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Properties

Label 2-504-56.13-c0-0-2
Degree $2$
Conductor $504$
Sign $1$
Analytic cond. $0.251528$
Root an. cond. $0.501526$
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 + 4-s − 7-s + 8-s − 14-s + 16-s − 2·23-s − 25-s − 28-s + 32-s − 2·46-s + 49-s − 50-s − 56-s + 64-s + 2·71-s − 2·79-s − 2·92-s + 98-s − 100-s − 112-s + 2·113-s + ⋯
L(s)  = 1  + 2-s + 4-s − 7-s + 8-s − 14-s + 16-s − 2·23-s − 25-s − 28-s + 32-s − 2·46-s + 49-s − 50-s − 56-s + 64-s + 2·71-s − 2·79-s − 2·92-s + 98-s − 100-s − 112-s + 2·113-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $1$
Analytic conductor: \(0.251528\)
Root analytic conductor: \(0.501526\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{504} (181, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.439893100\)
\(L(\frac12)\) \(\approx\) \(1.439893100\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 \)
7 \( 1 + T \)
good5 \( 1 + T^{2} \)
11 \( ( 1 - T )( 1 + T ) \)
13 \( 1 + T^{2} \)
17 \( ( 1 - T )( 1 + T ) \)
19 \( 1 + T^{2} \)
23 \( ( 1 + T )^{2} \)
29 \( ( 1 - T )( 1 + T ) \)
31 \( ( 1 - T )( 1 + T ) \)
37 \( ( 1 - T )( 1 + T ) \)
41 \( ( 1 - T )( 1 + T ) \)
43 \( ( 1 - T )( 1 + T ) \)
47 \( ( 1 - T )( 1 + T ) \)
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 \( ( 1 - T )( 1 + T ) \)
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

−11.36883216382947901268222888836, −10.26281683024352818903610460981, −9.666087849294455038237157313933, −8.255353598993736633003017504851, −7.28424641630970624002389000280, −6.29115887013593533422292284790, −5.65539355150237745385542105144, −4.28631963116974234825053477736, −3.43363451204020501313795054491, −2.15848280631622720996979725155, 2.15848280631622720996979725155, 3.43363451204020501313795054491, 4.28631963116974234825053477736, 5.65539355150237745385542105144, 6.29115887013593533422292284790, 7.28424641630970624002389000280, 8.255353598993736633003017504851, 9.666087849294455038237157313933, 10.26281683024352818903610460981, 11.36883216382947901268222888836

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