Properties

Degree 1
Conductor $ 13 \cdot 89 $
Sign $0.616 + 0.787i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.841 − 0.540i)2-s + (−0.415 + 0.909i)3-s + (0.415 + 0.909i)4-s + (0.142 − 0.989i)5-s + (0.841 − 0.540i)6-s + (−0.142 + 0.989i)7-s + (0.142 − 0.989i)8-s + (−0.654 − 0.755i)9-s + (−0.654 + 0.755i)10-s + (0.142 + 0.989i)11-s − 12-s + (0.654 − 0.755i)14-s + (0.841 + 0.540i)15-s + (−0.654 + 0.755i)16-s + (0.841 − 0.540i)17-s + (0.142 + 0.989i)18-s + ⋯
L(s,χ)  = 1  + (−0.841 − 0.540i)2-s + (−0.415 + 0.909i)3-s + (0.415 + 0.909i)4-s + (0.142 − 0.989i)5-s + (0.841 − 0.540i)6-s + (−0.142 + 0.989i)7-s + (0.142 − 0.989i)8-s + (−0.654 − 0.755i)9-s + (−0.654 + 0.755i)10-s + (0.142 + 0.989i)11-s − 12-s + (0.654 − 0.755i)14-s + (0.841 + 0.540i)15-s + (−0.654 + 0.755i)16-s + (0.841 − 0.540i)17-s + (0.142 + 0.989i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.616 + 0.787i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.616 + 0.787i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(1157\)    =    \(13 \cdot 89\)
\( \varepsilon \)  =  $0.616 + 0.787i$
motivic weight  =  \(0\)
character  :  $\chi_{1157} (311, \cdot )$
Sato-Tate  :  $\mu(22)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 1157,\ (0:\ ),\ 0.616 + 0.787i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(0.6877647899 + 0.3350466698i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(0.6877647899 + 0.3350466698i\)
\(L(\chi,1)\)  \(\approx\)  \(0.6455738596 + 0.06974632895i\)
\(L(1,\chi)\)  \(\approx\)  \(0.6455738596 + 0.06974632895i\)

Euler product

\[\begin{aligned}L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]
\[\begin{aligned}L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−21.04725177454440163413043222189, −20.0385788525492514390931508675, −19.16511148777868333103610785624, −18.855263783568805002327594504586, −18.179896622590770562070342953649, −17.20674667353892474619595691915, −16.81240615032964342910551056586, −16.11070243792409470314244869424, −14.70802564903699875048212768360, −14.34808454292447457366576876458, −13.55560743196455879951039390751, −12.56103900716405335992849758306, −11.39299629543053314142800038962, −10.702952780356448837705836397714, −10.3919038378776754141699956750, −9.12413025094176994602356069816, −8.042802610933860926177282408776, −7.55355190627359562516377885387, −6.66185268739073074356866617711, −6.19281792252395558576307441048, −5.37047578836134523121986418306, −3.788313521800750263818881500102, −2.65106567746473809621652519307, −1.56424015368395491336813140144, −0.58012672674578037440963755462, 0.910212799754344930957829586561, 2.130280058071637415754188946, 3.08812374995696646007122244548, 4.22956675930024182389350227504, 4.9891765768237007109133309604, 5.88489434176896806360448404096, 7.00604230256321162685962513002, 8.19758600669969560600348248229, 8.916408130823812780930650172610, 9.64488492116707025797235074261, 9.91625096890257999087014574792, 11.28306163073550274683110533191, 11.74234327031619089846391651028, 12.547120574493429595884762019190, 13.160588114586523708219211825915, 14.75554219570585279818281665428, 15.42475778491292318026811400924, 16.22296217572497908674715572337, 16.72979516003423061960745990285, 17.663286598414674261571713959495, 18.00046893489535546479698873585, 19.26610884666393594919346038926, 19.884274434433823558297626923904, 20.69952724602857349583918966980, 21.345748285384718742595364823548

Graph of the $Z$-function along the critical line