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Properties

Label 1-1157-1157.1000-r0-0-0
Degree $1$
Conductor $1157$
Sign $0.289 - 0.957i$
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  + (−0.841 + 0.540i)2-s + (0.909 − 0.415i)3-s + (0.415 − 0.909i)4-s + (−0.142 − 0.989i)5-s + (−0.540 + 0.841i)6-s + (0.989 − 0.142i)7-s + (0.142 + 0.989i)8-s + (0.654 − 0.755i)9-s + (0.654 + 0.755i)10-s + (0.142 − 0.989i)11-s − i·12-s + (−0.755 + 0.654i)14-s + (−0.540 − 0.841i)15-s + (−0.654 − 0.755i)16-s + (−0.841 − 0.540i)17-s + (−0.142 + 0.989i)18-s + ⋯
L(s)  = 1  + (−0.841 + 0.540i)2-s + (0.909 − 0.415i)3-s + (0.415 − 0.909i)4-s + (−0.142 − 0.989i)5-s + (−0.540 + 0.841i)6-s + (0.989 − 0.142i)7-s + (0.142 + 0.989i)8-s + (0.654 − 0.755i)9-s + (0.654 + 0.755i)10-s + (0.142 − 0.989i)11-s − i·12-s + (−0.755 + 0.654i)14-s + (−0.540 − 0.841i)15-s + (−0.654 − 0.755i)16-s + (−0.841 − 0.540i)17-s + (−0.142 + 0.989i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.289 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.289 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1157\)    =    \(13 \cdot 89\)
Sign: $0.289 - 0.957i$
Analytic conductor: \(5.37308\)
Root analytic conductor: \(5.37308\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1157} (1000, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1157,\ (0:\ ),\ 0.289 - 0.957i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.309794919 - 0.9727285024i\)
\(L(\frac12)\) \(\approx\) \(1.309794919 - 0.9727285024i\)
\(L(1)\) \(\approx\) \(1.089101740 - 0.2954206023i\)
\(L(1)\) \(\approx\) \(1.089101740 - 0.2954206023i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
89 \( 1 \)
good2 \( 1 + (-0.841 + 0.540i)T \)
3 \( 1 + (0.909 - 0.415i)T \)
5 \( 1 + (-0.142 - 0.989i)T \)
7 \( 1 + (0.989 - 0.142i)T \)
11 \( 1 + (0.142 - 0.989i)T \)
17 \( 1 + (-0.841 - 0.540i)T \)
19 \( 1 + (0.755 + 0.654i)T \)
23 \( 1 + (0.755 + 0.654i)T \)
29 \( 1 + (0.989 - 0.142i)T \)
31 \( 1 + (-0.755 + 0.654i)T \)
37 \( 1 - iT \)
41 \( 1 + (0.909 + 0.415i)T \)
43 \( 1 + (0.989 + 0.142i)T \)
47 \( 1 + (0.415 - 0.909i)T \)
53 \( 1 + (-0.415 - 0.909i)T \)
59 \( 1 + (-0.909 - 0.415i)T \)
61 \( 1 + (-0.281 + 0.959i)T \)
67 \( 1 + (-0.415 - 0.909i)T \)
71 \( 1 + (-0.142 + 0.989i)T \)
73 \( 1 + (0.654 + 0.755i)T \)
79 \( 1 + (0.654 + 0.755i)T \)
83 \( 1 + (-0.540 + 0.841i)T \)
97 \( 1 + (0.142 + 0.989i)T \)
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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

−21.22339181024062783558306617246, −20.50212027637053229098762128039, −19.93715137408863668516583261351, −19.172606434583050011962676580582, −18.41948436092260012577678826900, −17.777870122473492731680366122711, −17.10227992432118631779158153285, −15.77885542411167086880605712467, −15.30004615910027635932146455206, −14.572694221481583577827299887350, −13.74571596323523713758764099358, −12.72601904213525404149781543545, −11.77720266459782363846697072105, −10.80974051163890648257290312171, −10.554158291656330724859959867001, −9.39907145130223299355962444468, −8.911109963929820866952706913368, −7.80166085095683807170962653462, −7.43793797487375649859255325908, −6.46747492774538898466068962580, −4.71545942588434787608155846824, −4.08574368466261429316980080785, −2.90248947694985130199911159425, −2.36620739279083534439950381500, −1.44388983941658229126795508702, 0.866959152172415184675742507855, 1.47154711731745398952795015497, 2.56086015381279318185309045900, 3.889468179049263876356821915265, 4.97012926802200355138219651615, 5.776031775284308843748507493420, 6.98130967174850514993358698125, 7.73008798404191565132850661962, 8.35553371429808956309853059945, 8.98158871235847795986756520139, 9.551450297741941376011651373820, 10.8375278008968828484885898515, 11.55966841052779701602877448707, 12.5183289050365962385734224018, 13.721359654187049804329800687, 14.035609007597414667320344588, 14.98298772920801144748106393999, 15.85647924410199291456114303004, 16.38265000354387282481362329259, 17.44960975689748004810175796966, 17.969773617438765873152269052422, 18.7848432370241307054814816725, 19.70036796055765597951171144704, 20.025906394084883023089913733258, 20.96182524501877294123981397489

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