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Properties

Label 2-1440-1.1-c3-0-19
Degree 22
Conductor 14401440
Sign 11
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 00

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: 11
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: 00
Selberg data: (2, 1440, ( :3/2), 1)(2,\ 1440,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 2.5511231832.551123183
L(12)L(\frac12) ≈\approx 2.5511231832.551123183
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

−9.131768526814424120851624440208, −8.396605074421235903446078454751, −7.59221297714023975808421267711, −6.67765118949623631963326555762, −5.87391649775497196484658335616, −4.99185961046917821845735536783, −4.14289445596208043436685320015, −2.94638384658623576411236088630, −1.94546405213263710566367196803, −0.804821176280024084810123713662, 0.804821176280024084810123713662, 1.94546405213263710566367196803, 2.94638384658623576411236088630, 4.14289445596208043436685320015, 4.99185961046917821845735536783, 5.87391649775497196484658335616, 6.67765118949623631963326555762, 7.59221297714023975808421267711, 8.396605074421235903446078454751, 9.131768526814424120851624440208

Graph of the ZZ-function along the critical line