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Properties

Label 2-2576-2576.1917-c0-0-0
Degree $2$
Conductor $2576$
Sign $0.655 + 0.755i$
Analytic cond. $1.28559$
Root an. cond. $1.13383$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.909 − 0.415i)2-s + (0.654 + 0.755i)4-s + (0.540 − 0.841i)7-s + (−0.281 − 0.959i)8-s + (−0.755 − 0.654i)9-s + (1.50 + 1.12i)11-s + (−0.841 + 0.540i)14-s + (−0.142 + 0.989i)16-s + (0.415 + 0.909i)18-s + (−0.898 − 1.64i)22-s + (−0.654 − 0.755i)23-s + (0.989 + 0.142i)25-s + (0.989 − 0.142i)28-s + (1.05 + 0.574i)29-s + (0.540 − 0.841i)32-s + ⋯
L(s)  = 1  + (−0.909 − 0.415i)2-s + (0.654 + 0.755i)4-s + (0.540 − 0.841i)7-s + (−0.281 − 0.959i)8-s + (−0.755 − 0.654i)9-s + (1.50 + 1.12i)11-s + (−0.841 + 0.540i)14-s + (−0.142 + 0.989i)16-s + (0.415 + 0.909i)18-s + (−0.898 − 1.64i)22-s + (−0.654 − 0.755i)23-s + (0.989 + 0.142i)25-s + (0.989 − 0.142i)28-s + (1.05 + 0.574i)29-s + (0.540 − 0.841i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.655 + 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.655 + 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2576\)    =    \(2^{4} \cdot 7 \cdot 23\)
Sign: $0.655 + 0.755i$
Analytic conductor: \(1.28559\)
Root analytic conductor: \(1.13383\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2576} (1917, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2576,\ (\ :0),\ 0.655 + 0.755i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8834741920\)
\(L(\frac12)\) \(\approx\) \(0.8834741920\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.909 + 0.415i)T \)
7 \( 1 + (-0.540 + 0.841i)T \)
23 \( 1 + (0.654 + 0.755i)T \)
good3 \( 1 + (0.755 + 0.654i)T^{2} \)
5 \( 1 + (-0.989 - 0.142i)T^{2} \)
11 \( 1 + (-1.50 - 1.12i)T + (0.281 + 0.959i)T^{2} \)
13 \( 1 + (-0.909 - 0.415i)T^{2} \)
17 \( 1 + (-0.841 + 0.540i)T^{2} \)
19 \( 1 + (0.540 - 0.841i)T^{2} \)
29 \( 1 + (-1.05 - 0.574i)T + (0.540 + 0.841i)T^{2} \)
31 \( 1 + (0.654 + 0.755i)T^{2} \)
37 \( 1 + (0.0303 + 0.424i)T + (-0.989 + 0.142i)T^{2} \)
41 \( 1 + (-0.142 + 0.989i)T^{2} \)
43 \( 1 + (-0.133 - 0.0498i)T + (0.755 + 0.654i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (1.56 - 0.340i)T + (0.909 - 0.415i)T^{2} \)
59 \( 1 + (0.909 + 0.415i)T^{2} \)
61 \( 1 + (-0.755 + 0.654i)T^{2} \)
67 \( 1 + (-1.40 + 1.05i)T + (0.281 - 0.959i)T^{2} \)
71 \( 1 + (0.822 + 0.118i)T + (0.959 + 0.281i)T^{2} \)
73 \( 1 + (0.841 + 0.540i)T^{2} \)
79 \( 1 + (-1.66 + 1.07i)T + (0.415 - 0.909i)T^{2} \)
83 \( 1 + (-0.989 + 0.142i)T^{2} \)
89 \( 1 + (-0.654 + 0.755i)T^{2} \)
97 \( 1 + (0.142 - 0.989i)T^{2} \)
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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

−9.058329900677583139066995630621, −8.350573621366902547206944229248, −7.55645003608071098844516365373, −6.70190636030175201402697746400, −6.37613531390922754277165771041, −4.80007471396118589255282765953, −4.02563333931480848594266652724, −3.18149126795356593056012641587, −1.94954709530121662842871513156, −0.965175466709925806036307509515, 1.19567143301288576721078809234, 2.29554786297037047117302809384, 3.26991960000953674922840830045, 4.68233482263304325660357602613, 5.58102467029515635954114258460, 6.14450821319450305509098185198, 6.84933162931024884403922239499, 8.071561020795397388364209144825, 8.335159849707294902699225654056, 9.018962630281474946282108401733

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