Properties

Label 2-624-39.38-c0-0-0
Degree $2$
Conductor $624$
Sign $1$
Analytic cond. $0.311416$
Root an. cond. $0.558047$
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  + 3-s + 9-s − 13-s − 25-s + 27-s − 39-s − 2·43-s + 49-s − 2·61-s − 75-s + 2·79-s + 81-s − 2·103-s − 117-s + ⋯
L(s)  = 1  + 3-s + 9-s − 13-s − 25-s + 27-s − 39-s − 2·43-s + 49-s − 2·61-s − 75-s + 2·79-s + 81-s − 2·103-s − 117-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(624\)    =    \(2^{4} \cdot 3 \cdot 13\)
Sign: $1$
Analytic conductor: \(0.311416\)
Root analytic conductor: \(0.558047\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{624} (545, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 624,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.225816668\)
\(L(\frac12)\) \(\approx\) \(1.225816668\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - T \)
13 \( 1 + T \)
good5 \( 1 + T^{2} \)
7 \( ( 1 - T )( 1 + T ) \)
11 \( 1 + T^{2} \)
17 \( ( 1 - T )( 1 + T ) \)
19 \( ( 1 - T )( 1 + T ) \)
23 \( ( 1 - T )( 1 + T ) \)
29 \( ( 1 - T )( 1 + T ) \)
31 \( ( 1 - T )( 1 + T ) \)
37 \( ( 1 - T )( 1 + T ) \)
41 \( 1 + T^{2} \)
43 \( ( 1 + T )^{2} \)
47 \( 1 + T^{2} \)
53 \( ( 1 - T )( 1 + T ) \)
59 \( 1 + T^{2} \)
61 \( ( 1 + T )^{2} \)
67 \( ( 1 - T )( 1 + T ) \)
71 \( 1 + T^{2} \)
73 \( ( 1 - T )( 1 + T ) \)
79 \( ( 1 - T )^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + T^{2} \)
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

−10.58854000030411290247536749754, −9.814424254235537691555677314700, −9.144904734611384843436817893333, −8.148673500168146739590215078163, −7.47452512277443153862657270921, −6.53388950650064825619885190318, −5.17766734250840591721776738378, −4.14053057453284190747804537777, −3.03441820712828679733037239221, −1.90380721482039247499740957655, 1.90380721482039247499740957655, 3.03441820712828679733037239221, 4.14053057453284190747804537777, 5.17766734250840591721776738378, 6.53388950650064825619885190318, 7.47452512277443153862657270921, 8.148673500168146739590215078163, 9.144904734611384843436817893333, 9.814424254235537691555677314700, 10.58854000030411290247536749754

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