Properties

Degree 1
Conductor $ 37 \cdot 109 $
Sign $0.380 + 0.924i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

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

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.380 + 0.924i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.380 + 0.924i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(4033\)    =    \(37 \cdot 109\)
\( \varepsilon \)  =  $0.380 + 0.924i$
motivic weight  =  \(0\)
character  :  $\chi_{4033} (235, \cdot )$
Sato-Tate  :  $\mu(36)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 4033,\ (0:\ ),\ 0.380 + 0.924i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(0.5934759441 + 0.3975148652i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(0.5934759441 + 0.3975148652i\)
\(L(\chi,1)\)  \(\approx\)  \(0.5877461416 + 0.08797792843i\)
\(L(1,\chi)\)  \(\approx\)  \(0.5877461416 + 0.08797792843i\)

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

−18.35617861415975305019234957664, −17.50009817994547081948997998044, −17.1554421823151688950930718566, −16.16560098700145099814538691422, −15.89069327643771971368326486176, −15.29394913450964720141619198161, −14.31331228149794360047281872133, −13.395351920640951349710961834417, −12.70601868161951748608503554285, −11.8955984743795184797069457041, −11.18477481139655608437059299385, −10.714847262074533017072139795019, −10.31938543099409374882072135617, −9.269474743386561564699980070518, −8.8244212095114777467924325558, −8.02545951767428501539984099940, −7.25682239699439731502300546444, −6.78897795943023254113926619263, −5.165956180504341006296178194626, −4.71168102820787652112196219188, −3.89221571117633231025103171250, −3.31684440072240855971069782812, −2.6020695271073021140212122961, −1.16180549005666011535336609645, −0.43270670256402469386163759299, 0.74972787004612896811064053875, 1.618707965566548129933386169334, 2.48917082040872972792486259713, 3.387830514293646856987022256115, 4.54800297184813458573039823846, 5.63716342926003788274801699268, 6.02703135533704023811178168805, 6.63564526678691117548643164310, 7.75644315563305052519385436969, 7.99714988854254803059227678193, 8.57238765442213725950770777896, 9.23279085101459308080668574083, 10.49968322155019754279470560924, 11.090280307864209988033664174, 11.576563808635368227851721205538, 12.34614421357199440212252940221, 13.1620564720495119405111212497, 13.88006090228969670788677289012, 14.84061971784485733420752826259, 15.15172212782103041773756083376, 16.018731458972602009483662458634, 16.55930819623183907149697992459, 17.26769017403387024922858434325, 18.105362593994375804551975768976, 18.79815505911776428765962852004

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