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Properties

Label 2-504-63.25-c1-0-16
Degree $2$
Conductor $504$
Sign $0.0464 + 0.998i$
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  + (−0.455 + 1.67i)3-s + (0.240 − 0.416i)5-s + (−1.92 − 1.81i)7-s + (−2.58 − 1.52i)9-s + (−1.69 − 2.92i)11-s + (−2.86 − 4.95i)13-s + (0.587 + 0.592i)15-s + (2.75 − 4.77i)17-s + (2.18 + 3.77i)19-s + (3.90 − 2.39i)21-s + (−1.81 + 3.14i)23-s + (2.38 + 4.12i)25-s + (3.71 − 3.62i)27-s + (1.53 − 2.65i)29-s − 9.34·31-s + ⋯
L(s)  = 1  + (−0.262 + 0.964i)3-s + (0.107 − 0.186i)5-s + (−0.728 − 0.684i)7-s + (−0.861 − 0.507i)9-s + (−0.509 − 0.882i)11-s + (−0.793 − 1.37i)13-s + (0.151 + 0.152i)15-s + (0.668 − 1.15i)17-s + (0.500 + 0.866i)19-s + (0.852 − 0.522i)21-s + (−0.378 + 0.654i)23-s + (0.476 + 0.825i)25-s + (0.715 − 0.698i)27-s + (0.284 − 0.492i)29-s − 1.67·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0464 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0464 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.0464 + 0.998i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ 0.0464 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.513207 - 0.489899i\)
\(L(\frac12)\) \(\approx\) \(0.513207 - 0.489899i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.455 - 1.67i)T \)
7 \( 1 + (1.92 + 1.81i)T \)
good5 \( 1 + (-0.240 + 0.416i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.69 + 2.92i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.86 + 4.95i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.75 + 4.77i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.18 - 3.77i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.81 - 3.14i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.53 + 2.65i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 9.34T + 31T^{2} \)
37 \( 1 + (-1.48 - 2.57i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (6.29 + 10.9i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.90 + 3.30i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 3.76T + 47T^{2} \)
53 \( 1 + (-5.57 + 9.66i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 8.42T + 59T^{2} \)
61 \( 1 + 7.28T + 61T^{2} \)
67 \( 1 - 2.57T + 67T^{2} \)
71 \( 1 + 3.94T + 71T^{2} \)
73 \( 1 + (0.862 - 1.49i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + 5.59T + 79T^{2} \)
83 \( 1 + (0.119 - 0.206i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-0.648 - 1.12i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (7.02 - 12.1i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.44085199013816242237309166369, −9.995322025086550017933570418912, −9.192199845973010481219263195118, −8.012295553231987815459430153916, −7.11974577444026066786727749241, −5.54150402424558265468803967940, −5.34791593025617367323327842956, −3.69295507559685786133043872662, −3.05063444533319828764308656305, −0.41741944176614696336509317038, 1.88697872487071402343956186528, 2.86717295558264901297975744603, 4.60725550014202034572964833539, 5.73647591396740137775192181183, 6.66579696092306082285160226741, 7.27692219007523721702458739651, 8.400460673249955688396048794059, 9.352319408103880772067488999405, 10.22532276936487375375016504658, 11.29556139529379426997211588528

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