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Properties

Label 1-1157-1157.687-r1-0-0
Degree $1$
Conductor $1157$
Sign $-0.0507 - 0.998i$
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  + (0.971 + 0.235i)2-s + (−0.888 + 0.458i)3-s + (0.888 + 0.458i)4-s + (−0.755 − 0.654i)5-s + (−0.971 + 0.235i)6-s + (0.189 − 0.981i)7-s + (0.755 + 0.654i)8-s + (0.580 − 0.814i)9-s + (−0.580 − 0.814i)10-s + (−0.189 − 0.981i)11-s − 12-s + (0.415 − 0.909i)14-s + (0.971 + 0.235i)15-s + (0.580 + 0.814i)16-s + (−0.235 − 0.971i)17-s + (0.755 − 0.654i)18-s + ⋯
L(s)  = 1  + (0.971 + 0.235i)2-s + (−0.888 + 0.458i)3-s + (0.888 + 0.458i)4-s + (−0.755 − 0.654i)5-s + (−0.971 + 0.235i)6-s + (0.189 − 0.981i)7-s + (0.755 + 0.654i)8-s + (0.580 − 0.814i)9-s + (−0.580 − 0.814i)10-s + (−0.189 − 0.981i)11-s − 12-s + (0.415 − 0.909i)14-s + (0.971 + 0.235i)15-s + (0.580 + 0.814i)16-s + (−0.235 − 0.971i)17-s + (0.755 − 0.654i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0507 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0507 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1157\)    =    \(13 \cdot 89\)
Sign: $-0.0507 - 0.998i$
Analytic conductor: \(124.336\)
Root analytic conductor: \(124.336\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1157} (687, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1157,\ (1:\ ),\ -0.0507 - 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.595369823 - 1.678576459i\)
\(L(\frac12)\) \(\approx\) \(1.595369823 - 1.678576459i\)
\(L(1)\) \(\approx\) \(1.355240148 - 0.1471708000i\)
\(L(1)\) \(\approx\) \(1.355240148 - 0.1471708000i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
89 \( 1 \)
good2 \( 1 + (0.971 + 0.235i)T \)
3 \( 1 + (-0.888 + 0.458i)T \)
5 \( 1 + (-0.755 - 0.654i)T \)
7 \( 1 + (0.189 - 0.981i)T \)
11 \( 1 + (-0.189 - 0.981i)T \)
17 \( 1 + (-0.235 - 0.971i)T \)
19 \( 1 + (0.814 + 0.580i)T \)
23 \( 1 + (0.995 + 0.0950i)T \)
29 \( 1 + (-0.327 - 0.945i)T \)
31 \( 1 + (0.909 + 0.415i)T \)
37 \( 1 + (0.866 + 0.5i)T \)
41 \( 1 + (-0.998 + 0.0475i)T \)
43 \( 1 + (0.327 - 0.945i)T \)
47 \( 1 + (-0.540 - 0.841i)T \)
53 \( 1 + (0.841 + 0.540i)T \)
59 \( 1 + (-0.458 + 0.888i)T \)
61 \( 1 + (0.928 - 0.371i)T \)
67 \( 1 + (-0.458 - 0.888i)T \)
71 \( 1 + (-0.945 - 0.327i)T \)
73 \( 1 + (-0.909 - 0.415i)T \)
79 \( 1 + (0.415 - 0.909i)T \)
83 \( 1 + (0.281 - 0.959i)T \)
97 \( 1 + (-0.189 + 0.981i)T \)
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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

−21.61464846622785613782671595730, −20.65268186563168183430471397163, −19.62874432832680917528934760769, −19.08762624112167668575217384783, −18.26444684330561157045714945911, −17.585031809612903737877869265099, −16.41580599922887184716203898399, −15.63272101420047452763507271258, −15.081892396972620592141774321750, −14.4239900057213557855884725438, −13.10091149900013833713847387577, −12.67756410418407820809200354869, −11.82050211319288556694756790622, −11.35634857036036665353076698767, −10.63996532982438084990098275235, −9.696655790020796607498519051872, −8.20545624844341679265970352235, −7.27077958154525541624892390650, −6.70326724644064222066481752137, −5.81726040863993170983403972051, −4.96956102297696943522982377830, −4.289424144860105819282201306159, −3.02238671266479465447497843262, −2.23211077022637488567143315037, −1.163618847410124771008751443233, 0.41617126881780088680829283332, 1.244825886641822920125360554, 3.11175627523126952717178025419, 3.802931793864468065139767666281, 4.66691312076871344113306886278, 5.17583502839337927485931335902, 6.12851972141649995407092704201, 7.12026736610835860435452540758, 7.74790259711013334087174379600, 8.82614207770857908529823430304, 10.09222769705014999936792433546, 10.92376524154781062622282096870, 11.67903291497732242362600804763, 11.982448614807107908764449405020, 13.291519408831738644702109128356, 13.58684056041689347558924251915, 14.80651017744006084466799846205, 15.58357060154405383024686302641, 16.28142449321916294679757583030, 16.69827839496681541454828424717, 17.368752452383156725980658518470, 18.57028215985282944812493958026, 19.59877364534208822560815099788, 20.61847395955271279078414164688, 20.77165847313743034006851673128

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