Properties

Label 2-2576-2576.1931-c0-0-1
Degree $2$
Conductor $2576$
Sign $0.923 + 0.382i$
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  + i·2-s − 4-s i·7-s i·8-s + i·9-s + (−1 − i)11-s + 14-s + 16-s − 18-s + (1 − i)22-s − 23-s i·25-s + i·28-s + (1 − i)29-s + i·32-s + ⋯
L(s)  = 1  + i·2-s − 4-s i·7-s i·8-s + i·9-s + (−1 − i)11-s + 14-s + 16-s − 18-s + (1 − i)22-s − 23-s i·25-s + i·28-s + (1 − i)29-s + i·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2576\)    =    \(2^{4} \cdot 7 \cdot 23\)
Sign: $0.923 + 0.382i$
Analytic conductor: \(1.28559\)
Root analytic conductor: \(1.13383\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2576} (1931, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2576,\ (\ :0),\ 0.923 + 0.382i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7995561419\)
\(L(\frac12)\) \(\approx\) \(0.7995561419\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
7 \( 1 + iT \)
23 \( 1 + T \)
good3 \( 1 - iT^{2} \)
5 \( 1 + iT^{2} \)
11 \( 1 + (1 + i)T + iT^{2} \)
13 \( 1 + iT^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + iT^{2} \)
29 \( 1 + (-1 + i)T - iT^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + (-1 + i)T - iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + (1 + i)T + iT^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + (-1 + i)T - iT^{2} \)
59 \( 1 + iT^{2} \)
61 \( 1 - iT^{2} \)
67 \( 1 + (-1 + i)T - iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + T^{2} \)
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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

−8.641733391622842512803578234823, −8.037840690588849735231721064313, −7.71152155471619014966592634645, −6.73939598176279258521949409333, −5.98056618051459514259576393545, −5.20151535626506576038568937217, −4.43218783167915691020119632906, −3.62451153843131893607091821528, −2.35340339881131061012398659399, −0.54561443765878412772075480958, 1.43849404969501717806531764121, 2.52640129837559874268583283810, 3.18891518029643972702879888569, 4.28816006298142302012808537098, 5.08090103509392992251434744153, 5.81079809231201425773421817525, 6.78811293983254083951472274207, 7.88301855804167046018481941808, 8.526296817586219861611537572121, 9.319754801670042508729742919002

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