Properties

Label 2-152-152.37-c0-0-1
Degree $2$
Conductor $152$
Sign $1$
Analytic cond. $0.0758578$
Root an. cond. $0.275423$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3-s + 4-s − 6-s − 7-s + 8-s − 12-s − 13-s − 14-s + 16-s − 17-s + 19-s + 21-s − 23-s − 24-s + 25-s − 26-s + 27-s − 28-s − 29-s + 32-s − 34-s + 2·37-s + 38-s + 39-s + 42-s − 46-s + ⋯
L(s)  = 1  + 2-s − 3-s + 4-s − 6-s − 7-s + 8-s − 12-s − 13-s − 14-s + 16-s − 17-s + 19-s + 21-s − 23-s − 24-s + 25-s − 26-s + 27-s − 28-s − 29-s + 32-s − 34-s + 2·37-s + 38-s + 39-s + 42-s − 46-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(152\)    =    \(2^{3} \cdot 19\)
Sign: $1$
Analytic conductor: \(0.0758578\)
Root analytic conductor: \(0.275423\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{152} (37, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 152,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7788472013\)
\(L(\frac12)\) \(\approx\) \(0.7788472013\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
19 \( 1 - T \)
good3 \( 1 + T + T^{2} \)
5 \( ( 1 - T )( 1 + T ) \)
7 \( 1 + T + T^{2} \)
11 \( ( 1 - T )( 1 + T ) \)
13 \( 1 + T + T^{2} \)
17 \( 1 + T + T^{2} \)
23 \( 1 + T + T^{2} \)
29 \( 1 + T + T^{2} \)
31 \( ( 1 - T )( 1 + T ) \)
37 \( ( 1 - T )^{2} \)
41 \( ( 1 - T )( 1 + T ) \)
43 \( ( 1 - T )( 1 + T ) \)
47 \( ( 1 - T )^{2} \)
53 \( 1 + T + T^{2} \)
59 \( 1 + T + T^{2} \)
61 \( ( 1 - T )( 1 + T ) \)
67 \( 1 + T + T^{2} \)
71 \( ( 1 - T )( 1 + T ) \)
73 \( 1 + T + T^{2} \)
79 \( ( 1 - T )( 1 + T ) \)
83 \( ( 1 - T )( 1 + T ) \)
89 \( ( 1 - T )( 1 + T ) \)
97 \( ( 1 - T )( 1 + T ) \)
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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

−13.03984448983643196413376115173, −12.29223974961841780502300444212, −11.48149381049958660012230602000, −10.56096522015558538979777339827, −9.420874252673057307646955909800, −7.50781507085966449620544772843, −6.46055524279898288954266410418, −5.63050345906219303013958354741, −4.44276652764072509073740472545, −2.81196093969486835641282132236, 2.81196093969486835641282132236, 4.44276652764072509073740472545, 5.63050345906219303013958354741, 6.46055524279898288954266410418, 7.50781507085966449620544772843, 9.420874252673057307646955909800, 10.56096522015558538979777339827, 11.48149381049958660012230602000, 12.29223974961841780502300444212, 13.03984448983643196413376115173

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