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Properties

Label 1-1157-1157.778-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: $\chi_{1157} (778, \cdot )$
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

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

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