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Properties

Label 2-2496-39.38-c0-0-1
Degree $2$
Conductor $2496$
Sign $1$
Analytic cond. $1.24566$
Root an. cond. $1.11609$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 9-s + 13-s − 25-s − 27-s − 39-s + 2·43-s + 49-s + 2·61-s + 75-s + 2·79-s + 81-s − 2·103-s + 117-s + ⋯
L(s)  = 1  − 3-s + 9-s + 13-s − 25-s − 27-s − 39-s + 2·43-s + 49-s + 2·61-s + 75-s + 2·79-s + 81-s − 2·103-s + 117-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2496\)    =    \(2^{6} \cdot 3 \cdot 13\)
Sign: $1$
Analytic conductor: \(1.24566\)
Root analytic conductor: \(1.11609\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{2496} (1793, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2496,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9066910356\)
\(L(\frac12)\) \(\approx\) \(0.9066910356\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + T \)
13 \( 1 - T \)
good5 \( 1 + T^{2} \)
7 \( ( 1 - T )( 1 + T ) \)
11 \( 1 + T^{2} \)
17 \( ( 1 - T )( 1 + T ) \)
19 \( ( 1 - T )( 1 + T ) \)
23 \( ( 1 - T )( 1 + T ) \)
29 \( ( 1 - T )( 1 + T ) \)
31 \( ( 1 - T )( 1 + T ) \)
37 \( ( 1 - T )( 1 + T ) \)
41 \( 1 + T^{2} \)
43 \( ( 1 - T )^{2} \)
47 \( 1 + T^{2} \)
53 \( ( 1 - T )( 1 + T ) \)
59 \( 1 + T^{2} \)
61 \( ( 1 - T )^{2} \)
67 \( ( 1 - T )( 1 + T ) \)
71 \( 1 + T^{2} \)
73 \( ( 1 - T )( 1 + T ) \)
79 \( ( 1 - T )^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + T^{2} \)
97 \( ( 1 - T )( 1 + T ) \)
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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

−9.220703985348687024835575760411, −8.282158654389640906495241771473, −7.48285790828679069135982857251, −6.68795395973771317664138866965, −5.92922750536923791224234033848, −5.39130907775536442430147302103, −4.30763893815973840512504000220, −3.67419019211473414565029050569, −2.22027027167738786278901006374, −0.979215445602694142476161031703, 0.979215445602694142476161031703, 2.22027027167738786278901006374, 3.67419019211473414565029050569, 4.30763893815973840512504000220, 5.39130907775536442430147302103, 5.92922750536923791224234033848, 6.68795395973771317664138866965, 7.48285790828679069135982857251, 8.282158654389640906495241771473, 9.220703985348687024835575760411

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