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Properties

Label 2-504-63.4-c1-0-5
Degree $2$
Conductor $504$
Sign $0.593 - 0.805i$
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  + (−1.21 + 1.22i)3-s − 0.481·5-s + (2.53 − 0.763i)7-s + (−0.0248 − 2.99i)9-s + 3.38·11-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 + (−2.15 + 4.04i)21-s + 3.62·23-s − 4.76·25-s + (3.71 + 3.62i)27-s + (1.53 + 2.65i)29-s + (4.67 + 8.09i)31-s + ⋯
L(s)  = 1  + (−0.704 + 0.710i)3-s − 0.215·5-s + (0.957 − 0.288i)7-s + (−0.00827 − 0.999i)9-s + 1.01·11-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 + (−0.469 + 0.882i)21-s + 0.756·23-s − 0.953·25-s + (0.715 + 0.698i)27-s + (0.284 + 0.492i)29-s + (0.839 + 1.45i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.593 - 0.805i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.593 - 0.805i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.593 - 0.805i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ 0.593 - 0.805i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.10037 + 0.555970i\)
\(L(\frac12)\) \(\approx\) \(1.10037 + 0.555970i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.21 - 1.22i)T \)
7 \( 1 + (-2.53 + 0.763i)T \)
good5 \( 1 + 0.481T + 5T^{2} \)
11 \( 1 - 3.38T + 11T^{2} \)
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 \( 1 - 3.62T + 23T^{2} \)
29 \( 1 + (-1.53 - 2.65i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4.67 - 8.09i)T + (-15.5 + 26.8i)T^{2} \)
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 \( 1 + (-1.88 + 3.26i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-5.57 + 9.66i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.21 + 7.29i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.64 + 6.31i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.28 + 2.22i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 3.94T + 71T^{2} \)
73 \( 1 + (0.862 - 1.49i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.79 + 4.84i)T + (-39.5 - 68.4i)T^{2} \)
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

−11.33147546251289551765032504121, −10.05211937282289036015427278174, −9.523412637174555157148309431953, −8.483630480535081273419686827173, −7.26179041016808669705392711878, −6.49571263249785033473144436843, −5.10405428660720499682558260837, −4.56497476214757147299284486265, −3.41677887481579616131045325736, −1.36816456744947388170719951542, 0.985915758282882827765597647637, 2.44862837209468926098984589226, 4.16756464279804003666447553276, 5.34221109628679268278903637713, 6.00098803802663380157183189475, 7.32424336412280455515859666420, 7.84549833377342961875956217047, 8.838810934930701107393336690848, 10.13162765219220704724359639114, 10.92099254243463127234786649848

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