Properties

Label 2-504-1.1-c1-0-0
Degree $2$
Conductor $504$
Sign $1$
Analytic cond. $4.02446$
Root an. cond. $2.00610$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s − 7-s + 4·11-s + 2·13-s + 6·17-s + 8·19-s − 25-s − 6·29-s + 8·31-s + 2·35-s − 2·37-s − 2·41-s − 4·43-s + 8·47-s + 49-s − 6·53-s − 8·55-s − 6·61-s − 4·65-s − 4·67-s + 8·71-s + 10·73-s − 4·77-s + 16·79-s − 8·83-s − 12·85-s + 6·89-s + ⋯
L(s)  = 1  − 0.894·5-s − 0.377·7-s + 1.20·11-s + 0.554·13-s + 1.45·17-s + 1.83·19-s − 1/5·25-s − 1.11·29-s + 1.43·31-s + 0.338·35-s − 0.328·37-s − 0.312·41-s − 0.609·43-s + 1.16·47-s + 1/7·49-s − 0.824·53-s − 1.07·55-s − 0.768·61-s − 0.496·65-s − 0.488·67-s + 0.949·71-s + 1.17·73-s − 0.455·77-s + 1.80·79-s − 0.878·83-s − 1.30·85-s + 0.635·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $1$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{504} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.325718303\)
\(L(\frac12)\) \(\approx\) \(1.325718303\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 + T \)
good5 \( 1 + 2 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−11.09014316837144741766145173789, −9.886707849578587066687733658491, −9.256691510852288615675806117776, −8.093537689649372229665039418958, −7.40397429472305265033368100557, −6.35373904523042773985731435784, −5.29340219049011126125230323142, −3.89550776474805329328185747620, −3.26735667825841356907489548421, −1.15037892508990574535308118020, 1.15037892508990574535308118020, 3.26735667825841356907489548421, 3.89550776474805329328185747620, 5.29340219049011126125230323142, 6.35373904523042773985731435784, 7.40397429472305265033368100557, 8.093537689649372229665039418958, 9.256691510852288615675806117776, 9.886707849578587066687733658491, 11.09014316837144741766145173789

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