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Properties

Label 1-34e2-1156.103-r1-0-0
Degree $1$
Conductor $1156$
Sign $-0.653 - 0.757i$
Analytic cond. $124.229$
Root an. cond. $124.229$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.0922 − 0.995i)3-s + (0.739 − 0.673i)5-s + (0.982 − 0.183i)7-s + (−0.982 + 0.183i)9-s + (0.850 − 0.526i)11-s + (0.739 + 0.673i)13-s + (−0.739 − 0.673i)15-s + (0.273 − 0.961i)19-s + (−0.273 − 0.961i)21-s + (0.982 − 0.183i)23-s + (0.0922 − 0.995i)25-s + (0.273 + 0.961i)27-s + (−0.850 + 0.526i)29-s + (−0.739 − 0.673i)31-s + (−0.602 − 0.798i)33-s + ⋯
L(s)  = 1  + (−0.0922 − 0.995i)3-s + (0.739 − 0.673i)5-s + (0.982 − 0.183i)7-s + (−0.982 + 0.183i)9-s + (0.850 − 0.526i)11-s + (0.739 + 0.673i)13-s + (−0.739 − 0.673i)15-s + (0.273 − 0.961i)19-s + (−0.273 − 0.961i)21-s + (0.982 − 0.183i)23-s + (0.0922 − 0.995i)25-s + (0.273 + 0.961i)27-s + (−0.850 + 0.526i)29-s + (−0.739 − 0.673i)31-s + (−0.602 − 0.798i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.653 - 0.757i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.653 - 0.757i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1156\)    =    \(2^{2} \cdot 17^{2}\)
Sign: $-0.653 - 0.757i$
Analytic conductor: \(124.229\)
Root analytic conductor: \(124.229\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1156} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1156,\ (1:\ ),\ -0.653 - 0.757i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.272080462 - 2.778198735i\)
\(L(\frac12)\) \(\approx\) \(1.272080462 - 2.778198735i\)
\(L(1)\) \(\approx\) \(1.222363077 - 0.7679392130i\)
\(L(1)\) \(\approx\) \(1.222363077 - 0.7679392130i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
17 \( 1 \)
good3 \( 1 + (-0.0922 - 0.995i)T \)
5 \( 1 + (0.739 - 0.673i)T \)
7 \( 1 + (0.982 - 0.183i)T \)
11 \( 1 + (0.850 - 0.526i)T \)
13 \( 1 + (0.739 + 0.673i)T \)
19 \( 1 + (0.273 - 0.961i)T \)
23 \( 1 + (0.982 - 0.183i)T \)
29 \( 1 + (-0.850 + 0.526i)T \)
31 \( 1 + (-0.739 - 0.673i)T \)
37 \( 1 + (-0.602 + 0.798i)T \)
41 \( 1 + (0.0922 + 0.995i)T \)
43 \( 1 + (-0.445 - 0.895i)T \)
47 \( 1 + (0.982 + 0.183i)T \)
53 \( 1 + (-0.982 + 0.183i)T \)
59 \( 1 + (-0.932 - 0.361i)T \)
61 \( 1 + (0.932 - 0.361i)T \)
67 \( 1 + (0.273 - 0.961i)T \)
71 \( 1 + (0.982 - 0.183i)T \)
73 \( 1 + (0.445 - 0.895i)T \)
79 \( 1 + (0.273 - 0.961i)T \)
83 \( 1 + (-0.0922 + 0.995i)T \)
89 \( 1 + (0.739 - 0.673i)T \)
97 \( 1 + (-0.982 + 0.183i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−21.29712440819258591097798727782, −20.77402284922033100313345710731, −20.14263144973687359476887506923, −18.954681483531169120678343375343, −18.1146342597886532785475324090, −17.425951175808341196420109487756, −16.93532601667304303326230478111, −15.812770039724769787124984469568, −15.03747656541903701355199771569, −14.47549780882116032548587589643, −13.93306398763063439193255361852, −12.73518825941062083251869259842, −11.670754650933612027652319532188, −10.96022446453207488257283831610, −10.41929026722901929462851344398, −9.467263824376854489766753260193, −8.86665796567393795750078941195, −7.82656013262824669881913004808, −6.7825064126895583701823146839, −5.68188954351005558410663041954, −5.29622830739821624053130856022, −4.060892254816522747257186686275, −3.36851554536531528089935511839, −2.19921539706805587475846026185, −1.22397186662027688417569956808, 0.64698988361238792319197950637, 1.41986957284659007642200594537, 2.04059583914987761054566239931, 3.37205083627840305484113004447, 4.643799289027059278226472419898, 5.40495370848773286091830279298, 6.31221177974015138978806908734, 7.0157459833364742478423190181, 8.05759427963266825805357035905, 8.865021395035113224967061436071, 9.3034408372022232407998765610, 10.92510564807628075500451427388, 11.31647731559281158700603489008, 12.20897620212020343040004761997, 13.11080290360556934626988400166, 13.767548229302663460719193189319, 14.22920389205171224469429282279, 15.24117695964440788716523668457, 16.676158246419760612978661891921, 16.91409733321526044064511789713, 17.727703853639363566773058964683, 18.451776522216822100976951208731, 19.12197706818461493775344940414, 20.22661046970668707378544952960, 20.58250450878171803692602697395

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