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Properties

Label 2-152-152.99-c0-0-0
Degree $2$
Conductor $152$
Sign $-0.378 + 0.925i$
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  + (−0.939 − 0.342i)2-s + (−0.326 − 1.85i)3-s + (0.766 + 0.642i)4-s + (−0.326 + 1.85i)6-s + (−0.500 − 0.866i)8-s + (−2.37 + 0.866i)9-s + (−0.173 − 0.300i)11-s + (0.939 − 1.62i)12-s + (0.173 + 0.984i)16-s + (0.939 + 0.342i)17-s + 2.53·18-s + (0.766 + 0.642i)19-s + (0.0603 + 0.342i)22-s + (−1.43 + 1.20i)24-s + (0.173 − 0.984i)25-s + ⋯
L(s)  = 1  + (−0.939 − 0.342i)2-s + (−0.326 − 1.85i)3-s + (0.766 + 0.642i)4-s + (−0.326 + 1.85i)6-s + (−0.500 − 0.866i)8-s + (−2.37 + 0.866i)9-s + (−0.173 − 0.300i)11-s + (0.939 − 1.62i)12-s + (0.173 + 0.984i)16-s + (0.939 + 0.342i)17-s + 2.53·18-s + (0.766 + 0.642i)19-s + (0.0603 + 0.342i)22-s + (−1.43 + 1.20i)24-s + (0.173 − 0.984i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.378 + 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.378 + 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(152\)    =    \(2^{3} \cdot 19\)
Sign: $-0.378 + 0.925i$
Analytic conductor: \(0.0758578\)
Root analytic conductor: \(0.275423\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{152} (99, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 152,\ (\ :0),\ -0.378 + 0.925i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3974292175\)
\(L(\frac12)\) \(\approx\) \(0.3974292175\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.939 + 0.342i)T \)
19 \( 1 + (-0.766 - 0.642i)T \)
good3 \( 1 + (0.326 + 1.85i)T + (-0.939 + 0.342i)T^{2} \)
5 \( 1 + (-0.173 + 0.984i)T^{2} \)
7 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.173 + 0.300i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.939 + 0.342i)T^{2} \)
17 \( 1 + (-0.939 - 0.342i)T + (0.766 + 0.642i)T^{2} \)
23 \( 1 + (-0.173 - 0.984i)T^{2} \)
29 \( 1 + (-0.766 + 0.642i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (-0.266 - 1.50i)T + (-0.939 + 0.342i)T^{2} \)
43 \( 1 + (0.766 - 0.642i)T + (0.173 - 0.984i)T^{2} \)
47 \( 1 + (-0.766 + 0.642i)T^{2} \)
53 \( 1 + (-0.173 - 0.984i)T^{2} \)
59 \( 1 + (0.326 + 0.118i)T + (0.766 + 0.642i)T^{2} \)
61 \( 1 + (-0.173 - 0.984i)T^{2} \)
67 \( 1 + (1.43 - 0.524i)T + (0.766 - 0.642i)T^{2} \)
71 \( 1 + (-0.173 + 0.984i)T^{2} \)
73 \( 1 + (-0.0603 - 0.342i)T + (-0.939 + 0.342i)T^{2} \)
79 \( 1 + (0.939 - 0.342i)T^{2} \)
83 \( 1 + (0.766 - 1.32i)T + (-0.5 - 0.866i)T^{2} \)
89 \( 1 + (0.173 - 0.984i)T + (-0.939 - 0.342i)T^{2} \)
97 \( 1 + (1.43 + 0.524i)T + (0.766 + 0.642i)T^{2} \)
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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

−12.61827844700886109987831941579, −11.97180342685846314469107451509, −11.15418693766925744628716813408, −9.906150846988889769290051670243, −8.364272804129913652850256181892, −7.83150068134229021638421772963, −6.77135032788205920240945245072, −5.80589266530993807496984830934, −2.93868752547714674847099105795, −1.39037118027808155937599020260, 3.20332236583521029241306119119, 4.91519720797899269806536893658, 5.78546712841777911210077602306, 7.39758519646285917014352048049, 8.818110884497844507019730665502, 9.537530903351788838902073427769, 10.30210052813183256643901398852, 11.12773911450365579764331666597, 11.99817812850752840166902379306, 14.02535236977326069349586725758

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