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Properties

Label 2-504-63.4-c1-0-12
Degree $2$
Conductor $504$
Sign $0.409 + 0.912i$
Analytic cond. $4.02446$
Root an. cond. $2.00610$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.816 + 1.52i)3-s − 1.78·5-s + (1.90 − 1.83i)7-s + (−1.66 − 2.49i)9-s − 5.61·11-s + (3.14 − 5.43i)13-s + (1.45 − 2.72i)15-s + (0.646 − 1.11i)17-s + (0.559 + 0.968i)19-s + (1.25 + 4.40i)21-s + 7.61·23-s − 1.81·25-s + (5.17 − 0.507i)27-s + (−1.57 − 2.72i)29-s + (−0.501 − 0.868i)31-s + ⋯
L(s)  = 1  + (−0.471 + 0.881i)3-s − 0.797·5-s + (0.718 − 0.695i)7-s + (−0.555 − 0.831i)9-s − 1.69·11-s + (0.870 − 1.50i)13-s + (0.376 − 0.703i)15-s + (0.156 − 0.271i)17-s + (0.128 + 0.222i)19-s + (0.274 + 0.961i)21-s + 1.58·23-s − 0.363·25-s + (0.995 − 0.0977i)27-s + (−0.292 − 0.506i)29-s + (−0.0900 − 0.156i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.409 + 0.912i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.409 + 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.409 + 0.912i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ 0.409 + 0.912i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.653921 - 0.423395i\)
\(L(\frac12)\) \(\approx\) \(0.653921 - 0.423395i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.816 - 1.52i)T \)
7 \( 1 + (-1.90 + 1.83i)T \)
good5 \( 1 + 1.78T + 5T^{2} \)
11 \( 1 + 5.61T + 11T^{2} \)
13 \( 1 + (-3.14 + 5.43i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.646 + 1.11i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.559 - 0.968i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 7.61T + 23T^{2} \)
29 \( 1 + (1.57 + 2.72i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.501 + 0.868i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (5.96 + 10.3i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-4.14 + 7.17i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.34 - 4.06i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.972 + 1.68i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.45 - 7.72i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.19 + 7.26i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.41 - 4.17i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.27 - 2.21i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 8.86T + 71T^{2} \)
73 \( 1 + (-5.67 + 9.83i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (6.72 - 11.6i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.60 + 2.77i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (0.404 + 0.700i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.10 - 1.91i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.82392587778453238570509987641, −10.25252355019612767544088858599, −8.951933901152635586573292548722, −7.932684585349227160897669598391, −7.42775562919790981752218701370, −5.71795845064231756823405033261, −5.10794698507455415125399878138, −4.00635348681558106473322689775, −3.03383664878383254448965098064, −0.50922066765189818902212596094, 1.59493931833482601741665417240, 2.94544470837748336980983702454, 4.64762997229404063179519588217, 5.43284609491052813145673222227, 6.58149776636005035717019721426, 7.53126580178434770315719778381, 8.240552547552207728057575406605, 8.990525473001052854344324683621, 10.58248934914323101921967139086, 11.31443775834954631296041025410

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