# lfunc_search downloaded from the LMFDB on 14 April 2026. # Search link: https://www.lmfdb.org/L/2/3267/297.139/c0-0 # Query "{'degree': 2, 'conductor': 3267, 'spectral_label': 'c0-0'}" returned 152 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. "2-3267-11.10-c0-0-0" 1.27688895220846 1.630445396272019 2 3267 "11.10" [] [[0.0, 0.0]] 0 true true false false -0.16254957192973932 0 0.895633476524245239422701637981 ["ModularForm/GL2/Q/holomorphic/3267/1/c/a/2782/3"] "2-3267-11.10-c0-0-1" 1.27688895220846 1.630445396272019 2 3267 "11.10" [] [[0.0, 0.0]] 0 true true false false -0.16254957192973932 0 0.987091595782885604660525575453 ["ModularForm/GL2/Q/holomorphic/3267/1/c/a/2782/4"] "2-3267-11.10-c0-0-2" 1.27688895220846 1.630445396272019 2 3267 "11.10" [] [[0.0, 0.0]] 0 true true false false 0.16254957192973932 0 1.62842713776413164374217909733 ["ModularForm/GL2/Q/holomorphic/3267/1/c/a/2782/1"] "2-3267-11.10-c0-0-3" 1.27688895220846 1.630445396272019 2 3267 "11.10" [] [[0.0, 0.0]] 0 true true false false 0.16254957192973932 0 1.80857182052585382954070619772 ["ModularForm/GL2/Q/holomorphic/3267/1/c/a/2782/2"] "2-3267-11.2-c0-0-0" 1.27688895220846 1.630445396272019 2 3267 "11.2" [] [[0.0, 0.0]] 0 true true false false -0.20282633529289096 0 0.43068129681576193715264096459 ["ModularForm/GL2/Q/holomorphic/3267/1/l/a/838/2"] "2-3267-11.2-c0-0-1" 1.27688895220846 1.630445396272019 2 3267 "11.2" [] [[0.0, 0.0]] 0 true true false false 0.13067242490553896 0 1.16619913820001205251355152875 ["ModularForm/GL2/Q/holomorphic/3267/1/l/a/838/3"] "2-3267-11.2-c0-0-2" 1.27688895220846 1.630445396272019 2 3267 "11.2" [] [[0.0, 0.0]] 0 true true false false 0.2667982807317552 0 1.17067295121211329376219747123 ["ModularForm/GL2/Q/holomorphic/3267/1/l/a/838/1"] "2-3267-11.2-c0-0-3" 1.27688895220846 1.630445396272019 2 3267 "11.2" [] [[0.0, 0.0]] 0 true true false false 0.1977881554512534 0 1.36417890653897213981761366211 ["ModularForm/GL2/Q/holomorphic/3267/1/l/a/838/4"] "2-3267-11.6-c0-0-0" 1.27688895220846 1.630445396272019 2 3267 "11.6" [] [[0.0, 0.0]] 0 true true false false -0.2667982807317552 0 0.34669964667033080075886463480 ["ModularForm/GL2/Q/holomorphic/3267/1/l/a/2998/1"] "2-3267-11.6-c0-0-1" 1.27688895220846 1.630445396272019 2 3267 "11.6" [] [[0.0, 0.0]] 0 true true false false -0.13067242490553896 0 0.886188653004514832453301012965 ["ModularForm/GL2/Q/holomorphic/3267/1/l/a/2998/3"] "2-3267-11.6-c0-0-2" 1.27688895220846 1.630445396272019 2 3267 "11.6" [] [[0.0, 0.0]] 0 true true false false -0.1977881554512534 0 0.924940453784855138517614883580 ["ModularForm/GL2/Q/holomorphic/3267/1/l/a/2998/4"] "2-3267-11.6-c0-0-3" 1.27688895220846 1.630445396272019 2 3267 "11.6" [] [[0.0, 0.0]] 0 true true false false 0.20282633529289096 0 1.22554977726817142538065734547 ["ModularForm/GL2/Q/holomorphic/3267/1/l/a/2998/2"] "2-3267-11.7-c0-0-0" 1.27688895220846 1.630445396272019 2 3267 "11.7" [] [[0.0, 0.0]] 0 true true false false 0.4605442630349697 0 0.15258689065323235842523057926 ["ModularForm/GL2/Q/holomorphic/3267/1/l/a/2944/1"] "2-3267-11.7-c0-0-1" 1.27688895220846 1.630445396272019 2 3267 "11.7" [] [[0.0, 0.0]] 0 true true false false -0.3020069503651106 0 0.63304588066824351742954666003 ["ModularForm/GL2/Q/holomorphic/3267/1/l/a/2944/3"] "2-3267-11.7-c0-0-2" 1.27688895220846 1.630445396272019 2 3267 "11.7" [] [[0.0, 0.0]] 0 true true false false -0.13550571056354052 0 0.983078397169420318390823652212 ["ModularForm/GL2/Q/holomorphic/3267/1/l/a/2944/2"] "2-3267-11.7-c0-0-3" 1.27688895220846 1.630445396272019 2 3267 "11.7" [] [[0.0, 0.0]] 0 true true false false -0.1084658622455321 0 1.18920143630124494389179214403 ["ModularForm/GL2/Q/holomorphic/3267/1/l/a/2944/4"] "2-3267-11.8-c0-0-0" 1.27688895220846 1.630445396272019 2 3267 "11.8" [] [[0.0, 0.0]] 0 true true false false 0.13550571056354052 0 1.25945269566180054280318475513 ["ModularForm/GL2/Q/holomorphic/3267/1/l/a/2296/2"] "2-3267-11.8-c0-0-1" 1.27688895220846 1.630445396272019 2 3267 "11.8" [] [[0.0, 0.0]] 0 true true false false 0.1084658622455321 0 1.46741829347561648117891561707 ["ModularForm/GL2/Q/holomorphic/3267/1/l/a/2296/4"] "2-3267-11.8-c0-0-2" 1.27688895220846 1.630445396272019 2 3267 "11.8" [] [[0.0, 0.0]] 0 true true false false 0.3020069503651106 0 1.76802792579404070325287092040 ["ModularForm/GL2/Q/holomorphic/3267/1/l/a/2296/3"] "2-3267-11.8-c0-0-3" 1.27688895220846 1.630445396272019 2 3267 "11.8" [] [[0.0, 0.0]] 0 true true false false -0.4605442630349697 0 2.09112482333349634024479048526 ["ModularForm/GL2/Q/holomorphic/3267/1/l/a/2296/1"] "2-3267-27.11-c0-0-0" 1.27688895220846 1.630445396272019 2 3267 "27.11" [] [[0.0, 0.0]] 0 true true false false 0.3148148148148148 0 0.28202531224940203087612249285 ["ModularForm/GL2/Q/holomorphic/3267/1/q/a/848/1"] "2-3267-27.14-c0-0-0" 1.27688895220846 1.630445396272019 2 3267 "27.14" [] [[0.0, 0.0]] 0 true true false false -0.3148148148148148 0 0.57480218064960379523749741278 ["ModularForm/GL2/Q/holomorphic/3267/1/q/a/122/1"] "2-3267-27.2-c0-0-0" 1.27688895220846 1.630445396272019 2 3267 "27.2" [] [[0.0, 0.0]] 0 true true false false 0.3148148148148148 0 1.29875893146418546164645615957 ["ModularForm/GL2/Q/holomorphic/3267/1/q/a/3026/1"] "2-3267-27.20-c0-0-0" 1.27688895220846 1.630445396272019 2 3267 "27.20" [] [[0.0, 0.0]] 0 true true false false -0.018518518518518517 0 1.19058572886936486759117684068 ["ModularForm/GL2/Q/holomorphic/3267/1/q/a/1937/1"] "2-3267-27.23-c0-0-0" 1.27688895220846 1.630445396272019 2 3267 "27.23" [] [[0.0, 0.0]] 0 true true false false 0.018518518518518517 0 1.25772394459459126218881973657 ["ModularForm/GL2/Q/holomorphic/3267/1/q/a/1211/1"] "2-3267-27.5-c0-0-0" 1.27688895220846 1.630445396272019 2 3267 "27.5" [] [[0.0, 0.0]] 0 true true false false -0.3148148148148148 0 2.34089851591561749872558152667 ["ModularForm/GL2/Q/holomorphic/3267/1/q/a/2300/1"] "2-3267-297.104-c0-0-0" 1.27688895220846 1.630445396272019 2 3267 "297.104" [] [[0.0, 0.0]] 0 true true false false -0.33498061430970294 0 0.35094893173131666264957077807 ["ModularForm/GL2/Q/holomorphic/3267/1/be/a/995/1"] "2-3267-297.106-c0-0-0" 1.27688895220846 1.630445396272019 2 3267 "297.106" [] [[0.0, 0.0]] 0 true true false false 0.2610310777126431 0 2.07790270737296179375248539311 ["ModularForm/GL2/Q/holomorphic/3267/1/bf/a/403/1"] "2-3267-297.112-c0-0-0" 1.27688895220846 1.630445396272019 2 3267 "297.112" [] [[0.0, 0.0]] 0 true true false false 0.3112356485881253 0 1.98202009993201904432425604784 ["ModularForm/GL2/Q/holomorphic/3267/1/bf/a/112/1"] "2-3267-297.113-c0-0-0" 1.27688895220846 1.630445396272019 2 3267 "297.113" [] [[0.0, 0.0]] 0 true true false false -0.47588142184737975 0 1.62508953193746238429495315226 ["ModularForm/GL2/Q/holomorphic/3267/1/be/a/2786/1"] "2-3267-297.119-c0-0-0" 1.27688895220846 1.630445396272019 2 3267 "297.119" [] [[0.0, 0.0]] 0 true true false false -0.33168605235696375 0 2.31684592081331610496428044213 ["ModularForm/GL2/Q/holomorphic/3267/1/be/a/2792/1"] "2-3267-297.13-c0-0-0" 1.27688895220846 1.630445396272019 2 3267 "297.13" [] [[0.0, 0.0]] 0 true true false false -0.022097684745208034 0 1.52313331061645885916056321115 ["ModularForm/GL2/Q/holomorphic/3267/1/bf/a/1201/1"] "2-3267-297.137-c0-0-0" 1.27688895220846 1.630445396272019 2 3267 "297.137" [] [[0.0, 0.0]] 0 true true false false -0.03868431801340668 0 1.29523977070590316367701773510 ["ModularForm/GL2/Q/holomorphic/3267/1/be/a/2810/1"] "2-3267-297.139-c0-0-0" 1.27688895220846 1.630445396272019 2 3267 "297.139" [] [[0.0, 0.0]] 0 true true false false 0.4647347814163468 0 0.60785079986049719637201268572 ["ModularForm/GL2/Q/holomorphic/3267/1/bf/a/2218/1"] "2-3267-297.14-c0-0-0" 1.27688895220846 1.630445396272019 2 3267 "297.14" [] [[0.0, 0.0]] 0 true true false false -0.47588142184737975 0 0.31107208434711384387016312711 ["ModularForm/GL2/Q/holomorphic/3267/1/be/a/608/1"] "2-3267-297.146-c0-0-0" 1.27688895220846 1.630445396272019 2 3267 "297.146" [] [[0.0, 0.0]] 0 true true false false 0.15374820778224987 0 1.52607123399347225913304649564 ["ModularForm/GL2/Q/holomorphic/3267/1/be/a/1334/1"] "2-3267-297.151-c0-0-0" 1.27688895220846 1.630445396272019 2 3267 "297.151" [] [[0.0, 0.0]] 0 true true false false -0.13140144808301346 0 1.40470458283148781529755115434 ["ModularForm/GL2/Q/holomorphic/3267/1/bf/a/1933/1"] "2-3267-297.158-c0-0-0" 1.27688895220846 1.630445396272019 2 3267 "297.158" [] [[0.0, 0.0]] 0 true true false false 0.1795851255510835 0 1.36922855986393513196576637478 ["ModularForm/GL2/Q/holomorphic/3267/1/be/a/1049/1"] "2-3267-297.160-c0-0-0" 1.27688895220846 1.630445396272019 2 3267 "297.160" [] [[0.0, 0.0]] 0 true true false false 0.022097684745208034 0 1.37042470119783972349165009494 ["ModularForm/GL2/Q/holomorphic/3267/1/bf/a/457/1"] "2-3267-297.178-c0-0-0" 1.27688895220846 1.630445396272019 2 3267 "297.178" [] [[0.0, 0.0]] 0 true true false false 0.1075319448844216 0 1.34658994825243286605514736893 ["ModularForm/GL2/Q/holomorphic/3267/1/bf/a/475/1"] "2-3267-297.184-c0-0-0" 1.27688895220846 1.630445396272019 2 3267 "297.184" [] [[0.0, 0.0]] 0 true true false false -0.2610310777126431 0 0.55940335244794128740168162423 ["ModularForm/GL2/Q/holomorphic/3267/1/bf/a/481/1"] "2-3267-297.185-c0-0-0" 1.27688895220846 1.630445396272019 2 3267 "297.185" [] [[0.0, 0.0]] 0 true true false false 0.37201765134674003 0 1.61515187155055594884412285846 ["ModularForm/GL2/Q/holomorphic/3267/1/be/a/3155/1"] "2-3267-297.191-c0-0-0" 1.27688895220846 1.630445396272019 2 3267 "297.191" [] [[0.0, 0.0]] 0 true true false false 0.47588142184737975 0 1.55878364820597287055738298212 ["ModularForm/GL2/Q/holomorphic/3267/1/be/a/2864/1"] "2-3267-297.193-c0-0-0" 1.27688895220846 1.630445396272019 2 3267 "297.193" [] [[0.0, 0.0]] 0 true true false false 0.22580138844891176 0 1.32626303071026643828651602744 ["ModularForm/GL2/Q/holomorphic/3267/1/bf/a/2272/1"] "2-3267-297.20-c0-0-0" 1.27688895220846 1.630445396272019 2 3267 "297.20" [] [[0.0, 0.0]] 0 true true false false 0.33498061430970294 0 1.18578131130032431433079430883 ["ModularForm/GL2/Q/holomorphic/3267/1/be/a/614/1"] "2-3267-297.203-c0-0-0" 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"2-3267-297.257-c0-0-0" 1.27688895220846 1.630445396272019 2 3267 "297.257" [] [[0.0, 0.0]] 0 true true false false -0.15374820778224987 0 0.875199310626575083151013802671 ["ModularForm/GL2/Q/holomorphic/3267/1/be/a/3227/1"] "2-3267-297.259-c0-0-0" 1.27688895220846 1.630445396272019 2 3267 "297.259" [] [[0.0, 0.0]] 0 true true false false 0.022097684745208034 0 0.73578757731614749582260975555 ["ModularForm/GL2/Q/holomorphic/3267/1/bf/a/2635/1"] "2-3267-297.277-c0-0-0" 1.27688895220846 1.630445396272019 2 3267 "297.277" [] [[0.0, 0.0]] 0 true true false false -0.22580138844891176 0 0.52770281301175657073652918369 ["ModularForm/GL2/Q/holomorphic/3267/1/bf/a/2653/1"] "2-3267-297.283-c0-0-0" 1.27688895220846 1.630445396272019 2 3267 "297.283" [] [[0.0, 0.0]] 0 true true false false -0.2610310777126431 0 1.15819903551258542318064340152 ["ModularForm/GL2/Q/holomorphic/3267/1/bf/a/2659/1"] "2-3267-297.284-c0-0-0" 1.27688895220846 1.630445396272019 2 3267 "297.284" [] [[0.0, 0.0]] 0 true true false false 0.03868431801340668 0 1.14734180366905901326333585131 ["ModularForm/GL2/Q/holomorphic/3267/1/be/a/2066/1"] "2-3267-297.290-c0-0-0" 1.27688895220846 1.630445396272019 2 3267 "297.290" [] [[0.0, 0.0]] 0 true true false false 0.14254808851404643 0 1.73029750469992573987548461156 ["ModularForm/GL2/Q/holomorphic/3267/1/be/a/1775/1"] "2-3267-297.292-c0-0-0" 1.27688895220846 1.630445396272019 2 3267 "297.292" [] [[0.0, 0.0]] 0 true true false false -0.1075319448844216 0 0.78825676687835719732502455731 ["ModularForm/GL2/Q/holomorphic/3267/1/bf/a/1183/1"] "2-3267-297.38-c0-0-0" 1.27688895220846 1.630445396272019 2 3267 "297.38" [] [[0.0, 0.0]] 0 true true false false -0.03868431801340668 0 1.29199103171926567048783057401 ["ModularForm/GL2/Q/holomorphic/3267/1/be/a/632/1"] "2-3267-297.40-c0-0-0" 1.27688895220846 1.630445396272019 2 3267 "297.40" [] [[0.0, 0.0]] 0 true true false false 0.13140144808301346 0 1.14478440896019368343774410713 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["ModularForm/GL2/Q/holomorphic/3267/1/w/b/820/1"] "2-3267-99.94-c0-0-2" 1.27688895220846 1.630445396272019 2 3267 "99.94" [] [[0.0, 0.0]] 0 true true false false 0.09617175881928211 0 1.26888333004811860866723755382 ["ModularForm/GL2/Q/holomorphic/3267/1/w/a/820/1"] # 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).