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Properties

Label 2-504-504.331-c0-0-1
Degree $2$
Conductor $504$
Sign $0.400 + 0.916i$
Analytic cond. $0.251528$
Root an. cond. $0.501526$
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  + (0.866 − 0.5i)2-s + (0.5 − 0.866i)3-s + (0.499 − 0.866i)4-s + i·5-s − 0.999i·6-s + (−0.866 + 0.5i)7-s − 0.999i·8-s + (−0.499 − 0.866i)9-s + (0.5 + 0.866i)10-s − 11-s + (−0.499 − 0.866i)12-s + (0.866 − 0.5i)13-s + (−0.499 + 0.866i)14-s + (0.866 + 0.5i)15-s + (−0.5 − 0.866i)16-s + (0.5 + 0.866i)17-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)2-s + (0.5 − 0.866i)3-s + (0.499 − 0.866i)4-s + i·5-s − 0.999i·6-s + (−0.866 + 0.5i)7-s − 0.999i·8-s + (−0.499 − 0.866i)9-s + (0.5 + 0.866i)10-s − 11-s + (−0.499 − 0.866i)12-s + (0.866 − 0.5i)13-s + (−0.499 + 0.866i)14-s + (0.866 + 0.5i)15-s + (−0.5 − 0.866i)16-s + (0.5 + 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.400 + 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.400 + 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.400 + 0.916i$
Analytic conductor: \(0.251528\)
Root analytic conductor: \(0.501526\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (331, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :0),\ 0.400 + 0.916i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.447292918\)
\(L(\frac12)\) \(\approx\) \(1.447292918\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
3 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (0.866 - 0.5i)T \)
good5 \( 1 - iT - T^{2} \)
11 \( 1 + T + T^{2} \)
13 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
17 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 - iT - T^{2} \)
29 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
41 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (-1.73 + i)T + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
89 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
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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

−10.98554035597745859829219114553, −10.36957256466345044438942135131, −9.367610731504623448786799079425, −8.085192026662523928167662802492, −7.17512174613870938274356961699, −6.10108977121439847664956105504, −5.73866994805074998174581767687, −3.63772600769685368671720852398, −3.08398360977115788243994605674, −1.95492285343896800922369380199, 2.65695008032465853986207158804, 3.72379737935164288748811031295, 4.67421432204081251838781987181, 5.37899297025946147604761781902, 6.61154952234864529310654398574, 7.71269792194538599988972282068, 8.660033629033128786595068180095, 9.296353273909206408190557074832, 10.50044606626603565681818140247, 11.24142186837053815079842872039

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