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Properties

Label 1-1157-1157.331-r1-0-0
Degree $1$
Conductor $1157$
Sign $0.922 + 0.385i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.922 + 0.385i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.922 + 0.385i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1157\)    =    \(13 \cdot 89\)
Sign: $0.922 + 0.385i$
Analytic conductor: \(124.336\)
Root analytic conductor: \(124.336\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1157} (331, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1157,\ (1:\ ),\ 0.922 + 0.385i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.430955681 + 0.2867578274i\)
\(L(\frac12)\) \(\approx\) \(1.430955681 + 0.2867578274i\)
\(L(1)\) \(\approx\) \(1.186158444 - 0.7830401941i\)
\(L(1)\) \(\approx\) \(1.186158444 - 0.7830401941i\)

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

−21.32121355209140971587910901197, −20.43390670249202475655582314032, −20.1399178904200664861466480693, −18.11393449523304605575493445123, −17.906179372620274757233754220076, −16.93416888255486328057723272232, −16.31570050336067183193355778114, −15.460364079472837044948420080514, −15.03949563610148338230621807822, −13.970560501807089183178750736695, −13.581197693330574420468983985952, −12.524636235016195885093085272475, −11.69859972265506395599191895224, −10.79681209823371539523460965743, −9.86174588148661005108851757300, −8.69003332032849605078336372927, −8.58439161689178915579902612260, −7.347824650490875601991317056290, −6.20207996503303340780105279833, −5.26901227562927757519899488325, −4.86929907325752148341886442950, −4.25713074766505731598408808346, −2.792015524115037916028036709426, −2.17434004754391938928860700867, −0.1986881858609531669397795997, 1.26867718980455646471772970538, 1.913263587080990172552887398201, 2.669661452119301919206844095262, 3.66150410308801782884373934189, 4.89562571257188209043134065026, 5.81429500686218834891859348050, 6.30191519716526038889351471835, 7.44929389698617582405734127376, 8.20945511067933560789066648875, 9.35910561966888362863485369358, 10.49296812204750855794041307913, 10.89660379283193982303688500853, 11.73302680356335299534761119814, 12.63434088456197321486612248697, 13.452410321024514602884131014903, 13.836210190987201367012657389762, 14.628042457136093115449038775564, 15.240486931863613260455232678187, 16.660404790322895462813885243185, 17.72853661930791820653905106296, 18.20093895107268877412506899754, 18.71737062452622123518473810612, 19.73774266922484328646710854998, 20.32265565451932540264184450893, 21.332546748501265741493636391271

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