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Properties

Label 2-7920-1.1-c1-0-33
Degree 22
Conductor 79207920
Sign 11
Analytic cond. 63.241563.2415
Root an. cond. 7.952457.95245
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  − 5-s + 3.41·7-s + 11-s − 1.41·13-s − 4.24·17-s + 6.82·19-s + 2.82·23-s + 25-s − 3.65·29-s − 0.828·31-s − 3.41·35-s + 7.65·37-s + 7.65·41-s − 5.07·43-s − 12.4·47-s + 4.65·49-s + 10.4·53-s − 55-s − 3.17·59-s + 3.17·61-s + 1.41·65-s + 8.48·67-s + 6.48·71-s − 7.07·73-s + 3.41·77-s + 16.4·79-s − 1.75·83-s + ⋯
L(s)  = 1  − 0.447·5-s + 1.29·7-s + 0.301·11-s − 0.392·13-s − 1.02·17-s + 1.56·19-s + 0.589·23-s + 0.200·25-s − 0.679·29-s − 0.148·31-s − 0.577·35-s + 1.25·37-s + 1.19·41-s − 0.773·43-s − 1.82·47-s + 0.665·49-s + 1.44·53-s − 0.134·55-s − 0.412·59-s + 0.406·61-s + 0.175·65-s + 1.03·67-s + 0.769·71-s − 0.827·73-s + 0.389·77-s + 1.85·79-s − 0.192·83-s + ⋯

Functional equation

Λ(s)=(7920s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 7920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(7920s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 7920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 79207920    =    24⋅32⋅5⋅112^{4} \cdot 3^{2} \cdot 5 \cdot 11
Sign: 11
Analytic conductor: 63.241563.2415
Root analytic conductor: 7.952457.95245
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 7920, ( :1/2), 1)(2,\ 7920,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.2585053882.258505388
L(12)L(\frac12) ≈\approx 2.2585053882.258505388
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
3 1 1
5 1+T 1 + T
11 1−T 1 - T
good7 1−3.41T+7T2 1 - 3.41T + 7T^{2}
13 1+1.41T+13T2 1 + 1.41T + 13T^{2}
17 1+4.24T+17T2 1 + 4.24T + 17T^{2}
19 1−6.82T+19T2 1 - 6.82T + 19T^{2}
23 1−2.82T+23T2 1 - 2.82T + 23T^{2}
29 1+3.65T+29T2 1 + 3.65T + 29T^{2}
31 1+0.828T+31T2 1 + 0.828T + 31T^{2}
37 1−7.65T+37T2 1 - 7.65T + 37T^{2}
41 1−7.65T+41T2 1 - 7.65T + 41T^{2}
43 1+5.07T+43T2 1 + 5.07T + 43T^{2}
47 1+12.4T+47T2 1 + 12.4T + 47T^{2}
53 1−10.4T+53T2 1 - 10.4T + 53T^{2}
59 1+3.17T+59T2 1 + 3.17T + 59T^{2}
61 1−3.17T+61T2 1 - 3.17T + 61T^{2}
67 1−8.48T+67T2 1 - 8.48T + 67T^{2}
71 1−6.48T+71T2 1 - 6.48T + 71T^{2}
73 1+7.07T+73T2 1 + 7.07T + 73T^{2}
79 1−16.4T+79T2 1 - 16.4T + 79T^{2}
83 1+1.75T+83T2 1 + 1.75T + 83T^{2}
89 1−2T+89T2 1 - 2T + 89T^{2}
97 1−3.17T+97T2 1 - 3.17T + 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

−7.915267519737862949108166486253, −7.22706338004649759920767944024, −6.62579518065423300431740768651, −5.57556387930043359208140976744, −4.98992372658770539455532675687, −4.39770839082835835666110504710, −3.59228961202000058700130130778, −2.63328086337728497804891513776, −1.72935820272930836841945637350, −0.77476658773066623187586469196, 0.77476658773066623187586469196, 1.72935820272930836841945637350, 2.63328086337728497804891513776, 3.59228961202000058700130130778, 4.39770839082835835666110504710, 4.98992372658770539455532675687, 5.57556387930043359208140976744, 6.62579518065423300431740768651, 7.22706338004649759920767944024, 7.915267519737862949108166486253

Graph of the ZZ-function along the critical line