Properties

Label 2-504-63.4-c1-0-15
Degree $2$
Conductor $504$
Sign $0.793 - 0.609i$
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  + (1.13 + 1.30i)3-s + 3.19·5-s + (2.61 − 0.415i)7-s + (−0.411 + 2.97i)9-s − 2.28·11-s + (−0.675 + 1.16i)13-s + (3.63 + 4.17i)15-s + (2.21 − 3.83i)17-s + (−3.69 − 6.39i)19-s + (3.51 + 2.93i)21-s − 6.46·23-s + 5.20·25-s + (−4.34 + 2.84i)27-s + (−1.06 − 1.83i)29-s + (0.316 + 0.547i)31-s + ⋯
L(s)  = 1  + (0.656 + 0.754i)3-s + 1.42·5-s + (0.987 − 0.157i)7-s + (−0.137 + 0.990i)9-s − 0.688·11-s + (−0.187 + 0.324i)13-s + (0.938 + 1.07i)15-s + (0.537 − 0.930i)17-s + (−0.847 − 1.46i)19-s + (0.767 + 0.641i)21-s − 1.34·23-s + 1.04·25-s + (−0.837 + 0.547i)27-s + (−0.197 − 0.341i)29-s + (0.0567 + 0.0983i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.793 - 0.609i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.793 - 0.609i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.793 - 0.609i$
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: \((2,\ 504,\ (\ :1/2),\ 0.793 - 0.609i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.13464 + 0.725133i\)
\(L(\frac12)\) \(\approx\) \(2.13464 + 0.725133i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.13 - 1.30i)T \)
7 \( 1 + (-2.61 + 0.415i)T \)
good5 \( 1 - 3.19T + 5T^{2} \)
11 \( 1 + 2.28T + 11T^{2} \)
13 \( 1 + (0.675 - 1.16i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.21 + 3.83i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.69 + 6.39i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 6.46T + 23T^{2} \)
29 \( 1 + (1.06 + 1.83i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.316 - 0.547i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.92 - 3.34i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (5.05 - 8.74i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.24 - 7.35i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (3.26 - 5.65i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.39 + 4.15i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.10 + 5.37i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.45 + 7.71i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.50 - 2.61i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 15.3T + 71T^{2} \)
73 \( 1 + (-4.36 + 7.56i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.938 + 1.62i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.00 + 5.19i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-2.65 - 4.59i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-7.44 - 12.8i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.82058796847319383090111753427, −9.936112661126913545098242858207, −9.452768168134696222200794945770, −8.432215272935697282251472728317, −7.61797900120591778485977729034, −6.26233222233332977356720683917, −5.11046319495355291926734193397, −4.54967196299154174876290678551, −2.81654674367955284402719369777, −1.93531218645993535706308008441, 1.68643486030447257641902232494, 2.28745852529603529364600623649, 3.89897919238425342571817919868, 5.59533407969309867720758689417, 5.96131208209388138575717497958, 7.35403556091859392890650862589, 8.200470076117943923599381903897, 8.828313863716764808800735587268, 10.11298271783415231770836370877, 10.46223130227856205197694929251

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