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Properties

Label 2-152-152.75-c1-0-2
Degree $2$
Conductor $152$
Sign $0.911 - 0.412i$
Analytic cond. $1.21372$
Root an. cond. $1.10169$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.491 − 1.32i)2-s + 0.754i·3-s + (−1.51 + 1.30i)4-s + 2.08i·5-s + (1 − 0.370i)6-s + 2.23i·7-s + (2.47 + 1.37i)8-s + 2.43·9-s + (2.76 − 1.02i)10-s − 0.602·11-s + (−0.982 − 1.14i)12-s − 1.29·13-s + (2.96 − 1.09i)14-s − 1.57·15-s + (0.602 − 3.95i)16-s + 2.20·17-s + ⋯
L(s)  = 1  + (−0.347 − 0.937i)2-s + 0.435i·3-s + (−0.758 + 0.651i)4-s + 0.934i·5-s + (0.408 − 0.151i)6-s + 0.845i·7-s + (0.874 + 0.484i)8-s + 0.810·9-s + (0.875 − 0.324i)10-s − 0.181·11-s + (−0.283 − 0.330i)12-s − 0.359·13-s + (0.792 − 0.293i)14-s − 0.406·15-s + (0.150 − 0.988i)16-s + 0.534·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.911 - 0.412i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.911 - 0.412i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(152\)    =    \(2^{3} \cdot 19\)
Sign: $0.911 - 0.412i$
Analytic conductor: \(1.21372\)
Root analytic conductor: \(1.10169\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{152} (75, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 152,\ (\ :1/2),\ 0.911 - 0.412i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.875077 + 0.188701i\)
\(L(\frac12)\) \(\approx\) \(0.875077 + 0.188701i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.491 + 1.32i)T \)
19 \( 1 + (2.60 - 3.49i)T \)
good3 \( 1 - 0.754iT - 3T^{2} \)
5 \( 1 - 2.08iT - 5T^{2} \)
7 \( 1 - 2.23iT - 7T^{2} \)
11 \( 1 + 0.602T + 11T^{2} \)
13 \( 1 + 1.29T + 13T^{2} \)
17 \( 1 - 2.20T + 17T^{2} \)
23 \( 1 + 6.19iT - 23T^{2} \)
29 \( 1 - 8.20T + 29T^{2} \)
31 \( 1 + 4.94T + 31T^{2} \)
37 \( 1 + 6.91T + 37T^{2} \)
41 \( 1 + 6.53iT - 41T^{2} \)
43 \( 1 + 0.191T + 43T^{2} \)
47 \( 1 + 0.223iT - 47T^{2} \)
53 \( 1 - 4.83T + 53T^{2} \)
59 \( 1 + 5.60iT - 59T^{2} \)
61 \( 1 + 11.7iT - 61T^{2} \)
67 \( 1 + 6.23iT - 67T^{2} \)
71 \( 1 - 12.4T + 71T^{2} \)
73 \( 1 + 8.27T + 73T^{2} \)
79 \( 1 - 15.4T + 79T^{2} \)
83 \( 1 - 2T + 83T^{2} \)
89 \( 1 + 7.44iT - 89T^{2} \)
97 \( 1 - 17.6iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.60518739040063856970631403497, −12.14072701900900184562306130108, −10.67015299131879292234125609555, −10.34500260600259022048616428589, −9.211922111736773109700400357502, −8.106946594336298833954352923397, −6.76156469240448764803047464206, −5.03325209808497486590217750068, −3.62452608792153191293929485673, −2.29484836554691148109004785515, 1.13869122475273519799121890031, 4.24245490928842528244775146859, 5.26848231983022702319267281737, 6.79435986403230220064945770324, 7.54205126417645562867620205356, 8.596620988200007934571582386152, 9.684982989298139044072424773271, 10.58884012295079380709126176543, 12.23847971756251053945209470683, 13.19840468775824722638659759645

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