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Properties

Label 2-152-152.123-c0-0-0
Degree $2$
Conductor $152$
Sign $0.486 - 0.873i$
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  + (0.173 + 0.984i)2-s + (0.266 − 0.223i)3-s + (−0.939 + 0.342i)4-s + (0.266 + 0.223i)6-s + (−0.5 − 0.866i)8-s + (−0.152 + 0.866i)9-s + (−0.766 − 1.32i)11-s + (−0.173 + 0.300i)12-s + (0.766 − 0.642i)16-s + (−0.173 − 0.984i)17-s − 0.879·18-s + (−0.939 + 0.342i)19-s + (1.17 − 0.984i)22-s + (−0.326 − 0.118i)24-s + (0.766 + 0.642i)25-s + ⋯
L(s)  = 1  + (0.173 + 0.984i)2-s + (0.266 − 0.223i)3-s + (−0.939 + 0.342i)4-s + (0.266 + 0.223i)6-s + (−0.5 − 0.866i)8-s + (−0.152 + 0.866i)9-s + (−0.766 − 1.32i)11-s + (−0.173 + 0.300i)12-s + (0.766 − 0.642i)16-s + (−0.173 − 0.984i)17-s − 0.879·18-s + (−0.939 + 0.342i)19-s + (1.17 − 0.984i)22-s + (−0.326 − 0.118i)24-s + (0.766 + 0.642i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.486 - 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.486 - 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(152\)    =    \(2^{3} \cdot 19\)
Sign: $0.486 - 0.873i$
Analytic conductor: \(0.0758578\)
Root analytic conductor: \(0.275423\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{152} (123, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 152,\ (\ :0),\ 0.486 - 0.873i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6638767189\)
\(L(\frac12)\) \(\approx\) \(0.6638767189\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.173 - 0.984i)T \)
19 \( 1 + (0.939 - 0.342i)T \)
good3 \( 1 + (-0.266 + 0.223i)T + (0.173 - 0.984i)T^{2} \)
5 \( 1 + (-0.766 - 0.642i)T^{2} \)
7 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.766 + 1.32i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (-0.173 - 0.984i)T^{2} \)
17 \( 1 + (0.173 + 0.984i)T + (-0.939 + 0.342i)T^{2} \)
23 \( 1 + (-0.766 + 0.642i)T^{2} \)
29 \( 1 + (0.939 + 0.342i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (1.43 - 1.20i)T + (0.173 - 0.984i)T^{2} \)
43 \( 1 + (-0.939 - 0.342i)T + (0.766 + 0.642i)T^{2} \)
47 \( 1 + (0.939 + 0.342i)T^{2} \)
53 \( 1 + (-0.766 + 0.642i)T^{2} \)
59 \( 1 + (-0.266 - 1.50i)T + (-0.939 + 0.342i)T^{2} \)
61 \( 1 + (-0.766 + 0.642i)T^{2} \)
67 \( 1 + (0.326 - 1.85i)T + (-0.939 - 0.342i)T^{2} \)
71 \( 1 + (-0.766 - 0.642i)T^{2} \)
73 \( 1 + (-1.17 + 0.984i)T + (0.173 - 0.984i)T^{2} \)
79 \( 1 + (-0.173 + 0.984i)T^{2} \)
83 \( 1 + (-0.939 + 1.62i)T + (-0.5 - 0.866i)T^{2} \)
89 \( 1 + (0.766 + 0.642i)T + (0.173 + 0.984i)T^{2} \)
97 \( 1 + (0.326 + 1.85i)T + (-0.939 + 0.342i)T^{2} \)
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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

−13.48963246185039348730714480293, −12.93554009645926971070636102265, −11.42047336298945611090097665128, −10.29414127559180691733208346014, −8.838630690342372699794015776849, −8.163037950155806671509928702442, −7.13243471349478150365941830909, −5.83413913608135674393486940162, −4.80112177596242387302279731740, −3.04413068870740298625400270153, 2.28876432896270193808624426026, 3.81922436752524545879087043351, 4.94318360401979582091211444118, 6.51769894007172285748150378118, 8.203525440819530783913216385003, 9.196965266464010512139947826405, 10.15973972192527861392425112046, 10.93018617005960186477121113874, 12.39560689713815246110174439136, 12.63752696247498362069275879494

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