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Properties

Label 2-504-63.16-c1-0-11
Degree $2$
Conductor $504$
Sign $0.983 + 0.179i$
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  + (−1.07 − 1.35i)3-s + 3.40·5-s + (−2.05 + 1.66i)7-s + (−0.668 + 2.92i)9-s + 5.39·11-s + (1.89 + 3.28i)13-s + (−3.67 − 4.61i)15-s + (0.411 + 0.713i)17-s + (0.233 − 0.404i)19-s + (4.47 + 0.993i)21-s − 5.49·23-s + 6.61·25-s + (4.68 − 2.25i)27-s + (0.400 − 0.693i)29-s + (4.95 − 8.57i)31-s + ⋯
L(s)  = 1  + (−0.623 − 0.781i)3-s + 1.52·5-s + (−0.778 + 0.628i)7-s + (−0.222 + 0.974i)9-s + 1.62·11-s + (0.525 + 0.910i)13-s + (−0.949 − 1.19i)15-s + (0.0999 + 0.173i)17-s + (0.0535 − 0.0928i)19-s + (0.976 + 0.216i)21-s − 1.14·23-s + 1.32·25-s + (0.901 − 0.433i)27-s + (0.0743 − 0.128i)29-s + (0.889 − 1.54i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 + 0.179i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.983 + 0.179i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.983 + 0.179i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (457, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ 0.983 + 0.179i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.50121 - 0.135629i\)
\(L(\frac12)\) \(\approx\) \(1.50121 - 0.135629i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.07 + 1.35i)T \)
7 \( 1 + (2.05 - 1.66i)T \)
good5 \( 1 - 3.40T + 5T^{2} \)
11 \( 1 - 5.39T + 11T^{2} \)
13 \( 1 + (-1.89 - 3.28i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.411 - 0.713i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.233 + 0.404i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 5.49T + 23T^{2} \)
29 \( 1 + (-0.400 + 0.693i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-4.95 + 8.57i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.34 + 7.52i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.84 - 3.19i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.36 - 7.55i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-5.24 - 9.09i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.71 + 8.17i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.830 + 1.43i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.474 + 0.821i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.269 - 0.466i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 3.86T + 71T^{2} \)
73 \( 1 + (-2.58 - 4.48i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.91 + 6.78i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.79 - 6.57i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (3.73 - 6.46i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (3.22 - 5.58i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.07378957059898551633623911294, −9.633764953542879882824581541818, −9.497037968231216151545447130536, −8.264700182515647103622792683322, −6.77938985016080366428257636495, −6.16833656554251154603865639287, −5.83095765397464138672938691738, −4.23287648193433732595550204102, −2.44472855682482202772286766445, −1.44275733564297716656015811972, 1.21549428455054403318045761979, 3.16369336201196783527743234476, 4.20063831315447654061738668251, 5.50338391607836355574144098843, 6.23543484400709285085329623712, 6.82576247747139425366352542159, 8.637017089573690915967367418660, 9.452386649145480555299511608643, 10.13631209725290364479746607564, 10.54435997853624848935352090937

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