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Properties

Label 2-4080-1.1-c1-0-32
Degree 22
Conductor 40804080
Sign 11
Analytic cond. 32.578932.5789
Root an. cond. 5.707795.70779
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 5-s + 4·7-s + 9-s − 4·11-s + 2·13-s + 15-s + 17-s + 4·19-s + 4·21-s + 25-s + 27-s − 2·29-s − 4·33-s + 4·35-s − 2·37-s + 2·39-s + 6·41-s + 8·43-s + 45-s − 8·47-s + 9·49-s + 51-s + 6·53-s − 4·55-s + 4·57-s + 14·61-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s + 1.51·7-s + 1/3·9-s − 1.20·11-s + 0.554·13-s + 0.258·15-s + 0.242·17-s + 0.917·19-s + 0.872·21-s + 1/5·25-s + 0.192·27-s − 0.371·29-s − 0.696·33-s + 0.676·35-s − 0.328·37-s + 0.320·39-s + 0.937·41-s + 1.21·43-s + 0.149·45-s − 1.16·47-s + 9/7·49-s + 0.140·51-s + 0.824·53-s − 0.539·55-s + 0.529·57-s + 1.79·61-s + ⋯

Functional equation

Λ(s)=(4080s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(4080s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 40804080    =    24⋅3⋅5⋅172^{4} \cdot 3 \cdot 5 \cdot 17
Sign: 11
Analytic conductor: 32.578932.5789
Root analytic conductor: 5.707795.70779
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 4080, ( :1/2), 1)(2,\ 4080,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 3.2386392873.238639287
L(12)L(\frac12) ≈\approx 3.2386392873.238639287
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−T 1 - T
5 1−T 1 - T
17 1−T 1 - T
good7 1−4T+pT2 1 - 4 T + p T^{2}
11 1+4T+pT2 1 + 4 T + p T^{2}
13 1−2T+pT2 1 - 2 T + p T^{2}
19 1−4T+pT2 1 - 4 T + p T^{2}
23 1+pT2 1 + p T^{2}
29 1+2T+pT2 1 + 2 T + p T^{2}
31 1+pT2 1 + p T^{2}
37 1+2T+pT2 1 + 2 T + p T^{2}
41 1−6T+pT2 1 - 6 T + p T^{2}
43 1−8T+pT2 1 - 8 T + p T^{2}
47 1+8T+pT2 1 + 8 T + p T^{2}
53 1−6T+pT2 1 - 6 T + p T^{2}
59 1+pT2 1 + p T^{2}
61 1−14T+pT2 1 - 14 T + p T^{2}
67 1+pT2 1 + p T^{2}
71 1+12T+pT2 1 + 12 T + p T^{2}
73 1+2T+pT2 1 + 2 T + p T^{2}
79 1+8T+pT2 1 + 8 T + p T^{2}
83 1+4T+pT2 1 + 4 T + p T^{2}
89 1−2T+pT2 1 - 2 T + p T^{2}
97 1−14T+pT2 1 - 14 T + p 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

−8.470770728506762982214955189573, −7.61120582262229803259959075798, −7.40482834259110305072425292563, −6.07222000001737998766751687749, −5.34826462308760317540143328088, −4.79115526258017019169544729755, −3.82256359171002894408181453537, −2.79516385947658142038022464125, −2.01160533963847894191314784515, −1.07784062898596240556871145354, 1.07784062898596240556871145354, 2.01160533963847894191314784515, 2.79516385947658142038022464125, 3.82256359171002894408181453537, 4.79115526258017019169544729755, 5.34826462308760317540143328088, 6.07222000001737998766751687749, 7.40482834259110305072425292563, 7.61120582262229803259959075798, 8.470770728506762982214955189573

Graph of the ZZ-function along the critical line