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Properties

Label 2-3520-1.1-c1-0-16
Degree 22
Conductor 35203520
Sign 11
Analytic cond. 28.107328.1073
Root an. cond. 5.301635.30163
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  + 5-s − 2·7-s − 3·9-s − 11-s + 4·13-s − 4·17-s + 25-s + 6·29-s − 2·35-s + 2·37-s + 6·41-s − 2·43-s − 3·45-s − 3·49-s + 10·53-s − 55-s − 12·59-s + 6·61-s + 6·63-s + 4·65-s + 12·67-s + 16·71-s + 4·73-s + 2·77-s − 4·79-s + 9·81-s − 2·83-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.755·7-s − 9-s − 0.301·11-s + 1.10·13-s − 0.970·17-s + 1/5·25-s + 1.11·29-s − 0.338·35-s + 0.328·37-s + 0.937·41-s − 0.304·43-s − 0.447·45-s − 3/7·49-s + 1.37·53-s − 0.134·55-s − 1.56·59-s + 0.768·61-s + 0.755·63-s + 0.496·65-s + 1.46·67-s + 1.89·71-s + 0.468·73-s + 0.227·77-s − 0.450·79-s + 81-s − 0.219·83-s + ⋯

Functional equation

Λ(s)=(3520s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(3520s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 35203520    =    26⋅5⋅112^{6} \cdot 5 \cdot 11
Sign: 11
Analytic conductor: 28.107328.1073
Root analytic conductor: 5.301635.30163
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 3520, ( :1/2), 1)(2,\ 3520,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.5530950111.553095011
L(12)L(\frac12) ≈\approx 1.5530950111.553095011
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
5 1−T 1 - T
11 1+T 1 + T
good3 1+pT2 1 + p T^{2}
7 1+2T+pT2 1 + 2 T + p T^{2}
13 1−4T+pT2 1 - 4 T + p T^{2}
17 1+4T+pT2 1 + 4 T + p T^{2}
19 1+pT2 1 + p T^{2}
23 1+pT2 1 + p T^{2}
29 1−6T+pT2 1 - 6 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+2T+pT2 1 + 2 T + p T^{2}
47 1+pT2 1 + p T^{2}
53 1−10T+pT2 1 - 10 T + p T^{2}
59 1+12T+pT2 1 + 12 T + p T^{2}
61 1−6T+pT2 1 - 6 T + p T^{2}
67 1−12T+pT2 1 - 12 T + p T^{2}
71 1−16T+pT2 1 - 16 T + p T^{2}
73 1−4T+pT2 1 - 4 T + p T^{2}
79 1+4T+pT2 1 + 4 T + p T^{2}
83 1+2T+pT2 1 + 2 T + p T^{2}
89 1−6T+pT2 1 - 6 T + p T^{2}
97 1+2T+pT2 1 + 2 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

−8.597344779719288907602638819473, −8.035940562653855821779418746516, −6.85667269899173093324578324851, −6.30961469790096626723393594297, −5.72971066068134741309412572095, −4.83284777784030043470160695571, −3.79836956390090775269445181604, −2.96374376382014471097320958505, −2.17666388867997363005053217178, −0.71531896275518796286339281459, 0.71531896275518796286339281459, 2.17666388867997363005053217178, 2.96374376382014471097320958505, 3.79836956390090775269445181604, 4.83284777784030043470160695571, 5.72971066068134741309412572095, 6.30961469790096626723393594297, 6.85667269899173093324578324851, 8.035940562653855821779418746516, 8.597344779719288907602638819473

Graph of the ZZ-function along the critical line