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Properties

Label 1-1157-1157.111-r1-0-0
Degree $1$
Conductor $1157$
Sign $0.0242 - 0.999i$
Analytic cond. $124.336$
Root an. cond. $124.336$
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  + (0.814 + 0.580i)2-s + (0.327 − 0.945i)3-s + (0.327 + 0.945i)4-s + (0.281 − 0.959i)5-s + (0.814 − 0.580i)6-s + (0.690 + 0.723i)7-s + (−0.281 + 0.959i)8-s + (−0.786 − 0.618i)9-s + (0.786 − 0.618i)10-s + (0.690 − 0.723i)11-s + 12-s + (0.142 + 0.989i)14-s + (−0.814 − 0.580i)15-s + (−0.786 + 0.618i)16-s + (−0.580 − 0.814i)17-s + (−0.281 − 0.959i)18-s + ⋯
L(s)  = 1  + (0.814 + 0.580i)2-s + (0.327 − 0.945i)3-s + (0.327 + 0.945i)4-s + (0.281 − 0.959i)5-s + (0.814 − 0.580i)6-s + (0.690 + 0.723i)7-s + (−0.281 + 0.959i)8-s + (−0.786 − 0.618i)9-s + (0.786 − 0.618i)10-s + (0.690 − 0.723i)11-s + 12-s + (0.142 + 0.989i)14-s + (−0.814 − 0.580i)15-s + (−0.786 + 0.618i)16-s + (−0.580 − 0.814i)17-s + (−0.281 − 0.959i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0242 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0242 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1157\)    =    \(13 \cdot 89\)
Sign: $0.0242 - 0.999i$
Analytic conductor: \(124.336\)
Root analytic conductor: \(124.336\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1157} (111, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1157,\ (1:\ ),\ 0.0242 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.658052197 - 2.723364883i\)
\(L(\frac12)\) \(\approx\) \(2.658052197 - 2.723364883i\)
\(L(1)\) \(\approx\) \(1.915392925 - 0.3491678288i\)
\(L(1)\) \(\approx\) \(1.915392925 - 0.3491678288i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
89 \( 1 \)
good2 \( 1 + (0.814 + 0.580i)T \)
3 \( 1 + (0.327 - 0.945i)T \)
5 \( 1 + (0.281 - 0.959i)T \)
7 \( 1 + (0.690 + 0.723i)T \)
11 \( 1 + (0.690 - 0.723i)T \)
17 \( 1 + (-0.580 - 0.814i)T \)
19 \( 1 + (-0.618 + 0.786i)T \)
23 \( 1 + (0.928 - 0.371i)T \)
29 \( 1 + (-0.235 - 0.971i)T \)
31 \( 1 + (-0.989 + 0.142i)T \)
37 \( 1 + (0.866 - 0.5i)T \)
41 \( 1 + (0.189 - 0.981i)T \)
43 \( 1 + (0.235 - 0.971i)T \)
47 \( 1 + (0.755 + 0.654i)T \)
53 \( 1 + (-0.654 - 0.755i)T \)
59 \( 1 + (0.945 - 0.327i)T \)
61 \( 1 + (-0.0475 - 0.998i)T \)
67 \( 1 + (-0.945 - 0.327i)T \)
71 \( 1 + (0.971 + 0.235i)T \)
73 \( 1 + (-0.989 + 0.142i)T \)
79 \( 1 + (-0.142 - 0.989i)T \)
83 \( 1 + (-0.909 - 0.415i)T \)
97 \( 1 + (0.690 + 0.723i)T \)
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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

−21.491873970180852265370060051642, −20.60305966931715804727671918732, −19.851794352618994735440132538138, −19.462029801714774997776375539149, −18.243792313364089888192911235525, −17.40364265396195018982610864979, −16.6098846101914280351841992864, −15.34142171545638800232704004586, −14.82441346255206197265616039458, −14.52291866447261130238849252822, −13.57081845504901380713089537591, −12.885126047887581785596596744060, −11.45286117894799579418414363573, −11.0588250031555580865178915340, −10.44922234882764316032130511747, −9.65025610001523433939056340980, −8.83726287972337312470511158758, −7.4243066302112135667086033997, −6.68060340938670326214606602027, −5.66357330811647094071891545238, −4.59936983618918626388450053014, −4.1270210715554551654556974207, −3.21084026187181614803895694329, −2.313335925091943317112472060405, −1.37815501818752894317529777695, 0.48618090877965034983094255573, 1.78225895371155049420926885720, 2.46259144886236698105005969745, 3.6868634117029532990329230921, 4.67321030630694343714400282671, 5.64997903687283928638915717393, 6.10960447156683893421156081843, 7.17980737885073646353135903689, 8.037575626559312626250799580275, 8.78314889291500629856584347667, 9.15922696077830113313835777429, 11.140296150085413619960658192848, 11.738885606035073372395868098443, 12.532126406505388185718025017327, 13.032832862430410626420535919264, 13.97138288026897782428420816782, 14.40871442570424844882713731498, 15.30954356955211418160770849445, 16.24436690956564904749651565980, 17.07757024733955412617927029230, 17.57111083243531837487275542889, 18.535002465689982295836904417799, 19.34025937341597129784636694449, 20.49106183523178431738185717801, 20.77605989531554283309488257922

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