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Properties

Label 1-1157-1157.1047-r1-0-0
Degree $1$
Conductor $1157$
Sign $0.601 + 0.798i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.601 + 0.798i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.601 + 0.798i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1157\)    =    \(13 \cdot 89\)
Sign: $0.601 + 0.798i$
Analytic conductor: \(124.336\)
Root analytic conductor: \(124.336\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1157} (1047, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1157,\ (1:\ ),\ 0.601 + 0.798i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.671685839 + 0.8334707816i\)
\(L(\frac12)\) \(\approx\) \(1.671685839 + 0.8334707816i\)
\(L(1)\) \(\approx\) \(1.294624867 - 0.2646034091i\)
\(L(1)\) \(\approx\) \(1.294624867 - 0.2646034091i\)

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

−20.84338505765329749366852661480, −20.302432819043777603700755034297, −19.504425710584495547515653570830, −18.47235528395504318805805922288, −17.87946397228602972674156952643, −17.17018902591807678331316948783, −16.04841097301565599929864211000, −15.33048112505727443424406114790, −14.84558168632318939502728047118, −14.05816308521102671827853890638, −13.35061360437955875042760958285, −12.43323967540941634335339666076, −11.95502530534396711814670720324, −10.8443873185862323855277019165, −9.47001634045834493838996268502, −8.650057412955949365972132445461, −7.88433799036740683791785192658, −7.446209362917603387130672081951, −6.799142955664438852173540578450, −5.493640871303621160823753205494, −4.40791225442190607305882413598, −4.00671057046661430269955712536, −2.860122705715232047449081244784, −1.70906218877147272190169816187, −0.31072502125806614896974890299, 1.01692356509169716162963367298, 2.38880324186155518463286801056, 2.88421593607523644738073143597, 3.99271327532820770475940349638, 4.53526129529909925501052930972, 5.31904603603312959409723521891, 6.645935039785132345908479718474, 8.01138520699223208335626314673, 8.54619351668343785426387904337, 9.163006665075429055524571871094, 10.45464992836108085158230961935, 11.06068811666595628008232923579, 11.45507077453379949471727120910, 12.68749867793508909566526908355, 13.31612704715927037172130262369, 14.3672012831835575943399853282, 14.78562470401933774985362129632, 15.51787903532238489739204308611, 16.21578603965147544775155501080, 17.54405847792089498686415672078, 18.61408086918242295832629591898, 18.99128797420791428285399297675, 19.935182784221663421837441368472, 20.32667885163162917304584181031, 21.205509269165075765116196447309

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