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Properties

Label 2-504-7.2-c3-0-12
Degree $2$
Conductor $504$
Sign $0.754 - 0.655i$
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  + (7.86 − 13.6i)5-s + (11.2 + 14.7i)7-s + (29.3 + 50.8i)11-s − 20.7·13-s + (21.4 + 37.1i)17-s + (−68.5 + 118. i)19-s + (14.5 − 25.1i)23-s + (−61.3 − 106. i)25-s − 8.68·29-s + (101. + 175. i)31-s + (289. − 37.2i)35-s + (7.63 − 13.2i)37-s − 117.·41-s − 101.·43-s + (294. − 509. i)47-s + ⋯
L(s)  = 1  + (0.703 − 1.21i)5-s + (0.606 + 0.794i)7-s + (0.804 + 1.39i)11-s − 0.442·13-s + (0.306 + 0.530i)17-s + (−0.828 + 1.43i)19-s + (0.131 − 0.228i)23-s + (−0.490 − 0.849i)25-s − 0.0555·29-s + (0.586 + 1.01i)31-s + (1.39 − 0.180i)35-s + (0.0339 − 0.0587i)37-s − 0.446·41-s − 0.361·43-s + (0.913 − 1.58i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.754 - 0.655i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.754 - 0.655i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.754 - 0.655i$
Analytic conductor: \(29.7369\)
Root analytic conductor: \(5.45316\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :3/2),\ 0.754 - 0.655i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.321137170\)
\(L(\frac12)\) \(\approx\) \(2.321137170\)
\(L(\frac{5}{2})\) 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 \( 1 + (-11.2 - 14.7i)T \)
good5 \( 1 + (-7.86 + 13.6i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-29.3 - 50.8i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 20.7T + 2.19e3T^{2} \)
17 \( 1 + (-21.4 - 37.1i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (68.5 - 118. i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-14.5 + 25.1i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + 8.68T + 2.43e4T^{2} \)
31 \( 1 + (-101. - 175. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-7.63 + 13.2i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 117.T + 6.89e4T^{2} \)
43 \( 1 + 101.T + 7.95e4T^{2} \)
47 \( 1 + (-294. + 509. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-202. - 350. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-5.43 - 9.41i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-447. + 774. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-351. - 609. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 1.16e3T + 3.57e5T^{2} \)
73 \( 1 + (-558. - 966. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (69.1 - 119. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 894.T + 5.71e5T^{2} \)
89 \( 1 + (340. - 590. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 246.T + 9.12e5T^{2} \)
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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

−10.34410820330759924657828786943, −9.693692258924744177730610267027, −8.780810466731318303178627866509, −8.211642591554575366220458028954, −6.89275726659484873276453924593, −5.76047847256786860143843505913, −4.98894158233280947469491320959, −4.08837818049494684339482484003, −2.12211944195448813385712110105, −1.41316802925359574368234406898, 0.76082751751777219566996095836, 2.35366933912685925847745338688, 3.40013851498859731774321667909, 4.64364848428848861888985461973, 5.95070633632738694030612145629, 6.71587499006207875660937092293, 7.51355633267812389489312065681, 8.674379498686067645861967136355, 9.625846877818139433078844799684, 10.57049961554746028558508496578

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