Properties

Label 1-1157-1157.453-r1-0-0
Degree $1$
Conductor $1157$
Sign $0.606 - 0.795i$
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  + (−0.371 + 0.928i)2-s + (0.723 − 0.690i)3-s + (−0.723 − 0.690i)4-s + (−0.909 + 0.415i)5-s + (0.371 + 0.928i)6-s + (−0.814 − 0.580i)7-s + (0.909 − 0.415i)8-s + (0.0475 − 0.998i)9-s + (−0.0475 − 0.998i)10-s + (0.814 − 0.580i)11-s − 12-s + (0.841 − 0.540i)14-s + (−0.371 + 0.928i)15-s + (0.0475 + 0.998i)16-s + (−0.928 + 0.371i)17-s + (0.909 + 0.415i)18-s + ⋯
L(s)  = 1  + (−0.371 + 0.928i)2-s + (0.723 − 0.690i)3-s + (−0.723 − 0.690i)4-s + (−0.909 + 0.415i)5-s + (0.371 + 0.928i)6-s + (−0.814 − 0.580i)7-s + (0.909 − 0.415i)8-s + (0.0475 − 0.998i)9-s + (−0.0475 − 0.998i)10-s + (0.814 − 0.580i)11-s − 12-s + (0.841 − 0.540i)14-s + (−0.371 + 0.928i)15-s + (0.0475 + 0.998i)16-s + (−0.928 + 0.371i)17-s + (0.909 + 0.415i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.606 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.606 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1157\)    =    \(13 \cdot 89\)
Sign: $0.606 - 0.795i$
Analytic conductor: \(124.336\)
Root analytic conductor: \(124.336\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1157} (453, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1157,\ (1:\ ),\ 0.606 - 0.795i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.316363182 - 0.6517383274i\)
\(L(\frac12)\) \(\approx\) \(1.316363182 - 0.6517383274i\)
\(L(1)\) \(\approx\) \(0.8934599547 + 0.02107686732i\)
\(L(1)\) \(\approx\) \(0.8934599547 + 0.02107686732i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
89 \( 1 \)
good2 \( 1 + (-0.371 + 0.928i)T \)
3 \( 1 + (0.723 - 0.690i)T \)
5 \( 1 + (-0.909 + 0.415i)T \)
7 \( 1 + (-0.814 - 0.580i)T \)
11 \( 1 + (0.814 - 0.580i)T \)
17 \( 1 + (-0.928 + 0.371i)T \)
19 \( 1 + (0.998 + 0.0475i)T \)
23 \( 1 + (0.888 - 0.458i)T \)
29 \( 1 + (-0.995 - 0.0950i)T \)
31 \( 1 + (0.540 + 0.841i)T \)
37 \( 1 + (0.866 + 0.5i)T \)
41 \( 1 + (0.971 + 0.235i)T \)
43 \( 1 + (0.995 - 0.0950i)T \)
47 \( 1 + (0.281 + 0.959i)T \)
53 \( 1 + (-0.959 - 0.281i)T \)
59 \( 1 + (0.690 - 0.723i)T \)
61 \( 1 + (-0.327 + 0.945i)T \)
67 \( 1 + (0.690 + 0.723i)T \)
71 \( 1 + (-0.0950 - 0.995i)T \)
73 \( 1 + (-0.540 - 0.841i)T \)
79 \( 1 + (0.841 - 0.540i)T \)
83 \( 1 + (-0.989 - 0.142i)T \)
97 \( 1 + (0.814 + 0.580i)T \)
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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

−20.96401141484198908761959395031, −20.20595959863124117942935484420, −19.83549701024519484253141586448, −19.157363282580918164833614451342, −18.49618076558285758719562895227, −17.31233914523036117594642401811, −16.52074879852283336368979265464, −15.766164901243373518616874591361, −15.13457781206658328972025512985, −14.12089009257636605990250282160, −13.13389915677617630585451925489, −12.59041943068371030615720715181, −11.54385267351914471326750101785, −11.12494620712962216681085197658, −9.8270536291574989550691785654, −9.2506130473997365569802918279, −8.90562895603580839983758964104, −7.79859390590205881027072520632, −7.0989720958879356473188994246, −5.42143769028251744837636148416, −4.42310467576516586252294582432, −3.834406960609225957373361851856, −3.00627791206949134453639689759, −2.16373140509565097816650796842, −0.81910157008866267138333476680, 0.45509726813511919427278871880, 1.209255617736235182672196275902, 2.856736438017436448463202837725, 3.70311988188837658854416593196, 4.46809770338462867461388439327, 6.08242968240169949732012001074, 6.649247577998757692145601652289, 7.327595469872840101725565257327, 7.97149729271812518328758701211, 8.936520069802644725508531298937, 9.43109654727970434297414178398, 10.61625355890945896833809297807, 11.47425738650818568829327381796, 12.65643043854408313520073081687, 13.32329565047920126755309023338, 14.180415547079793734140794523394, 14.69109733451275630910636488112, 15.61355450376922482067194679278, 16.191844108242341775224715632, 17.10665126535080466402872440764, 17.929619085052140812978502325595, 18.82198363029680085497624318590, 19.33909554613011157682505018989, 19.7658104204041053469581620909, 20.60905080865909272941777282366

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