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Properties

Label 2-504-56.37-c1-0-35
Degree $2$
Conductor $504$
Sign $0.157 + 0.987i$
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  + (1.33 − 0.465i)2-s + (1.56 − 1.24i)4-s + (−0.937 − 0.541i)5-s + (−1.62 − 2.09i)7-s + (1.51 − 2.38i)8-s + (−1.50 − 0.286i)10-s + (4.52 − 2.61i)11-s + 4.43i·13-s + (−3.13 − 2.03i)14-s + (0.910 − 3.89i)16-s + (−3.65 − 6.33i)17-s + (2.42 + 1.39i)19-s + (−2.14 + 0.317i)20-s + (4.82 − 5.59i)22-s + (−1.51 + 2.62i)23-s + ⋯
L(s)  = 1  + (0.944 − 0.329i)2-s + (0.783 − 0.621i)4-s + (−0.419 − 0.242i)5-s + (−0.612 − 0.790i)7-s + (0.535 − 0.844i)8-s + (−0.475 − 0.0906i)10-s + (1.36 − 0.787i)11-s + 1.23i·13-s + (−0.838 − 0.544i)14-s + (0.227 − 0.973i)16-s + (−0.886 − 1.53i)17-s + (0.555 + 0.320i)19-s + (−0.478 + 0.0708i)20-s + (1.02 − 1.19i)22-s + (−0.315 + 0.546i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.157 + 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.157 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.157 + 0.987i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ 0.157 + 0.987i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.74495 - 1.48888i\)
\(L(\frac12)\) \(\approx\) \(1.74495 - 1.48888i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.33 + 0.465i)T \)
3 \( 1 \)
7 \( 1 + (1.62 + 2.09i)T \)
good5 \( 1 + (0.937 + 0.541i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-4.52 + 2.61i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 4.43iT - 13T^{2} \)
17 \( 1 + (3.65 + 6.33i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.42 - 1.39i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.51 - 2.62i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 3.06iT - 29T^{2} \)
31 \( 1 + (-3.20 - 5.55i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-7.85 - 4.53i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 6.05T + 41T^{2} \)
43 \( 1 - 10.7iT - 43T^{2} \)
47 \( 1 + (4.54 - 7.86i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.71 - 0.989i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-6.01 + 3.47i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.85 + 3.95i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.26 - 3.03i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 5.53T + 71T^{2} \)
73 \( 1 + (-1.5 - 2.59i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.621 + 1.07i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 12.1iT - 83T^{2} \)
89 \( 1 + (-3.02 + 5.24i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 4.82T + 97T^{2} \)
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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

−11.27875206565747110860925122716, −9.757865759011835911012233186818, −9.303114115246498866715545913017, −7.78674812918103154713667286230, −6.69499453567982789425334549951, −6.24143173065739438288089793085, −4.61778355320389059943985254080, −4.04502846352284785206602634657, −2.92989920035163267488480749590, −1.12261681376626241489783070779, 2.18973414188269638262773665181, 3.47866373349350059599490430126, 4.29212328686286872902700912093, 5.65391681045631555406055246166, 6.36404029603350258753019920328, 7.26294869432963978813473942241, 8.274213409621568587705730544348, 9.247174998034686610474587855747, 10.41690815581929862412075519656, 11.38715506732860674758702659821

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