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Properties

Label 2-504-7.4-c3-0-19
Degree $2$
Conductor $504$
Sign $0.764 - 0.644i$
Analytic cond. $29.7369$
Root an. cond. $5.45316$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (9.47 + 16.4i)5-s + (12.8 + 13.3i)7-s + (27.3 − 47.3i)11-s + 62.0·13-s + (61.2 − 106. i)17-s + (6.25 + 10.8i)19-s + (−37.2 − 64.4i)23-s + (−117. + 203. i)25-s + 232.·29-s + (−5.18 + 8.97i)31-s + (−98.3 + 337. i)35-s + (122. + 213. i)37-s − 238.·41-s − 92.9·43-s + (−242. − 420. i)47-s + ⋯
L(s)  = 1  + (0.847 + 1.46i)5-s + (0.691 + 0.722i)7-s + (0.750 − 1.29i)11-s + 1.32·13-s + (0.873 − 1.51i)17-s + (0.0755 + 0.130i)19-s + (−0.337 − 0.584i)23-s + (−0.937 + 1.62i)25-s + 1.48·29-s + (−0.0300 + 0.0520i)31-s + (−0.474 + 1.62i)35-s + (0.546 + 0.946i)37-s − 0.909·41-s − 0.329·43-s + (−0.753 − 1.30i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.764 - 0.644i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.764 - 0.644i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.764 - 0.644i$
Analytic conductor: \(29.7369\)
Root analytic conductor: \(5.45316\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :3/2),\ 0.764 - 0.644i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.909142175\)
\(L(\frac12)\) \(\approx\) \(2.909142175\)
\(L(\frac{5}{2})\) 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 \( 1 + (-12.8 - 13.3i)T \)
good5 \( 1 + (-9.47 - 16.4i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-27.3 + 47.3i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 62.0T + 2.19e3T^{2} \)
17 \( 1 + (-61.2 + 106. i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-6.25 - 10.8i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (37.2 + 64.4i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 232.T + 2.43e4T^{2} \)
31 \( 1 + (5.18 - 8.97i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-122. - 213. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 238.T + 6.89e4T^{2} \)
43 \( 1 + 92.9T + 7.95e4T^{2} \)
47 \( 1 + (242. + 420. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (189. - 327. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (91.3 - 158. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (198. + 343. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-130. + 226. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 874.T + 3.57e5T^{2} \)
73 \( 1 + (76.2 - 131. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (286. + 496. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 317.T + 5.71e5T^{2} \)
89 \( 1 + (47.5 + 82.2i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 1.60e3T + 9.12e5T^{2} \)
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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

−10.68372513710866930980779221056, −9.812385546518230981889878890492, −8.812575426974898781210394576051, −8.006030771108731950691930187733, −6.61299343672263374274991603916, −6.17234714581658155175782455431, −5.16559147638986253528683569828, −3.44544986249857791030813660834, −2.69043505861010437259388504958, −1.25470336873743662621237450862, 1.23654600600957647931219618380, 1.63879671344782716582063094039, 3.88610815060206595653706798304, 4.63628537771285177821855095803, 5.65730255809041125085500706357, 6.60556479414391815742547546943, 7.982451527519297244808341430412, 8.552058947585866343801241048485, 9.590481099762643339280617196614, 10.20206275385217787104221171429

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