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Properties

Label 1-1157-1157.245-r1-0-0
Degree $1$
Conductor $1157$
Sign $-0.835 - 0.549i$
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  + (0.0950 − 0.995i)2-s + (0.981 − 0.189i)3-s + (−0.981 − 0.189i)4-s + (0.281 − 0.959i)5-s + (−0.0950 − 0.995i)6-s + (0.971 − 0.235i)7-s + (−0.281 + 0.959i)8-s + (0.928 − 0.371i)9-s + (−0.928 − 0.371i)10-s + (−0.971 − 0.235i)11-s − 12-s + (−0.142 − 0.989i)14-s + (0.0950 − 0.995i)15-s + (0.928 + 0.371i)16-s + (0.995 − 0.0950i)17-s + (−0.281 − 0.959i)18-s + ⋯
L(s)  = 1  + (0.0950 − 0.995i)2-s + (0.981 − 0.189i)3-s + (−0.981 − 0.189i)4-s + (0.281 − 0.959i)5-s + (−0.0950 − 0.995i)6-s + (0.971 − 0.235i)7-s + (−0.281 + 0.959i)8-s + (0.928 − 0.371i)9-s + (−0.928 − 0.371i)10-s + (−0.971 − 0.235i)11-s − 12-s + (−0.142 − 0.989i)14-s + (0.0950 − 0.995i)15-s + (0.928 + 0.371i)16-s + (0.995 − 0.0950i)17-s + (−0.281 − 0.959i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.835 - 0.549i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.835 - 0.549i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1157\)    =    \(13 \cdot 89\)
Sign: $-0.835 - 0.549i$
Analytic conductor: \(124.336\)
Root analytic conductor: \(124.336\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1157} (245, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1157,\ (1:\ ),\ -0.835 - 0.549i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.165048899 - 3.893906803i\)
\(L(\frac12)\) \(\approx\) \(1.165048899 - 3.893906803i\)
\(L(1)\) \(\approx\) \(1.252488767 - 1.281388297i\)
\(L(1)\) \(\approx\) \(1.252488767 - 1.281388297i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
89 \( 1 \)
good2 \( 1 + (0.0950 - 0.995i)T \)
3 \( 1 + (0.981 - 0.189i)T \)
5 \( 1 + (0.281 - 0.959i)T \)
7 \( 1 + (0.971 - 0.235i)T \)
11 \( 1 + (-0.971 - 0.235i)T \)
17 \( 1 + (0.995 - 0.0950i)T \)
19 \( 1 + (0.371 + 0.928i)T \)
23 \( 1 + (0.786 + 0.618i)T \)
29 \( 1 + (0.723 - 0.690i)T \)
31 \( 1 + (0.989 - 0.142i)T \)
37 \( 1 + (0.866 + 0.5i)T \)
41 \( 1 + (0.945 - 0.327i)T \)
43 \( 1 + (-0.723 - 0.690i)T \)
47 \( 1 + (0.755 + 0.654i)T \)
53 \( 1 + (-0.654 - 0.755i)T \)
59 \( 1 + (0.189 - 0.981i)T \)
61 \( 1 + (-0.888 - 0.458i)T \)
67 \( 1 + (0.189 + 0.981i)T \)
71 \( 1 + (-0.690 + 0.723i)T \)
73 \( 1 + (-0.989 + 0.142i)T \)
79 \( 1 + (-0.142 - 0.989i)T \)
83 \( 1 + (0.909 + 0.415i)T \)
97 \( 1 + (-0.971 + 0.235i)T \)
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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

−21.34970603170943547190624563729, −20.99645395904499421046317945717, −19.7372859477633051761448249931, −18.81744076545019634189623909765, −18.2318054724206964621450404190, −17.75077286345241734769916849528, −16.62670358068516958917698099701, −15.66424586379100415113564633751, −15.072138753644362455503635881582, −14.55900299949480565549762024133, −13.89216043098139132148023089563, −13.22138217403129640076151085895, −12.22385947695577590074719383023, −10.86330696958697745307605657931, −10.184556842922272223517037886089, −9.29867186295129990678384993012, −8.45666992616534661366432255237, −7.65748757659385213834386791243, −7.236072583923559662190025354125, −6.10968192840347147566861557107, −5.055401680706187080119483694702, −4.44551043188080066297235248980, −3.11423452818910846932230934686, −2.58378585004964797790013811268, −1.13507600039839099938089051227, 0.80527309664615538071013479683, 1.40104875970841027573490864484, 2.35585279413208686404029651570, 3.25601139263915489165274960761, 4.28452063783678023977938333016, 4.98788447853010305290402973627, 5.83684242714320435660743525159, 7.69682256979474386370497381471, 8.068239082981136472918036976634, 8.8185139181055987220674889591, 9.78929281683011771943975800376, 10.25989115238673771021327837548, 11.45294817110255681500870098032, 12.24062972770384118340039869432, 12.96858079190029888783468194724, 13.70221205627929201885708299319, 14.1732936741758521574039426653, 15.09063387862828629276798754444, 16.08221010584974836956911785257, 17.19709590649918104267546090410, 17.85090203075646537306747033781, 18.76670055393737886243447060511, 19.239226456938887796092041961239, 20.361615538451237521140479178000, 20.74279011154387301940051392656

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