Properties

Label 2-504-63.16-c1-0-22
Degree $2$
Conductor $504$
Sign $-0.902 - 0.430i$
Analytic cond. $4.02446$
Root an. cond. $2.00610$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.5 − 0.866i)3-s + 5-s + (−2.5 − 0.866i)7-s + (1.5 + 2.59i)9-s − 3·11-s + (−0.5 − 0.866i)13-s + (−1.5 − 0.866i)15-s + (−1.5 − 2.59i)17-s + (−2.5 + 4.33i)19-s + (3 + 3.46i)21-s + 23-s − 4·25-s − 5.19i·27-s + (−4.5 + 7.79i)29-s + (−2 + 3.46i)31-s + ⋯
L(s)  = 1  + (−0.866 − 0.499i)3-s + 0.447·5-s + (−0.944 − 0.327i)7-s + (0.5 + 0.866i)9-s − 0.904·11-s + (−0.138 − 0.240i)13-s + (−0.387 − 0.223i)15-s + (−0.363 − 0.630i)17-s + (−0.573 + 0.993i)19-s + (0.654 + 0.755i)21-s + 0.208·23-s − 0.800·25-s − 0.999i·27-s + (−0.835 + 1.44i)29-s + (−0.359 + 0.622i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.902 - 0.430i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.902 - 0.430i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.902 - 0.430i$
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: \(1\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ -0.902 - 0.430i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.5 + 0.866i)T \)
7 \( 1 + (2.5 + 0.866i)T \)
good5 \( 1 - T + 5T^{2} \)
11 \( 1 + 3T + 11T^{2} \)
13 \( 1 + (0.5 + 0.866i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.5 + 2.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.5 - 4.33i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - T + 23T^{2} \)
29 \( 1 + (4.5 - 7.79i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (2 - 3.46i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.5 - 4.33i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.5 + 6.06i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.5 - 2.59i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (4 + 6.92i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.5 + 7.79i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2 + 3.46i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1 + 1.73i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6 - 10.3i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 8T + 71T^{2} \)
73 \( 1 + (-6.5 - 11.2i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (4 + 6.92i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-6.5 + 11.2i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-4.5 + 7.79i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-8.5 + 14.7i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.34836231675990158122457594446, −9.881707600079495059170256717032, −8.572544378603291298885114852962, −7.41804678390037314637467274775, −6.71919860045487598999978081608, −5.74977613807558241122157740740, −4.97504431922675873586174400249, −3.41768171506507303580053096014, −1.91450305252295693519187425443, 0, 2.33548856750677211636087214042, 3.79088052368507204221615201809, 4.93342821910264251036151756285, 5.94798816023374966874477759482, 6.50394113790698887435063618559, 7.74334515630327033354897986943, 9.148442425590475418981790287437, 9.663956534569453107209949519381, 10.56511679564692285714522832656

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