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Properties

Label 2-7600-1.1-c1-0-16
Degree 22
Conductor 76007600
Sign 11
Analytic cond. 60.686360.6863
Root an. cond. 7.790147.79014
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.23·3-s − 4.47·7-s − 1.47·9-s − 3.23·13-s − 6.47·17-s − 19-s − 5.52·21-s − 2·23-s − 5.52·27-s + 2·29-s + 1.52·31-s + 4.76·37-s − 4.00·39-s + 3.52·41-s + 0.472·43-s − 12.4·47-s + 13.0·49-s − 8.00·51-s + 11.2·53-s − 1.23·57-s + 10.4·59-s − 4.47·61-s + 6.58·63-s − 1.23·67-s − 2.47·69-s + 1.52·71-s + 6.47·73-s + ⋯
L(s)  = 1  + 0.713·3-s − 1.69·7-s − 0.490·9-s − 0.897·13-s − 1.56·17-s − 0.229·19-s − 1.20·21-s − 0.417·23-s − 1.06·27-s + 0.371·29-s + 0.274·31-s + 0.783·37-s − 0.640·39-s + 0.550·41-s + 0.0720·43-s − 1.81·47-s + 1.85·49-s − 1.12·51-s + 1.54·53-s − 0.163·57-s + 1.36·59-s − 0.572·61-s + 0.829·63-s − 0.151·67-s − 0.297·69-s + 0.181·71-s + 0.757·73-s + ⋯

Functional equation

Λ(s)=(7600s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(7600s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 76007600    =    24⋅52⋅192^{4} \cdot 5^{2} \cdot 19
Sign: 11
Analytic conductor: 60.686360.6863
Root analytic conductor: 7.790147.79014
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 7600, ( :1/2), 1)(2,\ 7600,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.97831953340.9783195334
L(12)L(\frac12) ≈\approx 0.97831953340.9783195334
L(32)L(\frac{3}{2}) 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
5 1 1
19 1+T 1 + T
good3 1−1.23T+3T2 1 - 1.23T + 3T^{2}
7 1+4.47T+7T2 1 + 4.47T + 7T^{2}
11 1+11T2 1 + 11T^{2}
13 1+3.23T+13T2 1 + 3.23T + 13T^{2}
17 1+6.47T+17T2 1 + 6.47T + 17T^{2}
23 1+2T+23T2 1 + 2T + 23T^{2}
29 1−2T+29T2 1 - 2T + 29T^{2}
31 1−1.52T+31T2 1 - 1.52T + 31T^{2}
37 1−4.76T+37T2 1 - 4.76T + 37T^{2}
41 1−3.52T+41T2 1 - 3.52T + 41T^{2}
43 1−0.472T+43T2 1 - 0.472T + 43T^{2}
47 1+12.4T+47T2 1 + 12.4T + 47T^{2}
53 1−11.2T+53T2 1 - 11.2T + 53T^{2}
59 1−10.4T+59T2 1 - 10.4T + 59T^{2}
61 1+4.47T+61T2 1 + 4.47T + 61T^{2}
67 1+1.23T+67T2 1 + 1.23T + 67T^{2}
71 1−1.52T+71T2 1 - 1.52T + 71T^{2}
73 1−6.47T+73T2 1 - 6.47T + 73T^{2}
79 1−6.47T+79T2 1 - 6.47T + 79T^{2}
83 1−14.9T+83T2 1 - 14.9T + 83T^{2}
89 1+6.94T+89T2 1 + 6.94T + 89T^{2}
97 1−4.18T+97T2 1 - 4.18T + 97T^{2}
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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

−7.951956100464347261885647380209, −7.10789445256684449422882857212, −6.51590854524446689831588539645, −6.00688224291904777763459890752, −5.02033174922278548344752523825, −4.12798405490171664212855804463, −3.43892332154625690865626492180, −2.61672370085233845960822954835, −2.22937810963164411499677669882, −0.43734075643188688902378163419, 0.43734075643188688902378163419, 2.22937810963164411499677669882, 2.61672370085233845960822954835, 3.43892332154625690865626492180, 4.12798405490171664212855804463, 5.02033174922278548344752523825, 6.00688224291904777763459890752, 6.51590854524446689831588539645, 7.10789445256684449422882857212, 7.951956100464347261885647380209

Graph of the ZZ-function along the critical line