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Properties

Label 2-2496-39.38-c0-0-1
Degree 22
Conductor 24962496
Sign 11
Analytic cond. 1.245661.24566
Root an. cond. 1.116091.11609
Motivic weight 00
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

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

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

Invariants

Degree: 22
Conductor: 24962496    =    26⋅3⋅132^{6} \cdot 3 \cdot 13
Sign: 11
Analytic conductor: 1.245661.24566
Root analytic conductor: 1.116091.11609
Motivic weight: 00
Rational: yes
Arithmetic: yes
Character: χ2496(1793,⋅)\chi_{2496} (1793, \cdot )
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 2496, ( :0), 1)(2,\ 2496,\ (\ :0),\ 1)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.90669103560.9066910356
L(12)L(\frac12) ≈\approx 0.90669103560.9066910356
L(1)L(1) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
3 1+T 1 + T
13 1−T 1 - T
good5 1+T2 1 + T^{2}
7 (1−T)(1+T) ( 1 - T )( 1 + T )
11 1+T2 1 + T^{2}
17 (1−T)(1+T) ( 1 - T )( 1 + T )
19 (1−T)(1+T) ( 1 - T )( 1 + T )
23 (1−T)(1+T) ( 1 - T )( 1 + T )
29 (1−T)(1+T) ( 1 - T )( 1 + T )
31 (1−T)(1+T) ( 1 - T )( 1 + T )
37 (1−T)(1+T) ( 1 - T )( 1 + T )
41 1+T2 1 + T^{2}
43 (1−T)2 ( 1 - T )^{2}
47 1+T2 1 + T^{2}
53 (1−T)(1+T) ( 1 - T )( 1 + T )
59 1+T2 1 + T^{2}
61 (1−T)2 ( 1 - T )^{2}
67 (1−T)(1+T) ( 1 - T )( 1 + T )
71 1+T2 1 + T^{2}
73 (1−T)(1+T) ( 1 - T )( 1 + T )
79 (1−T)2 ( 1 - T )^{2}
83 1+T2 1 + T^{2}
89 1+T2 1 + T^{2}
97 (1−T)(1+T) ( 1 - T )( 1 + T )
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(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 ZZ-function along the critical line