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Properties

Label 2-504-63.16-c1-0-16
Degree $2$
Conductor $504$
Sign $0.515 + 0.856i$
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  + (1.52 − 0.812i)3-s − 3.79·5-s + (2.59 − 0.525i)7-s + (1.67 − 2.48i)9-s + 4.51·11-s + (0.588 + 1.01i)13-s + (−5.81 + 3.08i)15-s + (−2.95 − 5.12i)17-s + (2.55 − 4.42i)19-s + (3.53 − 2.91i)21-s − 4.18·23-s + 9.43·25-s + (0.545 − 5.16i)27-s + (2.11 − 3.65i)29-s + (3.12 − 5.40i)31-s + ⋯
L(s)  = 1  + (0.883 − 0.469i)3-s − 1.69·5-s + (0.980 − 0.198i)7-s + (0.559 − 0.828i)9-s + 1.36·11-s + (0.163 + 0.282i)13-s + (−1.50 + 0.797i)15-s + (−0.717 − 1.24i)17-s + (0.586 − 1.01i)19-s + (0.772 − 0.635i)21-s − 0.871·23-s + 1.88·25-s + (0.105 − 0.994i)27-s + (0.392 − 0.679i)29-s + (0.560 − 0.971i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.515 + 0.856i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.515 + 0.856i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.515 + 0.856i$
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: \((2,\ 504,\ (\ :1/2),\ 0.515 + 0.856i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.44615 - 0.817236i\)
\(L(\frac12)\) \(\approx\) \(1.44615 - 0.817236i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.52 + 0.812i)T \)
7 \( 1 + (-2.59 + 0.525i)T \)
good5 \( 1 + 3.79T + 5T^{2} \)
11 \( 1 - 4.51T + 11T^{2} \)
13 \( 1 + (-0.588 - 1.01i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.95 + 5.12i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.55 + 4.42i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 4.18T + 23T^{2} \)
29 \( 1 + (-2.11 + 3.65i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.12 + 5.40i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.87 - 6.70i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.754 - 1.30i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.01 - 8.68i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.11 - 1.93i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6.49 - 11.2i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (6.19 - 10.7i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.729 + 1.26i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.813 - 1.40i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 8.48T + 71T^{2} \)
73 \( 1 + (-3.72 - 6.46i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.920 - 1.59i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.307 - 0.532i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (1.25 - 2.17i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-2.36 + 4.10i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.18519250542660508371364116128, −9.607278254355516265610037449612, −8.765980165894184633390603528111, −8.045242120812027369151352147581, −7.32792901821978356126772718479, −6.59341277911822438298682645000, −4.54565446913994212800013863175, −4.07882170991144289727239792062, −2.78565067331642433849305780461, −1.05148729461789996231270740699, 1.72821344745935968289087023886, 3.65778219321076395855021593230, 3.92631421732015883650932302318, 5.08304625758281439145941250066, 6.74672321916882861902709160050, 7.78971784332374308135131843233, 8.394638956315315346912866594217, 8.906706545203438787701864160394, 10.31109156609223744706163795488, 11.03587125850945836308237696023

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