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Properties

Label 2-7920-1.1-c1-0-28
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 + 0.585·7-s − 11-s + 1.41·13-s − 4.24·17-s + 1.17·19-s + 2.82·23-s + 25-s − 7.65·29-s + 4.82·31-s + 0.585·35-s − 3.65·37-s + 3.65·41-s + 9.07·43-s − 4.48·47-s − 6.65·49-s + 6.48·53-s − 55-s + 8.82·59-s + 8.82·61-s + 1.41·65-s − 8.48·67-s + 10.4·71-s + 7.07·73-s − 0.585·77-s − 0.485·79-s + 10.2·83-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.221·7-s − 0.301·11-s + 0.392·13-s − 1.02·17-s + 0.268·19-s + 0.589·23-s + 0.200·25-s − 1.42·29-s + 0.867·31-s + 0.0990·35-s − 0.601·37-s + 0.571·41-s + 1.38·43-s − 0.654·47-s − 0.950·49-s + 0.890·53-s − 0.134·55-s + 1.14·59-s + 1.13·61-s + 0.175·65-s − 1.03·67-s + 1.24·71-s + 0.827·73-s − 0.0667·77-s − 0.0545·79-s + 1.12·83-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 2.1791799962.179179996
L(12)L(\frac12) ≈\approx 2.1791799962.179179996
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 1−0.585T+7T2 1 - 0.585T + 7T^{2}
13 1−1.41T+13T2 1 - 1.41T + 13T^{2}
17 1+4.24T+17T2 1 + 4.24T + 17T^{2}
19 1−1.17T+19T2 1 - 1.17T + 19T^{2}
23 1−2.82T+23T2 1 - 2.82T + 23T^{2}
29 1+7.65T+29T2 1 + 7.65T + 29T^{2}
31 1−4.82T+31T2 1 - 4.82T + 31T^{2}
37 1+3.65T+37T2 1 + 3.65T + 37T^{2}
41 1−3.65T+41T2 1 - 3.65T + 41T^{2}
43 1−9.07T+43T2 1 - 9.07T + 43T^{2}
47 1+4.48T+47T2 1 + 4.48T + 47T^{2}
53 1−6.48T+53T2 1 - 6.48T + 53T^{2}
59 1−8.82T+59T2 1 - 8.82T + 59T^{2}
61 1−8.82T+61T2 1 - 8.82T + 61T^{2}
67 1+8.48T+67T2 1 + 8.48T + 67T^{2}
71 1−10.4T+71T2 1 - 10.4T + 71T^{2}
73 1−7.07T+73T2 1 - 7.07T + 73T^{2}
79 1+0.485T+79T2 1 + 0.485T + 79T^{2}
83 1−10.2T+83T2 1 - 10.2T + 83T^{2}
89 1+2T+89T2 1 + 2T + 89T^{2}
97 1−8.82T+97T2 1 - 8.82T + 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

−7.86742359342845289971225402962, −7.08494715646822316997389985779, −6.50654914424554425856181397563, −5.70387726442982119002732516346, −5.12527526823527625714200989903, −4.32246301787294433728658447599, −3.53046985465691196970212048177, −2.56220393342996253562323725020, −1.86125905996384875704005040948, −0.73290838982016040638422844127, 0.73290838982016040638422844127, 1.86125905996384875704005040948, 2.56220393342996253562323725020, 3.53046985465691196970212048177, 4.32246301787294433728658447599, 5.12527526823527625714200989903, 5.70387726442982119002732516346, 6.50654914424554425856181397563, 7.08494715646822316997389985779, 7.86742359342845289971225402962

Graph of the ZZ-function along the critical line