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Properties

Label 1-1157-1157.1043-r0-0-0
Degree $1$
Conductor $1157$
Sign $0.941 - 0.338i$
Analytic cond. $5.37308$
Root an. cond. $5.37308$
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  + (0.235 − 0.971i)2-s + (−0.888 + 0.458i)3-s + (−0.888 − 0.458i)4-s + (−0.654 + 0.755i)5-s + (0.235 + 0.971i)6-s + (0.981 + 0.189i)7-s + (−0.654 + 0.755i)8-s + (0.580 − 0.814i)9-s + (0.580 + 0.814i)10-s + (0.981 − 0.189i)11-s + 12-s + (0.415 − 0.909i)14-s + (0.235 − 0.971i)15-s + (0.580 + 0.814i)16-s + (0.235 + 0.971i)17-s + (−0.654 − 0.755i)18-s + ⋯
L(s)  = 1  + (0.235 − 0.971i)2-s + (−0.888 + 0.458i)3-s + (−0.888 − 0.458i)4-s + (−0.654 + 0.755i)5-s + (0.235 + 0.971i)6-s + (0.981 + 0.189i)7-s + (−0.654 + 0.755i)8-s + (0.580 − 0.814i)9-s + (0.580 + 0.814i)10-s + (0.981 − 0.189i)11-s + 12-s + (0.415 − 0.909i)14-s + (0.235 − 0.971i)15-s + (0.580 + 0.814i)16-s + (0.235 + 0.971i)17-s + (−0.654 − 0.755i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.941 - 0.338i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.941 - 0.338i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1157\)    =    \(13 \cdot 89\)
Sign: $0.941 - 0.338i$
Analytic conductor: \(5.37308\)
Root analytic conductor: \(5.37308\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1157} (1043, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1157,\ (0:\ ),\ 0.941 - 0.338i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.113328604 - 0.1938737413i\)
\(L(\frac12)\) \(\approx\) \(1.113328604 - 0.1938737413i\)
\(L(1)\) \(\approx\) \(0.8617658952 - 0.1988425400i\)
\(L(1)\) \(\approx\) \(0.8617658952 - 0.1988425400i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
89 \( 1 \)
good2 \( 1 + (0.235 - 0.971i)T \)
3 \( 1 + (-0.888 + 0.458i)T \)
5 \( 1 + (-0.654 + 0.755i)T \)
7 \( 1 + (0.981 + 0.189i)T \)
11 \( 1 + (0.981 - 0.189i)T \)
17 \( 1 + (0.235 + 0.971i)T \)
19 \( 1 + (0.580 - 0.814i)T \)
23 \( 1 + (-0.995 - 0.0950i)T \)
29 \( 1 + (-0.327 - 0.945i)T \)
31 \( 1 + (0.415 - 0.909i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (0.0475 + 0.998i)T \)
43 \( 1 + (-0.327 + 0.945i)T \)
47 \( 1 + (0.841 - 0.540i)T \)
53 \( 1 + (0.841 + 0.540i)T \)
59 \( 1 + (-0.888 - 0.458i)T \)
61 \( 1 + (0.928 - 0.371i)T \)
67 \( 1 + (-0.888 + 0.458i)T \)
71 \( 1 + (-0.327 + 0.945i)T \)
73 \( 1 + (0.415 - 0.909i)T \)
79 \( 1 + (0.415 - 0.909i)T \)
83 \( 1 + (-0.959 - 0.281i)T \)
97 \( 1 + (0.981 + 0.189i)T \)
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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

−21.50380247061960376612126993588, −20.64542901129060024976354131027, −19.72568448108895481404130018531, −18.694357095549601887277645132848, −17.97704817084167993702429286961, −17.3645391223313905743739503282, −16.59226950785086988583293598144, −16.145089933432334601085161982181, −15.279809332298059831338829549348, −14.13020943889271834785616323301, −13.82662881042527340798305838237, −12.43218864453938870773827688687, −12.1546821034730829268232689999, −11.436067877650707731949094709722, −10.24167525236870732438596400859, −9.0921775910089517815874069240, −8.330634363376991069150261610378, −7.40274199825841740760670680529, −7.07967716396494976915255933199, −5.75300827948645847939726235903, −5.215089530430520312118421097496, −4.40681715629490586472902467617, −3.674394142082135004008317349236, −1.674184487186794118144555157050, −0.73700070698589032034407075175, 0.84985896820277606982472576129, 1.940769817113309400851858262691, 3.20340236044924632825905635325, 4.11551123921053049822152604694, 4.56051954830223764382259043080, 5.745063035819872797277771564622, 6.40825920849843936526823320389, 7.70612270816672664545263991895, 8.62461288890902743781150800777, 9.68683952119746988966802552104, 10.39732444556992457532901149704, 11.22320915419228416849043988803, 11.69886435045446806259936260596, 12.07739057404795164268239772401, 13.33611509660760819102816257042, 14.32117522014165432547059771818, 14.96578095292319989268961286017, 15.5525431453374460577032926656, 16.86532109947047097570710740784, 17.52261851689808620451421853297, 18.24335639700922216869494886710, 18.90672810100100223010197675663, 19.78010819706218193672281208963, 20.520273677479123034952569533955, 21.471309164737774981036177046554

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