# lfunc_search downloaded from the LMFDB on 25 May 2026. # Search link: https://www.lmfdb.org/L/rational/16/336^8/1.1 # Query "{'degree': 16, 'conductor': 162447943996702457856, 'rational': True}" returned 41 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, $\epsilon$, $r$, First zero, Origin] # For more details, see the definitions at the bottom of the file. "16-336e8-1.1-c1e8-0-0" 1.6379784821282062 2684.9183014954187 16 162447943996702457856 "1.1" [] [[0.5, 0.0], [0.5, 0.0], [0.5, 0.0], [0.5, 0.0], [0.5, 0.0], [0.5, 0.0], [0.5, 0.0], [0.5, 0.0]] 1 false 1 0 0.17256259048464793447344690168 ["ModularForm/GL2/Q/holomorphic/336/2/bj/f"] "16-336e8-1.1-c1e8-0-1" 1.6379784821282062 2684.9183014954187 16 162447943996702457856 "1.1" [] [[0.5, 0.0], [0.5, 0.0], [0.5, 0.0], [0.5, 0.0], [0.5, 0.0], [0.5, 0.0], [0.5, 0.0], [0.5, 0.0]] 1 false 1 0 0.32952280190290071339497563229 ["ModularForm/GL2/Q/holomorphic/336/2/h/b"] "16-336e8-1.1-c1e8-0-2" 1.6379784821282062 2684.9183014954187 16 162447943996702457856 "1.1" [] [[0.5, 0.0], [0.5, 0.0], [0.5, 0.0], [0.5, 0.0], [0.5, 0.0], [0.5, 0.0], [0.5, 0.0], [0.5, 0.0]] 1 false 1 0 0.33789138564938135690419200069 ["ModularForm/GL2/Q/holomorphic/336/2/k/c"] "16-336e8-1.1-c1e8-0-3" 1.6379784821282062 2684.9183014954187 16 162447943996702457856 "1.1" [] [[0.5, 0.0], [0.5, 0.0], [0.5, 0.0], [0.5, 0.0], [0.5, 0.0], [0.5, 0.0], [0.5, 0.0], [0.5, 0.0]] 1 false 1 0 0.40497073075807073476602007324 ["ModularForm/GL2/Q/holomorphic/336/2/bj/e"] "16-336e8-1.1-c1e8-0-4" 1.6379784821282062 2684.9183014954187 16 162447943996702457856 "1.1" [] [[0.5, 0.0], [0.5, 0.0], [0.5, 0.0], [0.5, 0.0], [0.5, 0.0], [0.5, 0.0], [0.5, 0.0], [0.5, 0.0]] 1 false 1 0 0.854359512765591322583423232134 ["ModularForm/GL2/Q/holomorphic/336/2/bj/g"] "16-336e8-1.1-c1e8-0-5" 1.6379784821282062 2684.9183014954187 16 162447943996702457856 "1.1" [] [[0.5, 0.0], [0.5, 0.0], [0.5, 0.0], [0.5, 0.0], [0.5, 0.0], [0.5, 0.0], [0.5, 0.0], [0.5, 0.0]] 1 false 1 0 1.08725530218116088116906686945 ["ModularForm/GL2/Q/holomorphic/336/2/bq/a"] "16-336e8-1.1-c2e8-0-0" 3.0257787233216473 49362214.499789484 16 162447943996702457856 "1.1" [] [[1.0, 0.0], [1.0, 0.0], [1.0, 0.0], [1.0, 0.0], [1.0, 0.0], [1.0, 0.0], [1.0, 0.0], [1.0, 0.0]] 2 false 1 0 0.02063459552541131717228540971 ["ModularForm/GL2/Q/holomorphic/336/3/bn/g"] "16-336e8-1.1-c2e8-0-1" 3.0257787233216473 49362214.499789484 16 162447943996702457856 "1.1" [] [[1.0, 0.0], [1.0, 0.0], [1.0, 0.0], [1.0, 0.0], [1.0, 0.0], [1.0, 0.0], [1.0, 0.0], [1.0, 0.0]] 2 false 1 0 0.05967790029463097520425644787 ["ModularForm/GL2/Q/holomorphic/336/3/f/d"] "16-336e8-1.1-c2e8-0-2" 3.0257787233216473 49362214.499789484 16 162447943996702457856 "1.1" [] [[1.0, 0.0], [1.0, 0.0], [1.0, 0.0], [1.0, 0.0], [1.0, 0.0], [1.0, 0.0], [1.0, 0.0], [1.0, 0.0]] 2 false 1 0 0.07153921392281587751635326347 ["ModularForm/GL2/Q/holomorphic/336/3/o/h"] "16-336e8-1.1-c2e8-0-3" 3.0257787233216473 49362214.499789484 16 162447943996702457856 "1.1" [] [[1.0, 0.0], [1.0, 0.0], [1.0, 0.0], [1.0, 0.0], [1.0, 0.0], [1.0, 0.0], [1.0, 0.0], [1.0, 0.0]] 2 false 1 0 0.07634347305207055111978536205 ["ModularForm/GL2/Q/holomorphic/336/3/bh/g"] "16-336e8-1.1-c2e8-0-4" 3.0257787233216473 49362214.499789484 16 162447943996702457856 "1.1" [] [[1.0, 0.0], [1.0, 0.0], [1.0, 0.0], [1.0, 0.0], [1.0, 0.0], [1.0, 0.0], [1.0, 0.0], [1.0, 0.0]] 2 false 1 0 0.16575834957746035508095499531 ["ModularForm/GL2/Q/holomorphic/336/3/bh/h"] "16-336e8-1.1-c2e8-0-5" 3.0257787233216473 49362214.499789484 16 162447943996702457856 "1.1" [] [[1.0, 0.0], [1.0, 0.0], [1.0, 0.0], [1.0, 0.0], [1.0, 0.0], [1.0, 0.0], [1.0, 0.0], [1.0, 0.0]] 2 false 1 0 0.25526860727854271981841384364 ["ModularForm/GL2/Q/holomorphic/336/3/o/g"] "16-336e8-1.1-c3e8-0-0" 4.452487143376032 23858480765.62257 16 162447943996702457856 "1.1" [] [[1.5, 0.0], [1.5, 0.0], [1.5, 0.0], [1.5, 0.0], [1.5, 0.0], [1.5, 0.0], [1.5, 0.0], [1.5, 0.0]] 3 false 1 0 0.04741685790870965054494493425 ["ModularForm/GL2/Q/holomorphic/336/4/q/m"] "16-336e8-1.1-c3e8-0-1" 4.452487143376032 23858480765.62257 16 162447943996702457856 "1.1" [] [[1.5, 0.0], [1.5, 0.0], [1.5, 0.0], [1.5, 0.0], [1.5, 0.0], [1.5, 0.0], [1.5, 0.0], [1.5, 0.0]] 3 false 1 0 0.12758573228161346390308600919 ["ModularForm/GL2/Q/holomorphic/336/4/b/e"] "16-336e8-1.1-c3e8-0-2" 4.452487143376032 23858480765.62257 16 162447943996702457856 "1.1" [] [[1.5, 0.0], [1.5, 0.0], [1.5, 0.0], [1.5, 0.0], [1.5, 0.0], [1.5, 0.0], [1.5, 0.0], [1.5, 0.0]] 3 false 1 0 0.19005030361721679367654047925 ["ModularForm/GL2/Q/holomorphic/336/4/bl/i"] "16-336e8-1.1-c3e8-0-3" 4.452487143376032 23858480765.62257 16 162447943996702457856 "1.1" [] [[1.5, 0.0], [1.5, 0.0], [1.5, 0.0], [1.5, 0.0], [1.5, 0.0], [1.5, 0.0], [1.5, 0.0], [1.5, 0.0]] 3 false 1 0 0.19338043719365937536594474021 ["ModularForm/GL2/Q/holomorphic/336/4/k/c"] "16-336e8-1.1-c3e8-0-4" 4.452487143376032 23858480765.62257 16 162447943996702457856 "1.1" [] [[1.5, 0.0], [1.5, 0.0], [1.5, 0.0], [1.5, 0.0], [1.5, 0.0], [1.5, 0.0], [1.5, 0.0], [1.5, 0.0]] 3 false 1 0 0.21379955380861685686789599257 ["ModularForm/GL2/Q/holomorphic/336/4/k/d"] "16-336e8-1.1-c3e8-0-5" 4.452487143376032 23858480765.62257 16 162447943996702457856 "1.1" [] [[1.5, 0.0], [1.5, 0.0], [1.5, 0.0], [1.5, 0.0], [1.5, 0.0], [1.5, 0.0], [1.5, 0.0], [1.5, 0.0]] 3 false 1 0 0.29668700864592663646851308280 ["ModularForm/GL2/Q/holomorphic/336/4/bl/j"] "16-336e8-1.1-c3e8-0-6" 4.452487143376032 23858480765.62257 16 162447943996702457856 "1.1" [] [[1.5, 0.0], [1.5, 0.0], [1.5, 0.0], [1.5, 0.0], [1.5, 0.0], [1.5, 0.0], [1.5, 0.0], [1.5, 0.0]] 3 false 1 0 0.53269738532334219620332243955 ["ModularForm/GL2/Q/holomorphic/336/4/b/f"] "16-336e8-1.1-c4e8-0-0" 5.893412220112531 2117722781204.1648 16 162447943996702457856 "1.1" [] [[2.0, 0.0], [2.0, 0.0], [2.0, 0.0], [2.0, 0.0], [2.0, 0.0], [2.0, 0.0], [2.0, 0.0], [2.0, 0.0]] 4 false 1 0 0.04706808273058101147840915473 ["ModularForm/GL2/Q/holomorphic/336/5/be/c"] "16-336e8-1.1-c4e8-0-1" 5.893412220112531 2117722781204.1648 16 162447943996702457856 "1.1" [] [[2.0, 0.0], [2.0, 0.0], [2.0, 0.0], [2.0, 0.0], [2.0, 0.0], [2.0, 0.0], [2.0, 0.0], [2.0, 0.0]] 4 false 1 0 0.05586003364618991920839544809 ["ModularForm/GL2/Q/holomorphic/336/5/m/a"] "16-336e8-1.1-c4e8-0-2" 5.893412220112531 2117722781204.1648 16 162447943996702457856 "1.1" [] [[2.0, 0.0], [2.0, 0.0], [2.0, 0.0], [2.0, 0.0], [2.0, 0.0], [2.0, 0.0], [2.0, 0.0], [2.0, 0.0]] 4 false 1 0 0.05816110734058843385438869840 ["ModularForm/GL2/Q/holomorphic/336/5/d/c"] "16-336e8-1.1-c4e8-0-3" 5.893412220112531 2117722781204.1648 16 162447943996702457856 "1.1" [] [[2.0, 0.0], [2.0, 0.0], [2.0, 0.0], [2.0, 0.0], [2.0, 0.0], [2.0, 0.0], [2.0, 0.0], [2.0, 0.0]] 4 false 1 0 0.10188821297811400113382153889 ["ModularForm/GL2/Q/holomorphic/336/5/m/b"] "16-336e8-1.1-c4e8-0-4" 5.893412220112531 2117722781204.1648 16 162447943996702457856 "1.1" [] [[2.0, 0.0], [2.0, 0.0], [2.0, 0.0], [2.0, 0.0], [2.0, 0.0], [2.0, 0.0], [2.0, 0.0], [2.0, 0.0]] 4 false 1 0 0.14689234702603351762027017388 ["ModularForm/GL2/Q/holomorphic/336/5/bh/g"] "16-336e8-1.1-c4e8-0-5" 5.893412220112531 2117722781204.1648 16 162447943996702457856 "1.1" [] [[2.0, 0.0], [2.0, 0.0], [2.0, 0.0], [2.0, 0.0], [2.0, 0.0], [2.0, 0.0], [2.0, 0.0], [2.0, 0.0]] 4 false 1 0 0.16658339305154487853736072031 ["ModularForm/GL2/Q/holomorphic/336/5/d/b"] "16-336e8-1.1-c4e8-0-6" 5.893412220112531 2117722781204.1648 16 162447943996702457856 "1.1" [] [[2.0, 0.0], [2.0, 0.0], [2.0, 0.0], [2.0, 0.0], [2.0, 0.0], [2.0, 0.0], [2.0, 0.0], [2.0, 0.0]] 4 false 1 0 0.24538362580347800627739969378 ["ModularForm/GL2/Q/holomorphic/336/5/be/d"] "16-336e8-1.1-c4e8-0-7" 5.893412220112531 2117722781204.1648 16 162447943996702457856 "1.1" [] [[2.0, 0.0], [2.0, 0.0], [2.0, 0.0], [2.0, 0.0], [2.0, 0.0], [2.0, 0.0], [2.0, 0.0], [2.0, 0.0]] 4 false 1 0 0.27787828364894186996853417022 ["ModularForm/GL2/Q/holomorphic/336/5/d/a"] "16-336e8-1.1-c4e8-0-8" 5.893412220112531 2117722781204.1648 16 162447943996702457856 "1.1" [] [[2.0, 0.0], [2.0, 0.0], [2.0, 0.0], [2.0, 0.0], [2.0, 0.0], [2.0, 0.0], [2.0, 0.0], [2.0, 0.0]] 4 false 1 0 0.46593934663759208875232131463 ["ModularForm/GL2/Q/holomorphic/336/5/m/c"] "16-336e8-1.1-c5e8-0-0" 7.340910260802914 71121128796963.92 16 162447943996702457856 "1.1" [] [[2.5, 0.0], [2.5, 0.0], [2.5, 0.0], [2.5, 0.0], [2.5, 0.0], [2.5, 0.0], [2.5, 0.0], [2.5, 0.0]] 5 false 1 0 0.01463244889711078318694673993 ["ModularForm/GL2/Q/holomorphic/336/6/q/i"] "16-336e8-1.1-c5e8-0-1" 7.340910260802914 71121128796963.92 16 162447943996702457856 "1.1" [] [[2.5, 0.0], [2.5, 0.0], [2.5, 0.0], [2.5, 0.0], [2.5, 0.0], [2.5, 0.0], [2.5, 0.0], [2.5, 0.0]] 5 false 1 0 0.11574406731437508867417981264 ["ModularForm/GL2/Q/holomorphic/336/6/q/k"] "16-336e8-1.1-c5e8-0-2" 7.340910260802914 71121128796963.92 16 162447943996702457856 "1.1" [] [[2.5, 0.0], [2.5, 0.0], [2.5, 0.0], [2.5, 0.0], [2.5, 0.0], [2.5, 0.0], [2.5, 0.0], [2.5, 0.0]] 5 false 1 0 0.12705840911659962893721265145 ["ModularForm/GL2/Q/holomorphic/336/6/q/j"] "16-336e8-1.1-c5e8-0-3" 7.340910260802914 71121128796963.92 16 162447943996702457856 "1.1" [] [[2.5, 0.0], [2.5, 0.0], [2.5, 0.0], [2.5, 0.0], [2.5, 0.0], [2.5, 0.0], [2.5, 0.0], [2.5, 0.0]] 5 false 1 0 0.23675268031149895703975885451 ["ModularForm/GL2/Q/holomorphic/336/6/k/c"] "16-336e8-1.1-c6e8-0-0" 8.791937903344742 1274540947456385.0 16 162447943996702457856 "1.1" [] [[3.0, 0.0], [3.0, 0.0], [3.0, 0.0], [3.0, 0.0], [3.0, 0.0], [3.0, 0.0], [3.0, 0.0], [3.0, 0.0]] 6 false 1 0 0.00226816940048678435372441355 ["ModularForm/GL2/Q/holomorphic/336/7/bh/b"] "16-336e8-1.1-c6e8-0-1" 8.791937903344742 1274540947456385.0 16 162447943996702457856 "1.1" [] [[3.0, 0.0], [3.0, 0.0], [3.0, 0.0], [3.0, 0.0], [3.0, 0.0], [3.0, 0.0], [3.0, 0.0], [3.0, 0.0]] 6 false 1 0 0.04204054354827005875215397158 ["ModularForm/GL2/Q/holomorphic/336/7/bh/c"] "16-336e8-1.1-c6e8-0-2" 8.791937903344742 1274540947456385.0 16 162447943996702457856 "1.1" [] [[3.0, 0.0], [3.0, 0.0], [3.0, 0.0], [3.0, 0.0], [3.0, 0.0], [3.0, 0.0], [3.0, 0.0], [3.0, 0.0]] 6 false 1 0 0.04765513977844235000000666712 ["ModularForm/GL2/Q/holomorphic/336/7/bh/a"] "16-336e8-1.1-c6e8-0-3" 8.791937903344742 1274540947456385.0 16 162447943996702457856 "1.1" [] [[3.0, 0.0], [3.0, 0.0], [3.0, 0.0], [3.0, 0.0], [3.0, 0.0], [3.0, 0.0], [3.0, 0.0], [3.0, 0.0]] 6 false 1 0 0.06025961772473725317310648839 ["ModularForm/GL2/Q/holomorphic/336/7/bh/d"] "16-336e8-1.1-c6e8-0-4" 8.791937903344742 1274540947456385.0 16 162447943996702457856 "1.1" [] [[3.0, 0.0], [3.0, 0.0], [3.0, 0.0], [3.0, 0.0], [3.0, 0.0], [3.0, 0.0], [3.0, 0.0], [3.0, 0.0]] 6 false 1 0 0.086328612429988019358875135868 ["ModularForm/GL2/Q/holomorphic/336/7/f/b"] "16-336e8-1.1-c6e8-0-5" 8.791937903344742 1274540947456385.0 16 162447943996702457856 "1.1" [] [[3.0, 0.0], [3.0, 0.0], [3.0, 0.0], [3.0, 0.0], [3.0, 0.0], [3.0, 0.0], [3.0, 0.0], [3.0, 0.0]] 6 false 1 0 0.096791445420399577041765581816 ["ModularForm/GL2/Q/holomorphic/336/7/f/c"] "16-336e8-1.1-c6e8-0-6" 8.791937903344742 1274540947456385.0 16 162447943996702457856 "1.1" [] [[3.0, 0.0], [3.0, 0.0], [3.0, 0.0], [3.0, 0.0], [3.0, 0.0], [3.0, 0.0], [3.0, 0.0], [3.0, 0.0]] 6 false 1 0 0.11918235866680235669085072348 ["ModularForm/GL2/Q/holomorphic/336/7/f/a"] "16-336e8-1.1-c6e8-0-7" 8.791937903344742 1274540947456385.0 16 162447943996702457856 "1.1" [] [[3.0, 0.0], [3.0, 0.0], [3.0, 0.0], [3.0, 0.0], [3.0, 0.0], [3.0, 0.0], [3.0, 0.0], [3.0, 0.0]] 6 false 1 0 0.21610767831528016357935567671 ["ModularForm/GL2/Q/holomorphic/336/7/bh/e"] "16-336e8-1.1-c6e8-0-8" 8.791937903344742 1274540947456385.0 16 162447943996702457856 "1.1" [] [[3.0, 0.0], [3.0, 0.0], [3.0, 0.0], [3.0, 0.0], [3.0, 0.0], [3.0, 0.0], [3.0, 0.0], [3.0, 0.0]] 6 false 1 0 0.35444726760883804682018252660 ["ModularForm/GL2/Q/holomorphic/336/7/bh/f"] # 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 $1$. #$\epsilon$ (root_number) -- # The **sign** of the functional equation of an analytic L-function, also called the **root number**, is the complex number $\varepsilon$ that appears in the functional equation of $\Lambda(s)=\varepsilon \overline{\Lambda}(1-s)$. The sign appears as the 4th entry in the quadruple # known as the Selberg data. #$r$ (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).