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Properties

Label 1-1157-1157.1051-r1-0-0
Degree $1$
Conductor $1157$
Sign $-0.943 + 0.330i$
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  + (−0.458 − 0.888i)2-s + (0.814 − 0.580i)3-s + (−0.580 + 0.814i)4-s + (0.989 + 0.142i)5-s + (−0.888 − 0.458i)6-s + (−0.928 + 0.371i)7-s + (0.989 + 0.142i)8-s + (0.327 − 0.945i)9-s + (−0.327 − 0.945i)10-s + (0.371 − 0.928i)11-s + i·12-s + (0.755 + 0.654i)14-s + (0.888 − 0.458i)15-s + (−0.327 − 0.945i)16-s + (−0.888 − 0.458i)17-s + (−0.989 + 0.142i)18-s + ⋯
L(s)  = 1  + (−0.458 − 0.888i)2-s + (0.814 − 0.580i)3-s + (−0.580 + 0.814i)4-s + (0.989 + 0.142i)5-s + (−0.888 − 0.458i)6-s + (−0.928 + 0.371i)7-s + (0.989 + 0.142i)8-s + (0.327 − 0.945i)9-s + (−0.327 − 0.945i)10-s + (0.371 − 0.928i)11-s + i·12-s + (0.755 + 0.654i)14-s + (0.888 − 0.458i)15-s + (−0.327 − 0.945i)16-s + (−0.888 − 0.458i)17-s + (−0.989 + 0.142i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.943 + 0.330i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.943 + 0.330i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1157\)    =    \(13 \cdot 89\)
Sign: $-0.943 + 0.330i$
Analytic conductor: \(124.336\)
Root analytic conductor: \(124.336\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1157} (1051, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1157,\ (1:\ ),\ -0.943 + 0.330i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2740773150 - 1.611046493i\)
\(L(\frac12)\) \(\approx\) \(-0.2740773150 - 1.611046493i\)
\(L(1)\) \(\approx\) \(0.8246902819 - 0.7156430269i\)
\(L(1)\) \(\approx\) \(0.8246902819 - 0.7156430269i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
89 \( 1 \)
good2 \( 1 + (-0.458 - 0.888i)T \)
3 \( 1 + (0.814 - 0.580i)T \)
5 \( 1 + (0.989 + 0.142i)T \)
7 \( 1 + (-0.928 + 0.371i)T \)
11 \( 1 + (0.371 - 0.928i)T \)
17 \( 1 + (-0.888 - 0.458i)T \)
19 \( 1 + (0.327 - 0.945i)T \)
23 \( 1 + (0.189 + 0.981i)T \)
29 \( 1 + (0.618 - 0.786i)T \)
31 \( 1 + (0.654 - 0.755i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (-0.995 - 0.0950i)T \)
43 \( 1 + (-0.618 - 0.786i)T \)
47 \( 1 + (-0.909 + 0.415i)T \)
53 \( 1 + (-0.415 + 0.909i)T \)
59 \( 1 + (0.580 - 0.814i)T \)
61 \( 1 + (0.690 - 0.723i)T \)
67 \( 1 + (-0.814 + 0.580i)T \)
71 \( 1 + (0.618 + 0.786i)T \)
73 \( 1 + (0.755 + 0.654i)T \)
79 \( 1 + (0.654 - 0.755i)T \)
83 \( 1 + (0.841 - 0.540i)T \)
97 \( 1 + (0.371 + 0.928i)T \)
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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

−21.52774920149655685759001926941, −20.59497321085459784746050477947, −19.84688731839804204852137492487, −19.329393343805865993043965261088, −18.23243222270669133077032177031, −17.62802391693342495902973418286, −16.553247135878963024088256117922, −16.350091119281373439538875965123, −15.268393925969712039054855999461, −14.61489245263321050974818481091, −13.9265846437908864673757710162, −13.2337541930389560791927981205, −12.47403557842642116745901361084, −10.583384585433054683689949577696, −10.19406971699794427454134687827, −9.51969013753428312180020354802, −8.86129204528038708174832412364, −8.09915648379993352771617706977, −6.82611220691029867165426285786, −6.5299231662445238475619213570, −5.239127200676319661625691920565, −4.523560212130412780314369225978, −3.505271467249516207710105571217, −2.2313115646931867381745996271, −1.320291605229529817278519852790, 0.33399541976948416371148258112, 1.33253642421378202991060211600, 2.38554376342623416934137834905, 2.924488238523539592085192654, 3.70867501304356339153934902041, 5.090273617792951265449596598778, 6.376414658423054656469213153430, 6.91168726841158233130354686913, 8.17221232191167442634056901092, 8.93444704095001420412895469732, 9.472953814205514975213035095514, 10.0719929614012630887759268795, 11.29202455897131822984771452952, 11.97227844616386985989169637674, 13.041907806023079658514153532213, 13.58187650656722765459905383820, 13.81550204650324663281735343858, 15.20046866033463944843212934331, 16.05744548934602591596467515748, 17.255431390481244233665060720, 17.65940653760841810437434682925, 18.77181414382002775445323138593, 18.95366542267445145466198941053, 19.85920419889379350472523156609, 20.47066968251663182646863732591

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