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Properties

Label 1-1157-1157.996-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  + (−0.540 + 0.841i)2-s + (0.909 + 0.415i)3-s + (−0.415 − 0.909i)4-s + (0.989 + 0.142i)5-s + (−0.841 + 0.540i)6-s + (−0.142 + 0.989i)7-s + (0.989 + 0.142i)8-s + (0.654 + 0.755i)9-s + (−0.654 + 0.755i)10-s + (−0.989 + 0.142i)11-s − i·12-s + (−0.755 − 0.654i)14-s + (0.841 + 0.540i)15-s + (−0.654 + 0.755i)16-s + (0.841 − 0.540i)17-s + (−0.989 + 0.142i)18-s + ⋯
L(s)  = 1  + (−0.540 + 0.841i)2-s + (0.909 + 0.415i)3-s + (−0.415 − 0.909i)4-s + (0.989 + 0.142i)5-s + (−0.841 + 0.540i)6-s + (−0.142 + 0.989i)7-s + (0.989 + 0.142i)8-s + (0.654 + 0.755i)9-s + (−0.654 + 0.755i)10-s + (−0.989 + 0.142i)11-s − i·12-s + (−0.755 − 0.654i)14-s + (0.841 + 0.540i)15-s + (−0.654 + 0.755i)16-s + (0.841 − 0.540i)17-s + (−0.989 + 0.142i)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: $\chi_{1157} (996, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1157,\ (1:\ ),\ -0.554 - 0.832i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.5010333303 + 0.9353178477i\)
\(L(\frac12)\) \(\approx\) \(-0.5010333303 + 0.9353178477i\)
\(L(1)\) \(\approx\) \(0.7896080749 + 0.6824766810i\)
\(L(1)\) \(\approx\) \(0.7896080749 + 0.6824766810i\)

Euler product

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

−20.596870081012350517807029380522, −19.87139735357987490673844907615, −19.08290429858424844813940313586, −18.35341712865986787907151169677, −17.80413932426163748829514224805, −16.82287770431292840994731364786, −16.30330567479754502081955175751, −14.86230789078887467571660328876, −14.01403542342843436568098869811, −13.4388914520854728703183694975, −12.83414195367432924630004997073, −12.155994130863651415269697074, −10.73748998920848605278969917020, −10.07958788045886203617101195993, −9.77195698872868509400255212905, −8.4377255230032352069899922703, −8.14170258086794052556548521559, −7.12771788284449203780964572497, −6.162402137551149178441198991710, −4.78606100889955825142061908953, −3.765024817239982010122588778906, −2.98127546596975141125501437571, −2.010134558920889225335413491836, −1.354565976739135895302745778652, −0.20014938833267021312967871699, 1.58188653087410438616241414919, 2.360724946252840552076048003123, 3.28535108116960129231659302766, 4.95007764326852167831014773338, 5.25385480470031134940728960921, 6.331916612425287681401536037932, 7.23833964817067443890140703196, 8.20647689243053945816458960067, 8.817496511334712871859409860435, 9.60732349374231797117177652194, 10.110278482204610219855566950942, 10.91986572428832397624133352623, 12.46925970663129452244068899034, 13.325841821934525768977865030476, 14.06165524123708091631531356201, 14.62276791521636108495890728582, 15.70535679959384627381359959743, 15.76490830739053360494750174033, 16.93285142768165234306285342421, 17.85885971189352725912550564757, 18.4421685964604588471345710612, 19.064164031999977234137703524645, 19.93544033027802282674212368444, 20.88547959946647395269209190160, 21.59416066081266347980311758553

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