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Properties

Label 1-1157-1157.865-r1-0-0
Degree $1$
Conductor $1157$
Sign $-0.554 - 0.832i$
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  + (−0.690 + 0.723i)2-s + (0.0475 − 0.998i)3-s + (−0.0475 − 0.998i)4-s + (−0.755 − 0.654i)5-s + (0.690 + 0.723i)6-s + (−0.945 + 0.327i)7-s + (0.755 + 0.654i)8-s + (−0.995 − 0.0950i)9-s + (0.995 − 0.0950i)10-s + (0.945 + 0.327i)11-s − 12-s + (0.415 − 0.909i)14-s + (−0.690 + 0.723i)15-s + (−0.995 + 0.0950i)16-s + (−0.723 + 0.690i)17-s + (0.755 − 0.654i)18-s + ⋯
L(s)  = 1  + (−0.690 + 0.723i)2-s + (0.0475 − 0.998i)3-s + (−0.0475 − 0.998i)4-s + (−0.755 − 0.654i)5-s + (0.690 + 0.723i)6-s + (−0.945 + 0.327i)7-s + (0.755 + 0.654i)8-s + (−0.995 − 0.0950i)9-s + (0.995 − 0.0950i)10-s + (0.945 + 0.327i)11-s − 12-s + (0.415 − 0.909i)14-s + (−0.690 + 0.723i)15-s + (−0.995 + 0.0950i)16-s + (−0.723 + 0.690i)17-s + (0.755 − 0.654i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.554 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.554 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1157\)    =    \(13 \cdot 89\)
Sign: $-0.554 - 0.832i$
Analytic conductor: \(124.336\)
Root analytic conductor: \(124.336\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1157} (865, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1157,\ (1:\ ),\ -0.554 - 0.832i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2261292774 - 0.4221399823i\)
\(L(\frac12)\) \(\approx\) \(0.2261292774 - 0.4221399823i\)
\(L(1)\) \(\approx\) \(0.5550365624 - 0.06738422953i\)
\(L(1)\) \(\approx\) \(0.5550365624 - 0.06738422953i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
89 \( 1 \)
good2 \( 1 + (-0.690 + 0.723i)T \)
3 \( 1 + (0.0475 - 0.998i)T \)
5 \( 1 + (-0.755 - 0.654i)T \)
7 \( 1 + (-0.945 + 0.327i)T \)
11 \( 1 + (0.945 + 0.327i)T \)
17 \( 1 + (-0.723 + 0.690i)T \)
19 \( 1 + (0.0950 - 0.995i)T \)
23 \( 1 + (-0.580 + 0.814i)T \)
29 \( 1 + (0.981 + 0.189i)T \)
31 \( 1 + (0.909 + 0.415i)T \)
37 \( 1 + (-0.866 + 0.5i)T \)
41 \( 1 + (0.458 - 0.888i)T \)
43 \( 1 + (-0.981 + 0.189i)T \)
47 \( 1 + (-0.540 - 0.841i)T \)
53 \( 1 + (0.841 + 0.540i)T \)
59 \( 1 + (0.998 - 0.0475i)T \)
61 \( 1 + (-0.786 - 0.618i)T \)
67 \( 1 + (0.998 + 0.0475i)T \)
71 \( 1 + (0.189 + 0.981i)T \)
73 \( 1 + (-0.909 - 0.415i)T \)
79 \( 1 + (0.415 - 0.909i)T \)
83 \( 1 + (0.281 - 0.959i)T \)
97 \( 1 + (0.945 - 0.327i)T \)
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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

−21.2557915832399950977652167257, −20.41572123887040692362986855028, −19.68971893616914616746478729496, −19.36797168777871874707598917418, −18.41487436519383898184292779408, −17.52145248449170047187410990685, −16.48965260018567430914765723891, −16.22734040518386213682472750434, −15.38570233527411408430614913131, −14.31393885372004834519478719992, −13.63948407484401680330693741716, −12.323671727477275036811734929386, −11.721932738780624666320240620180, −10.9986027626988983740200645815, −10.168701480038865247294260682579, −9.72863399355680532291448907428, −8.72817279594764299407275957501, −8.08093430217253696563283726019, −6.89450212566832321049178320286, −6.23523180161091537411945965449, −4.523190977700593284759365870945, −3.89602593307245625994052112595, −3.22774547477042831012688487766, −2.44162616233404353235208908124, −0.71429437935931090032439210806, 0.200536022770066767745179567646, 1.1115620279725450881113532000, 2.105736671420783065332338196037, 3.44800509894720337509936193456, 4.65040179120525024689145523330, 5.679427263765064086232070266975, 6.68988187793849001171341816355, 6.93782351397719749807528683893, 8.083452961524614339461077081425, 8.740402285039337003254810141728, 9.27183998978706059868993697866, 10.38800862329040040416077155308, 11.62740050692143960349630822050, 12.06505418589300780374384111799, 13.12132811182620277789531606523, 13.710435509565371783121380197955, 14.80609027234405407274295071880, 15.57938813204864370039114960166, 16.14291540588703512090756668445, 17.27272446912922829639368112977, 17.4548838392362065112204977395, 18.57752587254375796622869722988, 19.38489685113070825480207181604, 19.71676821719544491086084884045, 20.16761478597049266212671746828

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