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Properties

Label 2-2940-1.1-c1-0-4
Degree 22
Conductor 29402940
Sign 11
Analytic cond. 23.476023.4760
Root an. cond. 4.845204.84520
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 + 9-s − 2·11-s − 4·13-s − 15-s − 2·17-s − 2·19-s + 4·23-s + 25-s + 27-s + 6·29-s + 2·31-s − 2·33-s + 10·37-s − 4·39-s + 10·41-s + 12·43-s − 45-s + 8·47-s − 2·51-s + 2·55-s − 2·57-s + 8·59-s + 2·61-s + 4·65-s − 12·67-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s + 1/3·9-s − 0.603·11-s − 1.10·13-s − 0.258·15-s − 0.485·17-s − 0.458·19-s + 0.834·23-s + 1/5·25-s + 0.192·27-s + 1.11·29-s + 0.359·31-s − 0.348·33-s + 1.64·37-s − 0.640·39-s + 1.56·41-s + 1.82·43-s − 0.149·45-s + 1.16·47-s − 0.280·51-s + 0.269·55-s − 0.264·57-s + 1.04·59-s + 0.256·61-s + 0.496·65-s − 1.46·67-s + ⋯

Functional equation

Λ(s)=(2940s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(2940s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 29402940    =    22⋅3⋅5⋅722^{2} \cdot 3 \cdot 5 \cdot 7^{2}
Sign: 11
Analytic conductor: 23.476023.4760
Root analytic conductor: 4.845204.84520
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 2940, ( :1/2), 1)(2,\ 2940,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.8503133311.850313331
L(12)L(\frac12) ≈\approx 1.8503133311.850313331
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
7 1 1
good11 1+2T+pT2 1 + 2 T + p T^{2}
13 1+4T+pT2 1 + 4 T + p T^{2}
17 1+2T+pT2 1 + 2 T + p T^{2}
19 1+2T+pT2 1 + 2 T + p T^{2}
23 1−4T+pT2 1 - 4 T + p T^{2}
29 1−6T+pT2 1 - 6 T + p T^{2}
31 1−2T+pT2 1 - 2 T + p T^{2}
37 1−10T+pT2 1 - 10 T + p T^{2}
41 1−10T+pT2 1 - 10 T + p T^{2}
43 1−12T+pT2 1 - 12 T + p T^{2}
47 1−8T+pT2 1 - 8 T + p T^{2}
53 1+pT2 1 + p T^{2}
59 1−8T+pT2 1 - 8 T + p T^{2}
61 1−2T+pT2 1 - 2 T + p T^{2}
67 1+12T+pT2 1 + 12 T + p T^{2}
71 1+10T+pT2 1 + 10 T + p T^{2}
73 1+4T+pT2 1 + 4 T + p T^{2}
79 1+pT2 1 + p T^{2}
83 1−12T+pT2 1 - 12 T + p T^{2}
89 1+2T+pT2 1 + 2 T + p T^{2}
97 1−8T+pT2 1 - 8 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

−8.836199545630867645438721125526, −7.82067649776875631788978756567, −7.51325888882796800062561076037, −6.62104364944396067563104383367, −5.67017038367597175775538292471, −4.60737339403970738856758018331, −4.18636216199252309981165234200, −2.80785568081983867387706785012, −2.44981952348987740648309403533, −0.798786744184690487080656808366, 0.798786744184690487080656808366, 2.44981952348987740648309403533, 2.80785568081983867387706785012, 4.18636216199252309981165234200, 4.60737339403970738856758018331, 5.67017038367597175775538292471, 6.62104364944396067563104383367, 7.51325888882796800062561076037, 7.82067649776875631788978756567, 8.836199545630867645438721125526

Graph of the ZZ-function along the critical line