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Properties

Label 2-2576-2576.1707-c0-0-0
Degree $2$
Conductor $2576$
Sign $0.868 + 0.495i$
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  + (−0.281 − 0.959i)2-s + (−0.841 + 0.540i)4-s + (−0.909 + 0.415i)7-s + (0.755 + 0.654i)8-s + (0.989 − 0.142i)9-s + (−0.125 − 0.0683i)11-s + (0.654 + 0.755i)14-s + (0.415 − 0.909i)16-s + (−0.415 − 0.909i)18-s + (−0.0303 + 0.139i)22-s + (0.142 − 0.989i)23-s + (−0.281 + 0.959i)25-s + (0.540 − 0.841i)28-s + (−0.373 + 1.71i)29-s + (−0.989 − 0.142i)32-s + ⋯
L(s)  = 1  + (−0.281 − 0.959i)2-s + (−0.841 + 0.540i)4-s + (−0.909 + 0.415i)7-s + (0.755 + 0.654i)8-s + (0.989 − 0.142i)9-s + (−0.125 − 0.0683i)11-s + (0.654 + 0.755i)14-s + (0.415 − 0.909i)16-s + (−0.415 − 0.909i)18-s + (−0.0303 + 0.139i)22-s + (0.142 − 0.989i)23-s + (−0.281 + 0.959i)25-s + (0.540 − 0.841i)28-s + (−0.373 + 1.71i)29-s + (−0.989 − 0.142i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.868 + 0.495i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.868 + 0.495i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2576\)    =    \(2^{4} \cdot 7 \cdot 23\)
Sign: $0.868 + 0.495i$
Analytic conductor: \(1.28559\)
Root analytic conductor: \(1.13383\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2576} (1707, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2576,\ (\ :0),\ 0.868 + 0.495i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9065301595\)
\(L(\frac12)\) \(\approx\) \(0.9065301595\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.281 + 0.959i)T \)
7 \( 1 + (0.909 - 0.415i)T \)
23 \( 1 + (-0.142 + 0.989i)T \)
good3 \( 1 + (-0.989 + 0.142i)T^{2} \)
5 \( 1 + (0.281 - 0.959i)T^{2} \)
11 \( 1 + (0.125 + 0.0683i)T + (0.540 + 0.841i)T^{2} \)
13 \( 1 + (0.755 - 0.654i)T^{2} \)
17 \( 1 + (0.415 - 0.909i)T^{2} \)
19 \( 1 + (-0.909 + 0.415i)T^{2} \)
29 \( 1 + (0.373 - 1.71i)T + (-0.909 - 0.415i)T^{2} \)
31 \( 1 + (-0.142 + 0.989i)T^{2} \)
37 \( 1 + (-1.50 - 1.12i)T + (0.281 + 0.959i)T^{2} \)
41 \( 1 + (-0.959 - 0.281i)T^{2} \)
43 \( 1 + (-1.59 + 0.114i)T + (0.989 - 0.142i)T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + (-1.64 - 0.613i)T + (0.755 + 0.654i)T^{2} \)
59 \( 1 + (0.755 - 0.654i)T^{2} \)
61 \( 1 + (0.989 + 0.142i)T^{2} \)
67 \( 1 + (-1.71 + 0.936i)T + (0.540 - 0.841i)T^{2} \)
71 \( 1 + (1.45 + 0.425i)T + (0.841 + 0.540i)T^{2} \)
73 \( 1 + (0.415 + 0.909i)T^{2} \)
79 \( 1 + (-0.234 + 0.512i)T + (-0.654 - 0.755i)T^{2} \)
83 \( 1 + (-0.281 - 0.959i)T^{2} \)
89 \( 1 + (0.142 + 0.989i)T^{2} \)
97 \( 1 + (-0.959 - 0.281i)T^{2} \)
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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

−9.245744763150350486550471556566, −8.523862878382314227205530772027, −7.53456449979400532822597629849, −6.87107499640625486243177252865, −5.84594918599768636660245320999, −4.87463337323974102009505796855, −4.01128477150435533719120355685, −3.20349212943044999963323428777, −2.32740412302312730898855476642, −1.09957177622836868470510122950, 0.841224836883373959412013045408, 2.37881822711253902814538213109, 3.97969007800787491547249625432, 4.21805624912677614767819683419, 5.51845014567383433632055914071, 6.12083656128680051287665395901, 6.97655710918116987946268896674, 7.51058414106652156187143384779, 8.152968533589795833851053616733, 9.267124394134742030645097422120

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