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Properties

Label 1-1157-1157.826-r1-0-0
Degree $1$
Conductor $1157$
Sign $-0.995 - 0.0980i$
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  + (−0.690 + 0.723i)2-s + (−0.0475 + 0.998i)3-s + (−0.0475 − 0.998i)4-s + (−0.755 − 0.654i)5-s + (−0.690 − 0.723i)6-s + (0.945 − 0.327i)7-s + (0.755 + 0.654i)8-s + (−0.995 − 0.0950i)9-s + (0.995 − 0.0950i)10-s + (0.945 + 0.327i)11-s + 12-s + (−0.415 + 0.909i)14-s + (0.690 − 0.723i)15-s + (−0.995 + 0.0950i)16-s + (−0.723 + 0.690i)17-s + (0.755 − 0.654i)18-s + ⋯
L(s)  = 1  + (−0.690 + 0.723i)2-s + (−0.0475 + 0.998i)3-s + (−0.0475 − 0.998i)4-s + (−0.755 − 0.654i)5-s + (−0.690 − 0.723i)6-s + (0.945 − 0.327i)7-s + (0.755 + 0.654i)8-s + (−0.995 − 0.0950i)9-s + (0.995 − 0.0950i)10-s + (0.945 + 0.327i)11-s + 12-s + (−0.415 + 0.909i)14-s + (0.690 − 0.723i)15-s + (−0.995 + 0.0950i)16-s + (−0.723 + 0.690i)17-s + (0.755 − 0.654i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.995 - 0.0980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.995 - 0.0980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1157\)    =    \(13 \cdot 89\)
Sign: $-0.995 - 0.0980i$
Analytic conductor: \(124.336\)
Root analytic conductor: \(124.336\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1157} (826, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1157,\ (1:\ ),\ -0.995 - 0.0980i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.03454496379 + 0.7027825087i\)
\(L(\frac12)\) \(\approx\) \(-0.03454496379 + 0.7027825087i\)
\(L(1)\) \(\approx\) \(0.5788823013 + 0.3509207686i\)
\(L(1)\) \(\approx\) \(0.5788823013 + 0.3509207686i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
89 \( 1 \)
good2 \( 1 + (-0.690 + 0.723i)T \)
3 \( 1 + (-0.0475 + 0.998i)T \)
5 \( 1 + (-0.755 - 0.654i)T \)
7 \( 1 + (0.945 - 0.327i)T \)
11 \( 1 + (0.945 + 0.327i)T \)
17 \( 1 + (-0.723 + 0.690i)T \)
19 \( 1 + (-0.0950 + 0.995i)T \)
23 \( 1 + (0.580 - 0.814i)T \)
29 \( 1 + (-0.981 - 0.189i)T \)
31 \( 1 + (-0.909 - 0.415i)T \)
37 \( 1 + (0.866 - 0.5i)T \)
41 \( 1 + (-0.458 + 0.888i)T \)
43 \( 1 + (0.981 - 0.189i)T \)
47 \( 1 + (-0.540 - 0.841i)T \)
53 \( 1 + (0.841 + 0.540i)T \)
59 \( 1 + (-0.998 + 0.0475i)T \)
61 \( 1 + (0.786 + 0.618i)T \)
67 \( 1 + (0.998 + 0.0475i)T \)
71 \( 1 + (0.189 + 0.981i)T \)
73 \( 1 + (-0.909 - 0.415i)T \)
79 \( 1 + (0.415 - 0.909i)T \)
83 \( 1 + (-0.281 + 0.959i)T \)
97 \( 1 + (0.945 - 0.327i)T \)
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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

−20.21924799647488323311038379995, −19.89918180136694987189863532224, −19.00041788890637194381199845730, −18.56798960778688175273500535122, −17.723110208345928508969006157873, −17.312896337777865059294227172708, −16.23883446670796130725147996207, −15.192974100282418906469380422263, −14.33974030513764654940917585253, −13.52721265039348175689500697634, −12.61901475621181163586655431721, −11.71542766642406065537112110336, −11.28109432844094822949378551715, −10.920264680754272361852019610293, −9.27104314256146867024277215517, −8.77907408497361731670325703976, −7.82244861061350785324632043905, −7.234213972805533103965827261841, −6.53750360839203794711812388470, −5.12029322551762271448306021615, −3.96906013341747928224393351688, −2.99530028101340923570700110354, −2.170750602803857738554463995696, −1.22424276595985493400337103675, −0.22812754738995767187313605265, 0.94961277904780674282937156908, 2.01325904346802954629449606060, 3.92040306159202306420422156806, 4.30459076646416075730753548845, 5.198817186999053167330686501820, 6.07077284030385788498951633090, 7.23073723808348992414276555589, 8.08175149472244281002462402055, 8.73616054167165265690511811508, 9.337513549075983732770713876896, 10.35207048243316846857307746872, 11.12964399460936742591977895984, 11.66568637210631232803124776884, 12.90854359759003955310755776429, 14.182969947897944042968634208956, 14.91894852398732983394998793476, 15.12556350672829838496353937989, 16.319665363741774389186759224457, 16.78074722926198501389840790943, 17.26101284971213892580783040810, 18.236216262723991208262393069, 19.23561552325692498712364157332, 20.20381496467362009738534619506, 20.29792738427274555153352109590, 21.35820594055000109787749806064

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