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Properties

Label 1-1157-1157.1109-r1-0-0
Degree $1$
Conductor $1157$
Sign $0.895 + 0.445i$
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  + (0.995 − 0.0950i)2-s + (0.828 − 0.560i)3-s + (0.981 − 0.189i)4-s + (0.281 + 0.959i)5-s + (0.771 − 0.636i)6-s + (0.520 + 0.853i)7-s + (0.959 − 0.281i)8-s + (0.371 − 0.928i)9-s + (0.371 + 0.928i)10-s + (0.235 + 0.971i)11-s + (0.707 − 0.707i)12-s + (0.599 + 0.800i)14-s + (0.771 + 0.636i)15-s + (0.928 − 0.371i)16-s + (0.0950 − 0.995i)17-s + (0.281 − 0.959i)18-s + ⋯
L(s)  = 1  + (0.995 − 0.0950i)2-s + (0.828 − 0.560i)3-s + (0.981 − 0.189i)4-s + (0.281 + 0.959i)5-s + (0.771 − 0.636i)6-s + (0.520 + 0.853i)7-s + (0.959 − 0.281i)8-s + (0.371 − 0.928i)9-s + (0.371 + 0.928i)10-s + (0.235 + 0.971i)11-s + (0.707 − 0.707i)12-s + (0.599 + 0.800i)14-s + (0.771 + 0.636i)15-s + (0.928 − 0.371i)16-s + (0.0950 − 0.995i)17-s + (0.281 − 0.959i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.895 + 0.445i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.895 + 0.445i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1157\)    =    \(13 \cdot 89\)
Sign: $0.895 + 0.445i$
Analytic conductor: \(124.336\)
Root analytic conductor: \(124.336\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1157} (1109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1157,\ (1:\ ),\ 0.895 + 0.445i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(7.656795311 + 1.797603696i\)
\(L(\frac12)\) \(\approx\) \(7.656795311 + 1.797603696i\)
\(L(1)\) \(\approx\) \(3.136694198 + 0.1823772152i\)
\(L(1)\) \(\approx\) \(3.136694198 + 0.1823772152i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
89 \( 1 \)
good2 \( 1 + (0.995 - 0.0950i)T \)
3 \( 1 + (0.828 - 0.560i)T \)
5 \( 1 + (0.281 + 0.959i)T \)
7 \( 1 + (0.520 + 0.853i)T \)
11 \( 1 + (0.235 + 0.971i)T \)
17 \( 1 + (0.0950 - 0.995i)T \)
19 \( 1 + (0.393 + 0.919i)T \)
23 \( 1 + (-0.118 + 0.992i)T \)
29 \( 1 + (-0.0237 - 0.999i)T \)
31 \( 1 + (0.800 - 0.599i)T \)
37 \( 1 + (0.258 - 0.965i)T \)
41 \( 1 + (0.436 + 0.899i)T \)
43 \( 1 + (-0.0237 + 0.999i)T \)
47 \( 1 + (-0.755 + 0.654i)T \)
53 \( 1 + (-0.755 - 0.654i)T \)
59 \( 1 + (0.828 + 0.560i)T \)
61 \( 1 + (0.304 - 0.952i)T \)
67 \( 1 + (-0.981 - 0.189i)T \)
71 \( 1 + (-0.690 - 0.723i)T \)
73 \( 1 + (-0.142 + 0.989i)T \)
79 \( 1 + (0.989 + 0.142i)T \)
83 \( 1 + (0.936 + 0.349i)T \)
97 \( 1 + (-0.235 + 0.971i)T \)
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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

−20.864994056710592224422823684657, −20.60289442106811133289015631402, −19.76388588944030669038701650624, −19.23086130864872737915213553375, −17.66447983746727625311316784300, −16.7419173397141802133921207493, −16.37223574597220720726779479663, −15.504625359829992879130092232262, −14.62713815044110847524094244322, −13.93236852378695984031285474676, −13.46374933672699237720068371907, −12.71125010207489299998638610938, −11.67384223732751769502872888430, −10.71423609361414921612599737710, −10.16738162014750678162768866740, −8.79061887066427204064793255410, −8.362735365073619186391681507341, −7.3990406056765771791341817800, −6.35033872611142872220846497404, −5.23675928697789209825195042477, −4.622648050642942078969219812113, −3.88354865920807357981570063093, −3.05040566641211494574763720513, −1.88479452472510934867578398585, −0.96544880523155914041258585340, 1.397710959953511117760574270433, 2.19713190805225550943951875182, 2.79396072169864402097069283969, 3.70394834268241786982521156683, 4.745521550860630724994669236922, 5.87005094073738673410082334222, 6.49717491012294436914616842109, 7.5547135944022317876316022264, 7.87184031294748575268731318072, 9.514818304073775060108226838546, 9.861582231077539515789131587164, 11.32570128984754361969870349999, 11.78514662130775480948994898383, 12.6108636651422087422570628280, 13.49511324607786595749992750206, 14.20161319603050745729424006782, 14.76174674027894994596727851016, 15.26617329259769346287901941636, 16.108086072396295894123913833548, 17.632831984103944212818314126447, 18.092702952251365575510436403403, 19.05082240630600266314891445313, 19.562641975428264662039901115423, 20.71179057293568624280953645941, 20.99157670229184645806053059620

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