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Properties

Label 2-720-1.1-c5-0-4
Degree 22
Conductor 720720
Sign 11
Analytic cond. 115.476115.476
Root an. cond. 10.745910.7459
Motivic weight 55
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 25·5-s − 218·7-s − 480·11-s − 622·13-s − 186·17-s + 1.20e3·19-s − 3.18e3·23-s + 625·25-s − 5.52e3·29-s − 9.35e3·31-s − 5.45e3·35-s + 5.61e3·37-s + 1.43e4·41-s + 370·43-s + 1.61e4·47-s + 3.07e4·49-s + 4.37e3·53-s − 1.20e4·55-s − 1.17e4·59-s + 1.32e4·61-s − 1.55e4·65-s + 1.15e4·67-s − 2.95e4·71-s + 3.36e4·73-s + 1.04e5·77-s − 3.12e4·79-s − 3.84e4·83-s + ⋯
L(s)  = 1  + 0.447·5-s − 1.68·7-s − 1.19·11-s − 1.02·13-s − 0.156·17-s + 0.765·19-s − 1.25·23-s + 1/5·25-s − 1.22·29-s − 1.74·31-s − 0.752·35-s + 0.674·37-s + 1.33·41-s + 0.0305·43-s + 1.06·47-s + 1.82·49-s + 0.213·53-s − 0.534·55-s − 0.439·59-s + 0.454·61-s − 0.456·65-s + 0.314·67-s − 0.695·71-s + 0.740·73-s + 2.01·77-s − 0.562·79-s − 0.612·83-s + ⋯

Functional equation

Λ(s)=(720s/2 ΓC(s) L(s)=( Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
Λ(s)=(720s/2 ΓC(s+5/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 720720    =    24⋅32⋅52^{4} \cdot 3^{2} \cdot 5
Sign: 11
Analytic conductor: 115.476115.476
Root analytic conductor: 10.745910.7459
Motivic weight: 55
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 720, ( :5/2), 1)(2,\ 720,\ (\ :5/2),\ 1)

Particular Values

L(3)L(3) ≈\approx 0.72579126540.7257912654
L(12)L(\frac12) ≈\approx 0.72579126540.7257912654
L(72)L(\frac{7}{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−p2T 1 - p^{2} T
good7 1+218T+p5T2 1 + 218 T + p^{5} T^{2}
11 1+480T+p5T2 1 + 480 T + p^{5} T^{2}
13 1+622T+p5T2 1 + 622 T + p^{5} T^{2}
17 1+186T+p5T2 1 + 186 T + p^{5} T^{2}
19 1−1204T+p5T2 1 - 1204 T + p^{5} T^{2}
23 1+3186T+p5T2 1 + 3186 T + p^{5} T^{2}
29 1+5526T+p5T2 1 + 5526 T + p^{5} T^{2}
31 1+9356T+p5T2 1 + 9356 T + p^{5} T^{2}
37 1−5618T+p5T2 1 - 5618 T + p^{5} T^{2}
41 1−14394T+p5T2 1 - 14394 T + p^{5} T^{2}
43 1−370T+p5T2 1 - 370 T + p^{5} T^{2}
47 1−16146T+p5T2 1 - 16146 T + p^{5} T^{2}
53 1−4374T+p5T2 1 - 4374 T + p^{5} T^{2}
59 1+11748T+p5T2 1 + 11748 T + p^{5} T^{2}
61 1−13202T+p5T2 1 - 13202 T + p^{5} T^{2}
67 1−11542T+p5T2 1 - 11542 T + p^{5} T^{2}
71 1+29532T+p5T2 1 + 29532 T + p^{5} T^{2}
73 1−33698T+p5T2 1 - 33698 T + p^{5} T^{2}
79 1+31208T+p5T2 1 + 31208 T + p^{5} T^{2}
83 1+38466T+p5T2 1 + 38466 T + p^{5} T^{2}
89 1+119514T+p5T2 1 + 119514 T + p^{5} T^{2}
97 1−94658T+p5T2 1 - 94658 T + p^{5} 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

−9.690816688234801809005562874992, −9.098566798818769925975626306144, −7.68980610564291871131544125167, −7.13290944209961654065967088343, −5.94194987005716777417466882588, −5.43562095897497227288491159121, −4.02619197900537234653869128669, −2.95440724832837064510945733095, −2.15999979740506238796366021183, −0.36848009909611478170979415846, 0.36848009909611478170979415846, 2.15999979740506238796366021183, 2.95440724832837064510945733095, 4.02619197900537234653869128669, 5.43562095897497227288491159121, 5.94194987005716777417466882588, 7.13290944209961654065967088343, 7.68980610564291871131544125167, 9.098566798818769925975626306144, 9.690816688234801809005562874992

Graph of the ZZ-function along the critical line