathbb C}(s+\nu_k) # \cdot L(s) = \varepsilon \overline{\Lambda}(1-s), # \] # where $N$ is an integer, $\Gamma_{\mathbb R}$ and $\Gamma_{\mathbb C}$ are defined in terms of the $\Gamma$-function, $\mathrm{Re}(\mu_j) = 0 \ \mathrm{or} \ 1$ (assuming Selberg's eigenvalue conjecture), and $\mathrm{Re}(\nu_k)$ is a positive integer # or half-integer, # \[ # \sum \mu_j + 2 \sum \nu_k \ \ \ \ \text{is real}, # \] # and $\varepsilon$ is the sign of the functional equation. # With those restrictions on the spectral parameters, the # data in the functional equation is specified uniquely. The integer $d = J + 2 K$ # is the degree of the L-function. The integer $N$ is the conductor (or level) # of the L-function. The pair $[J,K]$ is the signature of the L-function. The parameters # in the functional equation can be used to make up the 4-tuple called the Selberg data. # The axioms of the Selberg class are less restrictive than # given above. # Note that the functional equation above has the central point at $s=1/2$, and relates $s\leftrightarrow 1-s$. # For many L-functions there is another normalization which is natural. The corresponding functional equation relates $s\leftrightarrow w+1-s$ for some positive integer $w$, # called the motivic weight of the L-function. The central point is at $s=(w+1)/2$, and the arithmetically normalized Dirichlet coefficients $a_n n^{w/2}$ are algebraic integers. #$\nu$ (nus) -- # All known analytic L-functions have a **functional equation** that can be written in the form # \[ # \Lambda(s) := N^{s/2} # \prod_{j=1}^J \Gamma_{\mathbb R}(s+\mu_j) \prod_{k=1}^K \Gamma_{\mathbb C}(s+\nu_k) # \cdot L(s) = \varepsilon \overline{\Lambda}(1-s), # \] # where $N$ is an integer, $\Gamma_{\mathbb R}$ and $\Gamma_{\mathbb C}$ are defined in terms of the $\Gamma$-function, $\mathrm{Re}(\mu_j) = 0 \ \mathrm{or} \ 1$ (assuming Selberg's eigenvalue conjecture), and $\mathrm{Re}(\nu_k)$ is a positive integer # or half-integer, # \[ # \sum \mu_j + 2 \sum \nu_k \ \ \ \ \text{is real}, # \] # and $\varepsilon$ is the sign of the functional equation. # With those restrictions on the spectral parameters, the # data in the functional equation is specified uniquely. The integer $d = J + 2 K$ # is the degree of the L-function. The integer $N$ is the conductor (or level) # of the L-function. The pair $[J,K]$ is the signature of the L-function. The parameters # in the functional equation can be used to make up the 4-tuple called the Selberg data. # The axioms of the Selberg class are less restrictive than # given above. # Note that the functional equation above has the central point at $s=1/2$, and relates $s\leftrightarrow 1-s$. # For many L-functions there is another normalization which is natural. The corresponding functional equation relates $s\leftrightarrow w+1-s$ for some positive integer $w$, # called the motivic weight of the L-function. The central point is at $s=(w+1)/2$, and the arithmetically normalized Dirichlet coefficients $a_n n^{w/2}$ are algebraic integers. #$w$ (motivic_weight) -- # The **motivic weight** (or **arithmetic weight**) of an arithmetic L-function with analytic normalization $L_{an}(s)=\sum_{n=1}^\infty a_nn^{-s}$ is the least nonnegative integer $w$ for which $a_nn^{w/2}$ is an algebraic integer for all $n\ge 1$. # If the L-function arises from a motive, then the weight of the motive has the # same parity as the motivic weight of the L-function, but the weight of the motive # could be larger. This apparent discrepancy comes from the fact that a Tate twist # increases the weight of the motive. This corresponds to the change of variables # $s \mapsto s + j$ in the L-function of the motive. #prim (primitive) -- # An L-function is primitive if it cannot be written as a product of nontrivial L-functions. The "trivial L-function" is the constant function $1$. #arith (algebraic) -- # An L-function $L(s) = \sum_{n=1}^{\infty} a_n n^{-s}$ is called **arithmetic** if its Dirichlet coefficients $a_n$ are algebraic numbers. #$\mathbb{Q}$ (rational) -- # A **rational** L-function $L(s)$ is an arithmetic L-function with coefficient field $\Q$; equivalently, its Euler product in the arithmetic normalization can be written as a product over rational primes # \[ # L(s)=\prod_pL_p(p^{-s})^{-1} # \] # with $L_p\in \Z[T]$. #self-dual (self_dual) -- # An L-function $L(s) = \sum_{n=1}^{\infty} \frac{a_n}{n^s}$ is called **self-dual** if its Dirichlet coefficients $a_n$ are real. #$\operatorname{Arg}(\epsilon)$ (root_angle) -- # The **root angle** of an L-function is the argument of its root number, as a real number $\alpha$ with $-0.5 < \alpha \le 0.5$. #$r$ (order_of_vanishing) -- # The **analytic rank** of an L-function $L(s)$ is its order of vanishing at its central point. # When the analytic rank $r$ is positive, the value listed in the LMFDB is typically an upper bound that is believed to be tight (in the sense that there are known to be $r$ zeroes located very near to the central point). #First zero (z1) -- # The **zeros** of an L-function $L(s)$ are the complex numbers $\rho$ for which $L(\rho)=0$. # Under the Riemann Hypothesis, every non-trivial zero $\rho$ lies on the critical line $\Re(s)=1/2$ (in the analytic normalization). # The **lowest zero** of an L-function $L(s)$ is the least $\gamma>0$ for which $L(1/2+i\gamma)=0$. Note that even when $L(1/2)=0$, the lowest zero is by definition a positive real number. #Origin (instance_urls) -- # L-functions arise from many different sources. Already in degree 2 we have examples of # L-functions associated with holomorphic cusp forms, with Maass forms, with elliptic curves, with characters of number fields (Hecke characters), and with 2-dimensional representations of the Galois group of a number field (Artin L-functions). # Sometimes an L-function may arise from more than one source. For example, the L-functions associated with elliptic curves are also associated with weight 2 cusp forms. A goal of the Langlands program ostensibly is to prove that any degree $d$ L-function is associated with an automorphic form on $\mathrm{GL}(d)$. Because of this representation theoretic genesis, one can associate an L-function not only to an automorphic representation but also to symmetric powers, or exterior powers of that representation, or to the tensor product of two representations (the Rankin-Selberg product of two L-functions).