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Properties

Label 2-504-56.19-c1-0-13
Degree $2$
Conductor $504$
Sign $0.944 + 0.327i$
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  + (−1.02 − 0.969i)2-s + (0.118 + 1.99i)4-s + (1.09 − 1.89i)5-s + (−1.40 + 2.24i)7-s + (1.81 − 2.16i)8-s + (−2.95 + 0.887i)10-s + (2.14 + 3.71i)11-s + 2.03·13-s + (3.61 − 0.951i)14-s + (−3.97 + 0.472i)16-s + (1.56 − 0.903i)17-s + (0.509 + 0.294i)19-s + (3.90 + 1.95i)20-s + (1.39 − 5.90i)22-s + (0.146 + 0.0846i)23-s + ⋯
L(s)  = 1  + (−0.727 − 0.685i)2-s + (0.0591 + 0.998i)4-s + (0.488 − 0.845i)5-s + (−0.529 + 0.848i)7-s + (0.641 − 0.767i)8-s + (−0.935 + 0.280i)10-s + (0.646 + 1.11i)11-s + 0.564·13-s + (0.967 − 0.254i)14-s + (−0.992 + 0.118i)16-s + (0.379 − 0.219i)17-s + (0.116 + 0.0674i)19-s + (0.873 + 0.437i)20-s + (0.297 − 1.25i)22-s + (0.0305 + 0.0176i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 + 0.327i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.944 + 0.327i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.944 + 0.327i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ 0.944 + 0.327i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.06853 - 0.180122i\)
\(L(\frac12)\) \(\approx\) \(1.06853 - 0.180122i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.02 + 0.969i)T \)
3 \( 1 \)
7 \( 1 + (1.40 - 2.24i)T \)
good5 \( 1 + (-1.09 + 1.89i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.14 - 3.71i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 2.03T + 13T^{2} \)
17 \( 1 + (-1.56 + 0.903i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.509 - 0.294i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.146 - 0.0846i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 0.439iT - 29T^{2} \)
31 \( 1 + (-2.82 - 4.88i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-7.72 - 4.46i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 10.6iT - 41T^{2} \)
43 \( 1 - 8.59T + 43T^{2} \)
47 \( 1 + (-5.47 + 9.48i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.53 - 2.61i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.730 - 0.421i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.22 - 7.31i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.87 + 11.9i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 9.72iT - 71T^{2} \)
73 \( 1 + (-11.0 + 6.39i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.784 + 0.453i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 11.4iT - 83T^{2} \)
89 \( 1 + (14.5 + 8.39i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 13.3iT - 97T^{2} \)
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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

−10.73233810029448909097913492456, −9.756379328600784981002078541063, −9.209178014337103956946247645393, −8.603712163655719811290309951894, −7.42350826093674541406632617764, −6.34477284703865048358712953335, −5.12254966905509787011690442044, −3.90683879209168790961183333120, −2.52602692664290202496126016328, −1.29936979949991485002226956853, 1.02159584082546463909094470399, 2.89752652334071943936892308572, 4.25099934902770506942406379415, 6.07199220773802967570301056456, 6.20039462703193304323758064599, 7.32756869460918928326498323104, 8.180093059089358436719893258102, 9.301239734676175827706347125894, 9.926019523670771846480645554989, 10.90386838190543431390183229166

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