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Properties

Label 2-7600-1.1-c1-0-76
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  + 2.41·3-s + 1.58·7-s + 2.82·9-s − 1.41·11-s − 0.171·13-s − 17-s + 19-s + 3.82·21-s + 9.24·23-s − 0.414·27-s − 5.82·29-s + 2.24·31-s − 3.41·33-s − 8.48·37-s − 0.414·39-s + 4.24·41-s + 10.2·43-s − 4.48·49-s − 2.41·51-s + 11.4·53-s + 2.41·57-s + 12.8·59-s + 5.75·61-s + 4.48·63-s + 13.2·67-s + 22.3·69-s + 10.5·71-s + ⋯
L(s)  = 1  + 1.39·3-s + 0.599·7-s + 0.942·9-s − 0.426·11-s − 0.0475·13-s − 0.242·17-s + 0.229·19-s + 0.835·21-s + 1.92·23-s − 0.0797·27-s − 1.08·29-s + 0.402·31-s − 0.594·33-s − 1.39·37-s − 0.0663·39-s + 0.662·41-s + 1.56·43-s − 0.640·49-s − 0.338·51-s + 1.57·53-s + 0.319·57-s + 1.67·59-s + 0.737·61-s + 0.565·63-s + 1.61·67-s + 2.68·69-s + 1.25·71-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 3.8206528193.820652819
L(12)L(\frac12) ≈\approx 3.8206528193.820652819
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 1−2.41T+3T2 1 - 2.41T + 3T^{2}
7 1−1.58T+7T2 1 - 1.58T + 7T^{2}
11 1+1.41T+11T2 1 + 1.41T + 11T^{2}
13 1+0.171T+13T2 1 + 0.171T + 13T^{2}
17 1+T+17T2 1 + T + 17T^{2}
23 1−9.24T+23T2 1 - 9.24T + 23T^{2}
29 1+5.82T+29T2 1 + 5.82T + 29T^{2}
31 1−2.24T+31T2 1 - 2.24T + 31T^{2}
37 1+8.48T+37T2 1 + 8.48T + 37T^{2}
41 1−4.24T+41T2 1 - 4.24T + 41T^{2}
43 1−10.2T+43T2 1 - 10.2T + 43T^{2}
47 1+47T2 1 + 47T^{2}
53 1−11.4T+53T2 1 - 11.4T + 53T^{2}
59 1−12.8T+59T2 1 - 12.8T + 59T^{2}
61 1−5.75T+61T2 1 - 5.75T + 61T^{2}
67 1−13.2T+67T2 1 - 13.2T + 67T^{2}
71 1−10.5T+71T2 1 - 10.5T + 71T^{2}
73 1−5.48T+73T2 1 - 5.48T + 73T^{2}
79 1+10.4T+79T2 1 + 10.4T + 79T^{2}
83 1+2.48T+83T2 1 + 2.48T + 83T^{2}
89 1+7.07T+89T2 1 + 7.07T + 89T^{2}
97 1−11.6T+97T2 1 - 11.6T + 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

−7.976584807959091217271511858297, −7.28557645294643647943438977413, −6.85083730224638853489088562510, −5.57449215302970781917805708784, −5.08528312481028799144300779478, −4.10994176468701710379574596263, −3.46783207734431914887480932850, −2.62057138152629032964333986600, −2.06754149125574246855682069547, −0.932759353704666590548488436690, 0.932759353704666590548488436690, 2.06754149125574246855682069547, 2.62057138152629032964333986600, 3.46783207734431914887480932850, 4.10994176468701710379574596263, 5.08528312481028799144300779478, 5.57449215302970781917805708784, 6.85083730224638853489088562510, 7.28557645294643647943438977413, 7.976584807959091217271511858297

Graph of the ZZ-function along the critical line