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Properties

Label 2-504-24.11-c1-0-8
Degree $2$
Conductor $504$
Sign $0.992 + 0.124i$
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  + (−0.386 − 1.36i)2-s + (−1.70 + 1.05i)4-s + 3.11·5-s + i·7-s + (2.08 + 1.90i)8-s + (−1.20 − 4.24i)10-s + 5.48i·11-s + 1.65i·13-s + (1.36 − 0.386i)14-s + (1.78 − 3.57i)16-s + 4.14i·17-s − 2.94·19-s + (−5.30 + 3.28i)20-s + (7.45 − 2.11i)22-s − 0.388·23-s + ⋯
L(s)  = 1  + (−0.273 − 0.961i)2-s + (−0.850 + 0.525i)4-s + 1.39·5-s + 0.377i·7-s + (0.738 + 0.674i)8-s + (−0.381 − 1.34i)10-s + 1.65i·11-s + 0.459i·13-s + (0.363 − 0.103i)14-s + (0.446 − 0.894i)16-s + 1.00i·17-s − 0.674·19-s + (−1.18 + 0.733i)20-s + (1.59 − 0.451i)22-s − 0.0809·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.124i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 + 0.124i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.992 + 0.124i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ 0.992 + 0.124i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.39367 - 0.0869769i\)
\(L(\frac12)\) \(\approx\) \(1.39367 - 0.0869769i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.386 + 1.36i)T \)
3 \( 1 \)
7 \( 1 - iT \)
good5 \( 1 - 3.11T + 5T^{2} \)
11 \( 1 - 5.48iT - 11T^{2} \)
13 \( 1 - 1.65iT - 13T^{2} \)
17 \( 1 - 4.14iT - 17T^{2} \)
19 \( 1 + 2.94T + 19T^{2} \)
23 \( 1 + 0.388T + 23T^{2} \)
29 \( 1 - 6.81T + 29T^{2} \)
31 \( 1 - 1.59iT - 31T^{2} \)
37 \( 1 + 10.6iT - 37T^{2} \)
41 \( 1 + 4.43iT - 41T^{2} \)
43 \( 1 + 4.18T + 43T^{2} \)
47 \( 1 - 11.5T + 47T^{2} \)
53 \( 1 - 7.45T + 53T^{2} \)
59 \( 1 + 11.5iT - 59T^{2} \)
61 \( 1 + 6.75iT - 61T^{2} \)
67 \( 1 - 5.03T + 67T^{2} \)
71 \( 1 + 15.2T + 71T^{2} \)
73 \( 1 + 16.4T + 73T^{2} \)
79 \( 1 - 2.01iT - 79T^{2} \)
83 \( 1 + 6.21iT - 83T^{2} \)
89 \( 1 - 5.93iT - 89T^{2} \)
97 \( 1 - 14.1T + 97T^{2} \)
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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

−10.51459123296541209917396580349, −10.19676686760056692144711093436, −9.277198816053047832802702057450, −8.673669960383941612903873257936, −7.34075556529056995739913830523, −6.18686836097619086221912764025, −5.07329339126434201653453307831, −4.06626547040515192454900099468, −2.36147561889322081469119837299, −1.78376803299552241050595117384, 1.00304124492795432521914135872, 2.91826798963269815188298474442, 4.56753731196681050827570195113, 5.69092225268678122724302629501, 6.17139704419573199033945631470, 7.18024206705195985684754130255, 8.403823509596157615903991911941, 8.947129685930587588523346004590, 10.07659260091236608463822360920, 10.44905149101085692866645809877

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