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Properties

Label 2-504-504.187-c1-0-89
Degree $2$
Conductor $504$
Sign $0.364 - 0.931i$
Analytic cond. $4.02446$
Root an. cond. $2.00610$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.366 − 1.36i)2-s + (−1.5 − 0.866i)3-s + (−1.73 − i)4-s − 1.73·5-s + (−1.73 + 1.73i)6-s + (0.866 − 2.5i)7-s + (−2 + 1.99i)8-s + (1.5 + 2.59i)9-s + (−0.633 + 2.36i)10-s − 5·11-s + (1.73 + 3i)12-s + (0.866 + 1.5i)13-s + (−3.09 − 2.09i)14-s + (2.59 + 1.49i)15-s + (1.99 + 3.46i)16-s + (1.5 − 0.866i)17-s + ⋯
L(s)  = 1  + (0.258 − 0.965i)2-s + (−0.866 − 0.499i)3-s + (−0.866 − 0.5i)4-s − 0.774·5-s + (−0.707 + 0.707i)6-s + (0.327 − 0.944i)7-s + (−0.707 + 0.707i)8-s + (0.5 + 0.866i)9-s + (−0.200 + 0.748i)10-s − 1.50·11-s + (0.499 + 0.866i)12-s + (0.240 + 0.416i)13-s + (−0.827 − 0.560i)14-s + (0.670 + 0.387i)15-s + (0.499 + 0.866i)16-s + (0.363 − 0.210i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.364 - 0.931i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.364 - 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.364 - 0.931i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (187, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ 0.364 - 0.931i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.366 + 1.36i)T \)
3 \( 1 + (1.5 + 0.866i)T \)
7 \( 1 + (-0.866 + 2.5i)T \)
good5 \( 1 + 1.73T + 5T^{2} \)
11 \( 1 + 5T + 11T^{2} \)
13 \( 1 + (-0.866 - 1.5i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.5 + 0.866i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.5 + 0.866i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 7iT - 23T^{2} \)
29 \( 1 + (-6.06 - 3.5i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.46 + 6i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.866 - 0.5i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (10.5 - 6.06i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.5 - 7.79i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.73 + 3i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (9.52 - 5.5i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (6 + 3.46i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.73 + 3i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4 + 6.92i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 10iT - 71T^{2} \)
73 \( 1 + (7.5 - 4.33i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.19 - 3i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (13.5 + 7.79i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-7.5 - 4.33i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.5 + 0.866i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.46230676361408135765234696220, −9.799997463325463764427131498253, −8.128096458749639245406321607611, −7.68142473406405608865775414407, −6.37695983558836267577290702337, −5.11176051229669124820939818973, −4.48178661076032301978022551723, −3.17162332148049511351785610652, −1.51280727774372652397456691677, 0, 3.07189618255850938355234576248, 4.41130104161250409471495465854, 5.16887307344073799371832340406, 5.93611552046455988341263105363, 6.94698456113061965405142441383, 8.213869281822172983139451905419, 8.493666143966824923620607180560, 9.986664326955648207358728606810, 10.64124792667946121924643316383

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