Properties

Degree 1
Conductor $ 37 \cdot 109 $
Sign $-0.548 + 0.836i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.939 − 0.342i)2-s + (0.766 + 0.642i)3-s + (0.766 − 0.642i)4-s + (−0.766 − 0.642i)5-s + (0.939 + 0.342i)6-s + (0.766 + 0.642i)7-s + (0.5 − 0.866i)8-s + (0.173 + 0.984i)9-s + (−0.939 − 0.342i)10-s + (−0.939 + 0.342i)11-s + 12-s + (−0.173 + 0.984i)13-s + (0.939 + 0.342i)14-s + (−0.173 − 0.984i)15-s + (0.173 − 0.984i)16-s + (−0.766 − 0.642i)17-s + ⋯
L(s,χ)  = 1  + (0.939 − 0.342i)2-s + (0.766 + 0.642i)3-s + (0.766 − 0.642i)4-s + (−0.766 − 0.642i)5-s + (0.939 + 0.342i)6-s + (0.766 + 0.642i)7-s + (0.5 − 0.866i)8-s + (0.173 + 0.984i)9-s + (−0.939 − 0.342i)10-s + (−0.939 + 0.342i)11-s + 12-s + (−0.173 + 0.984i)13-s + (0.939 + 0.342i)14-s + (−0.173 − 0.984i)15-s + (0.173 − 0.984i)16-s + (−0.766 − 0.642i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.548 + 0.836i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.548 + 0.836i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(4033\)    =    \(37 \cdot 109\)
\( \varepsilon \)  =  $-0.548 + 0.836i$
motivic weight  =  \(0\)
character  :  $\chi_{4033} (432, \cdot )$
Sato-Tate  :  $\mu(18)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 4033,\ (0:\ ),\ -0.548 + 0.836i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(0.8826991923 + 1.633590092i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(0.8826991923 + 1.633590092i\)
\(L(\chi,1)\)  \(\approx\)  \(1.708627356 + 0.2057396761i\)
\(L(1,\chi)\)  \(\approx\)  \(1.708627356 + 0.2057396761i\)

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

−18.089329424235787944463731829217, −17.72243356508108346005031412366, −16.810448141536606318333356790131, −15.84265421984880700889156622121, −15.16187198125727879146439031118, −14.941139864013243922649495665073, −14.17248614399847854463559898790, −13.39673577253323979909362957334, −13.0781183807496629542960903268, −12.21142899695636400976501656685, −11.531345163214914195838647352, −10.67424803943690799477902798730, −10.38531697637990073046711118150, −8.67492226543823349595255647343, −8.24626441561571513001707237365, −7.57594894432450814151116064740, −7.19604004695538727049328969963, −6.35777213302450466980106337672, −5.55090380603066629831529407988, −4.49839303174808785347083478743, −3.93853275681447577158316234308, −3.16942412433855515302160045573, −2.483047680219960818342864580553, −1.764436235841491944625126021344, −0.272366503368497746188901269, 1.58977380480552780645019228073, 2.16759520416311623751084501055, 2.86837933847654067662936933300, 3.93326807145554095039111446247, 4.44357908489808073088295236804, 4.89996345887251042645255765413, 5.637680518659489165650924493688, 6.738396371873330927204190461356, 7.77937313731959757383972288101, 8.10589122275016845913022066440, 9.10415952534833772959260296813, 9.63898577488052087921175697360, 10.720412509409754005708612442617, 11.10705223940451049292093968457, 12.07027226820447328584286831619, 12.395142056372078485292095330204, 13.35696114638258379505785651983, 13.99055858752498433707905271250, 14.58468379239410104828144717959, 15.39681988575208890005909138483, 15.65050122179980909988037396325, 16.2715190698639453310189464212, 17.102328868420241681224842893004, 18.33243623993877169775746554427, 18.936933435393652107281127456532

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