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Properties

Label 2-152-152.35-c0-0-0
Degree $2$
Conductor $152$
Sign $0.158 - 0.987i$
Analytic cond. $0.0758578$
Root an. cond. $0.275423$
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  + (0.766 + 0.642i)2-s + (−1.43 + 0.524i)3-s + (0.173 + 0.984i)4-s + (−1.43 − 0.524i)6-s + (−0.500 + 0.866i)8-s + (1.03 − 0.866i)9-s + (0.939 − 1.62i)11-s + (−0.766 − 1.32i)12-s + (−0.939 + 0.342i)16-s + (−0.766 − 0.642i)17-s + 1.34·18-s + (0.173 + 0.984i)19-s + (1.76 − 0.642i)22-s + (0.266 − 1.50i)24-s + (−0.939 − 0.342i)25-s + ⋯
L(s)  = 1  + (0.766 + 0.642i)2-s + (−1.43 + 0.524i)3-s + (0.173 + 0.984i)4-s + (−1.43 − 0.524i)6-s + (−0.500 + 0.866i)8-s + (1.03 − 0.866i)9-s + (0.939 − 1.62i)11-s + (−0.766 − 1.32i)12-s + (−0.939 + 0.342i)16-s + (−0.766 − 0.642i)17-s + 1.34·18-s + (0.173 + 0.984i)19-s + (1.76 − 0.642i)22-s + (0.266 − 1.50i)24-s + (−0.939 − 0.342i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.158 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.158 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(152\)    =    \(2^{3} \cdot 19\)
Sign: $0.158 - 0.987i$
Analytic conductor: \(0.0758578\)
Root analytic conductor: \(0.275423\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{152} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 152,\ (\ :0),\ 0.158 - 0.987i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6408940441\)
\(L(\frac12)\) \(\approx\) \(0.6408940441\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.766 - 0.642i)T \)
19 \( 1 + (-0.173 - 0.984i)T \)
good3 \( 1 + (1.43 - 0.524i)T + (0.766 - 0.642i)T^{2} \)
5 \( 1 + (0.939 + 0.342i)T^{2} \)
7 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.939 + 1.62i)T + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (-0.766 - 0.642i)T^{2} \)
17 \( 1 + (0.766 + 0.642i)T + (0.173 + 0.984i)T^{2} \)
23 \( 1 + (0.939 - 0.342i)T^{2} \)
29 \( 1 + (-0.173 + 0.984i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (0.326 - 0.118i)T + (0.766 - 0.642i)T^{2} \)
43 \( 1 + (0.173 - 0.984i)T + (-0.939 - 0.342i)T^{2} \)
47 \( 1 + (-0.173 + 0.984i)T^{2} \)
53 \( 1 + (0.939 - 0.342i)T^{2} \)
59 \( 1 + (1.43 + 1.20i)T + (0.173 + 0.984i)T^{2} \)
61 \( 1 + (0.939 - 0.342i)T^{2} \)
67 \( 1 + (-0.266 + 0.223i)T + (0.173 - 0.984i)T^{2} \)
71 \( 1 + (0.939 + 0.342i)T^{2} \)
73 \( 1 + (-1.76 + 0.642i)T + (0.766 - 0.642i)T^{2} \)
79 \( 1 + (-0.766 + 0.642i)T^{2} \)
83 \( 1 + (0.173 + 0.300i)T + (-0.5 + 0.866i)T^{2} \)
89 \( 1 + (-0.939 - 0.342i)T + (0.766 + 0.642i)T^{2} \)
97 \( 1 + (-0.266 - 0.223i)T + (0.173 + 0.984i)T^{2} \)
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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

−13.58028381674417588593838814410, −12.28021438899587653387694813070, −11.53417152729868146212698528655, −10.92273636431200115756053665644, −9.389139201469049750242244703370, −8.061747327129448460845458977144, −6.46149551819120856720807159243, −5.95939940282955441739391670155, −4.80514628386457119246701430492, −3.60482179663070603773393341866, 1.81909768811786175071966088324, 4.19419389257057914864481064603, 5.21030569718209570977292750569, 6.42903125424806765413335232275, 7.10497112277540841963935825572, 9.313763336944063215364157161759, 10.38483817057921223197165049852, 11.35805099377802497669281295246, 12.01403000114572407670640642645, 12.71908359797808511645762764996

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