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Properties

Label 1-1157-1157.58-r0-0-0
Degree $1$
Conductor $1157$
Sign $0.202 + 0.979i$
Analytic cond. $5.37308$
Root an. cond. $5.37308$
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  + (0.945 + 0.327i)2-s + (−0.992 − 0.118i)3-s + (0.786 + 0.618i)4-s + (−0.841 + 0.540i)5-s + (−0.899 − 0.436i)6-s + (−0.739 − 0.672i)7-s + (0.540 + 0.841i)8-s + (0.971 + 0.235i)9-s + (−0.971 + 0.235i)10-s + (−0.998 − 0.0475i)11-s + (−0.707 − 0.707i)12-s + (−0.479 − 0.877i)14-s + (0.899 − 0.436i)15-s + (0.235 + 0.971i)16-s + (0.945 − 0.327i)17-s + (0.841 + 0.540i)18-s + ⋯
L(s)  = 1  + (0.945 + 0.327i)2-s + (−0.992 − 0.118i)3-s + (0.786 + 0.618i)4-s + (−0.841 + 0.540i)5-s + (−0.899 − 0.436i)6-s + (−0.739 − 0.672i)7-s + (0.540 + 0.841i)8-s + (0.971 + 0.235i)9-s + (−0.971 + 0.235i)10-s + (−0.998 − 0.0475i)11-s + (−0.707 − 0.707i)12-s + (−0.479 − 0.877i)14-s + (0.899 − 0.436i)15-s + (0.235 + 0.971i)16-s + (0.945 − 0.327i)17-s + (0.841 + 0.540i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.202 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.202 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1157\)    =    \(13 \cdot 89\)
Sign: $0.202 + 0.979i$
Analytic conductor: \(5.37308\)
Root analytic conductor: \(5.37308\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1157} (58, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1157,\ (0:\ ),\ 0.202 + 0.979i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.058826109 + 0.8626787433i\)
\(L(\frac12)\) \(\approx\) \(1.058826109 + 0.8626787433i\)
\(L(1)\) \(\approx\) \(1.060382902 + 0.3281002111i\)
\(L(1)\) \(\approx\) \(1.060382902 + 0.3281002111i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
89 \( 1 \)
good2 \( 1 + (0.945 + 0.327i)T \)
3 \( 1 + (-0.992 - 0.118i)T \)
5 \( 1 + (-0.841 + 0.540i)T \)
7 \( 1 + (-0.739 - 0.672i)T \)
11 \( 1 + (-0.998 - 0.0475i)T \)
17 \( 1 + (0.945 - 0.327i)T \)
19 \( 1 + (0.853 - 0.520i)T \)
23 \( 1 + (-0.0237 - 0.999i)T \)
29 \( 1 + (0.952 - 0.304i)T \)
31 \( 1 + (0.479 + 0.877i)T \)
37 \( 1 + (-0.258 + 0.965i)T \)
41 \( 1 + (-0.393 + 0.919i)T \)
43 \( 1 + (-0.952 - 0.304i)T \)
47 \( 1 + (-0.142 + 0.989i)T \)
53 \( 1 + (-0.989 + 0.142i)T \)
59 \( 1 + (0.118 + 0.992i)T \)
61 \( 1 + (0.636 + 0.771i)T \)
67 \( 1 + (0.618 + 0.786i)T \)
71 \( 1 + (0.888 - 0.458i)T \)
73 \( 1 + (-0.281 + 0.959i)T \)
79 \( 1 + (0.281 - 0.959i)T \)
83 \( 1 + (-0.0713 - 0.997i)T \)
97 \( 1 + (0.998 - 0.0475i)T \)
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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

−21.232039981212211123794084032296, −20.60416086417323722837258033865, −19.58722388261036224564078437597, −18.91396204851552759508961250884, −18.249222850360062977237374487673, −16.938043304229475482015294453345, −16.19604278429558720174367450102, −15.661746672224080065447285445895, −15.236752315936518838547529431281, −13.93025963756153317868351461474, −12.90048693417828753393239434236, −12.4775259210888881530271993889, −11.84799665060010949625718309, −11.18484778729803421592592245282, −10.14864187973291535872223048927, −9.60502265538406108086500377562, −8.09898713892463596578531583618, −7.26386852330506840300823476895, −6.28029142688038174250655078734, −5.3199791333905926013941018706, −5.11916685019288647025764174383, −3.79525424242472350809823581855, −3.24655180293838779866580239740, −1.83645556001386922124948680701, −0.58358094122315325465707413043, 0.95430948072527179934408867882, 2.77246067701670815139487230066, 3.34901961501024115907842649888, 4.50875403076738504531118072861, 5.04510184406324182283732958658, 6.191817407467159312433208288710, 6.81743037229830291874587007527, 7.47715499983572141332961135271, 8.214080778817705837648018430301, 10.096473687408083541046777100185, 10.48712651431238283687895522409, 11.47479733687913400140326744840, 12.04740947143120848715909495738, 12.81808739909494279675601266536, 13.55695403814368723767499116654, 14.396203651112319440238608166383, 15.49451854192617418411133944979, 16.00760189358199232583471525130, 16.461124603674309220656099337276, 17.41611274850887567673089651647, 18.352940953662762924437831389246, 19.10159071324433301796265326792, 20.03961561269540925619913862001, 20.821905692870331341537920281474, 21.751439076707874305469403445627

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