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Properties

Label 2-1440-1.1-c3-0-47
Degree 22
Conductor 14401440
Sign −1-1
Analytic cond. 84.962784.9627
Root an. cond. 9.217529.21752
Motivic weight 33
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5·5-s − 8·7-s − 4·11-s − 6·13-s + 2·17-s − 16·19-s + 60·23-s + 25·25-s + 142·29-s − 176·31-s − 40·35-s − 214·37-s + 278·41-s − 68·43-s − 116·47-s − 279·49-s + 350·53-s − 20·55-s − 684·59-s − 394·61-s − 30·65-s + 108·67-s + 96·71-s − 398·73-s + 32·77-s + 136·79-s − 436·83-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.431·7-s − 0.109·11-s − 0.128·13-s + 0.0285·17-s − 0.193·19-s + 0.543·23-s + 1/5·25-s + 0.909·29-s − 1.01·31-s − 0.193·35-s − 0.950·37-s + 1.05·41-s − 0.241·43-s − 0.360·47-s − 0.813·49-s + 0.907·53-s − 0.0490·55-s − 1.50·59-s − 0.826·61-s − 0.0572·65-s + 0.196·67-s + 0.160·71-s − 0.638·73-s + 0.0473·77-s + 0.193·79-s − 0.576·83-s + ⋯

Functional equation

Λ(s)=(1440s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(1440s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 14401440    =    25⋅32⋅52^{5} \cdot 3^{2} \cdot 5
Sign: −1-1
Analytic conductor: 84.962784.9627
Root analytic conductor: 9.217529.21752
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 1440, ( :3/2), −1)(2,\ 1440,\ (\ :3/2),\ -1)

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
L(52)L(\frac{5}{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−pT 1 - p T
good7 1+8T+p3T2 1 + 8 T + p^{3} T^{2}
11 1+4T+p3T2 1 + 4 T + p^{3} T^{2}
13 1+6T+p3T2 1 + 6 T + p^{3} T^{2}
17 1−2T+p3T2 1 - 2 T + p^{3} T^{2}
19 1+16T+p3T2 1 + 16 T + p^{3} T^{2}
23 1−60T+p3T2 1 - 60 T + p^{3} T^{2}
29 1−142T+p3T2 1 - 142 T + p^{3} T^{2}
31 1+176T+p3T2 1 + 176 T + p^{3} T^{2}
37 1+214T+p3T2 1 + 214 T + p^{3} T^{2}
41 1−278T+p3T2 1 - 278 T + p^{3} T^{2}
43 1+68T+p3T2 1 + 68 T + p^{3} T^{2}
47 1+116T+p3T2 1 + 116 T + p^{3} T^{2}
53 1−350T+p3T2 1 - 350 T + p^{3} T^{2}
59 1+684T+p3T2 1 + 684 T + p^{3} T^{2}
61 1+394T+p3T2 1 + 394 T + p^{3} T^{2}
67 1−108T+p3T2 1 - 108 T + p^{3} T^{2}
71 1−96T+p3T2 1 - 96 T + p^{3} T^{2}
73 1+398T+p3T2 1 + 398 T + p^{3} T^{2}
79 1−136T+p3T2 1 - 136 T + p^{3} T^{2}
83 1+436T+p3T2 1 + 436 T + p^{3} T^{2}
89 1−750T+p3T2 1 - 750 T + p^{3} T^{2}
97 1−82T+p3T2 1 - 82 T + p^{3} T^{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

−8.914560998707932233712388363964, −7.959652385438611044154332111268, −7.05594487313866128175776368674, −6.32548120016501812629175971678, −5.46358371477024199769776913894, −4.59389700579660561897702384657, −3.46162776702748594952527898423, −2.54368728189513980580303591760, −1.37576435598610081062675045585, 0, 1.37576435598610081062675045585, 2.54368728189513980580303591760, 3.46162776702748594952527898423, 4.59389700579660561897702384657, 5.46358371477024199769776913894, 6.32548120016501812629175971678, 7.05594487313866128175776368674, 7.959652385438611044154332111268, 8.914560998707932233712388363964

Graph of the ZZ-function along the critical line