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Properties

Label 2-504-63.5-c1-0-2
Degree $2$
Conductor $504$
Sign $0.476 - 0.879i$
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  + (−0.559 + 1.63i)3-s + (−1.82 − 3.15i)5-s + (1.57 + 2.12i)7-s + (−2.37 − 1.83i)9-s + (1.21 + 0.699i)11-s + (2.97 + 1.71i)13-s + (6.19 − 1.22i)15-s + (2.41 + 4.18i)17-s + (7.23 + 4.17i)19-s + (−4.36 + 1.39i)21-s + (−7.73 + 4.46i)23-s + (−4.14 + 7.18i)25-s + (4.33 − 2.86i)27-s + (5.02 − 2.89i)29-s − 6.11i·31-s + ⋯
L(s)  = 1  + (−0.322 + 0.946i)3-s + (−0.815 − 1.41i)5-s + (0.594 + 0.803i)7-s + (−0.791 − 0.610i)9-s + (0.365 + 0.210i)11-s + (0.825 + 0.476i)13-s + (1.59 − 0.315i)15-s + (0.586 + 1.01i)17-s + (1.66 + 0.958i)19-s + (−0.952 + 0.303i)21-s + (−1.61 + 0.930i)23-s + (−0.829 + 1.43i)25-s + (0.833 − 0.552i)27-s + (0.932 − 0.538i)29-s − 1.09i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.476 - 0.879i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.476 - 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.476 - 0.879i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ 0.476 - 0.879i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.00961 + 0.601534i\)
\(L(\frac12)\) \(\approx\) \(1.00961 + 0.601534i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.559 - 1.63i)T \)
7 \( 1 + (-1.57 - 2.12i)T \)
good5 \( 1 + (1.82 + 3.15i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.21 - 0.699i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.97 - 1.71i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.41 - 4.18i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-7.23 - 4.17i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (7.73 - 4.46i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-5.02 + 2.89i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 6.11iT - 31T^{2} \)
37 \( 1 + (2.89 - 5.00i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.802 - 1.38i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.22 - 3.85i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 3.02T + 47T^{2} \)
53 \( 1 + (-4.40 + 2.54i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 - 6.42T + 59T^{2} \)
61 \( 1 + 8.99iT - 61T^{2} \)
67 \( 1 + 5.06T + 67T^{2} \)
71 \( 1 - 4.20iT - 71T^{2} \)
73 \( 1 + (0.745 - 0.430i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 - 2.62T + 79T^{2} \)
83 \( 1 + (-3.82 - 6.62i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-4.44 + 7.69i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (9.63 - 5.56i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.36965315401434775558389226349, −9.978056061346676483617844378264, −9.344233833065067813609028621893, −8.321948478047952818591150912517, −7.986144908321357403283818549788, −6.00156746227840795361089575452, −5.38938405783861933987433320755, −4.31341123766731098683189437018, −3.64980397416520119006349184987, −1.39631871589176711591193052050, 0.869770103220210903332122092077, 2.74066450345689753962150478333, 3.72972564991734316348661583223, 5.22659918545472301570007199467, 6.45605699640889374800548373214, 7.24232282847148744203318060624, 7.67053363118731699616946130735, 8.696834116208806359654953258391, 10.36300761069465645446601126803, 10.80477851330868001239726348039

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