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Properties

Label 2-5760-1.1-c1-0-27
Degree 22
Conductor 57605760
Sign 11
Analytic cond. 45.993845.9938
Root an. cond. 6.781876.78187
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 − 3.12·7-s + 4·11-s + 7.12·13-s + 1.12·17-s − 1.12·19-s + 5.12·23-s + 25-s + 2·29-s − 3.12·31-s − 3.12·35-s + 3.12·37-s − 6.24·41-s − 4·43-s − 5.12·47-s + 2.75·49-s + 12.2·53-s + 4·55-s + 10.2·59-s + 6·61-s + 7.12·65-s − 8·67-s − 10.2·71-s − 8.24·73-s − 12.4·77-s + 15.1·79-s − 12·83-s + ⋯
L(s)  = 1  + 0.447·5-s − 1.18·7-s + 1.20·11-s + 1.97·13-s + 0.272·17-s − 0.257·19-s + 1.06·23-s + 0.200·25-s + 0.371·29-s − 0.560·31-s − 0.527·35-s + 0.513·37-s − 0.975·41-s − 0.609·43-s − 0.747·47-s + 0.393·49-s + 1.68·53-s + 0.539·55-s + 1.33·59-s + 0.768·61-s + 0.883·65-s − 0.977·67-s − 1.21·71-s − 0.965·73-s − 1.42·77-s + 1.70·79-s − 1.31·83-s + ⋯

Functional equation

Λ(s)=(5760s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 5760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(5760s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 5760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 57605760    =    27⋅32⋅52^{7} \cdot 3^{2} \cdot 5
Sign: 11
Analytic conductor: 45.993845.9938
Root analytic conductor: 6.781876.78187
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 5760, ( :1/2), 1)(2,\ 5760,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.3424206312.342420631
L(12)L(\frac12) ≈\approx 2.3424206312.342420631
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
good7 1+3.12T+7T2 1 + 3.12T + 7T^{2}
11 1−4T+11T2 1 - 4T + 11T^{2}
13 1−7.12T+13T2 1 - 7.12T + 13T^{2}
17 1−1.12T+17T2 1 - 1.12T + 17T^{2}
19 1+1.12T+19T2 1 + 1.12T + 19T^{2}
23 1−5.12T+23T2 1 - 5.12T + 23T^{2}
29 1−2T+29T2 1 - 2T + 29T^{2}
31 1+3.12T+31T2 1 + 3.12T + 31T^{2}
37 1−3.12T+37T2 1 - 3.12T + 37T^{2}
41 1+6.24T+41T2 1 + 6.24T + 41T^{2}
43 1+4T+43T2 1 + 4T + 43T^{2}
47 1+5.12T+47T2 1 + 5.12T + 47T^{2}
53 1−12.2T+53T2 1 - 12.2T + 53T^{2}
59 1−10.2T+59T2 1 - 10.2T + 59T^{2}
61 1−6T+61T2 1 - 6T + 61T^{2}
67 1+8T+67T2 1 + 8T + 67T^{2}
71 1+10.2T+71T2 1 + 10.2T + 71T^{2}
73 1+8.24T+73T2 1 + 8.24T + 73T^{2}
79 1−15.1T+79T2 1 - 15.1T + 79T^{2}
83 1+12T+83T2 1 + 12T + 83T^{2}
89 1+2.24T+89T2 1 + 2.24T + 89T^{2}
97 1−8.24T+97T2 1 - 8.24T + 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

−8.439914725970747038292246958927, −7.09911388697602465908604396890, −6.65054274666194798336827535162, −6.08610715828858184376044710782, −5.46412072991822693014752692089, −4.28061382099855004078045895753, −3.57033675440102626384612038426, −3.03038358013232828235457994536, −1.70823334303374909577272147217, −0.864942721513197752730434794887, 0.864942721513197752730434794887, 1.70823334303374909577272147217, 3.03038358013232828235457994536, 3.57033675440102626384612038426, 4.28061382099855004078045895753, 5.46412072991822693014752692089, 6.08610715828858184376044710782, 6.65054274666194798336827535162, 7.09911388697602465908604396890, 8.439914725970747038292246958927

Graph of the ZZ-function along the critical line