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Properties

Label 2-3520-1.1-c1-0-6
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  − 3-s − 5-s − 7-s − 2·9-s − 11-s + 2·13-s + 15-s − 5·17-s + 7·19-s + 21-s − 6·23-s + 25-s + 5·27-s + 29-s − 5·31-s + 33-s + 35-s − 11·37-s − 2·39-s + 2·41-s − 4·43-s + 2·45-s + 6·47-s − 6·49-s + 5·51-s + 53-s + 55-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s − 0.377·7-s − 2/3·9-s − 0.301·11-s + 0.554·13-s + 0.258·15-s − 1.21·17-s + 1.60·19-s + 0.218·21-s − 1.25·23-s + 1/5·25-s + 0.962·27-s + 0.185·29-s − 0.898·31-s + 0.174·33-s + 0.169·35-s − 1.80·37-s − 0.320·39-s + 0.312·41-s − 0.609·43-s + 0.298·45-s + 0.875·47-s − 6/7·49-s + 0.700·51-s + 0.137·53-s + 0.134·55-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 0.79448447120.7944844712
L(12)L(\frac12) ≈\approx 0.79448447120.7944844712
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+T+pT2 1 + T + p T^{2}
7 1+T+pT2 1 + T + p T^{2}
13 1−2T+pT2 1 - 2 T + p T^{2}
17 1+5T+pT2 1 + 5 T + p T^{2}
19 1−7T+pT2 1 - 7 T + p T^{2}
23 1+6T+pT2 1 + 6 T + p T^{2}
29 1−T+pT2 1 - T + p T^{2}
31 1+5T+pT2 1 + 5 T + p T^{2}
37 1+11T+pT2 1 + 11 T + p T^{2}
41 1−2T+pT2 1 - 2 T + p T^{2}
43 1+4T+pT2 1 + 4 T + p T^{2}
47 1−6T+pT2 1 - 6 T + p T^{2}
53 1−T+pT2 1 - T + p T^{2}
59 1−10T+pT2 1 - 10 T + p T^{2}
61 1+5T+pT2 1 + 5 T + p T^{2}
67 1−8T+pT2 1 - 8 T + p T^{2}
71 1−9T+pT2 1 - 9 T + p T^{2}
73 1+6T+pT2 1 + 6 T + p T^{2}
79 1+10T+pT2 1 + 10 T + p T^{2}
83 1−6T+pT2 1 - 6 T + p T^{2}
89 1−17T+pT2 1 - 17 T + p T^{2}
97 1−16T+pT2 1 - 16 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

−8.608740568697848080864230626915, −7.82448368747913391276430948485, −7.01608618441703671864583600001, −6.31376271112230576625519630596, −5.55058267506529268713362678327, −4.92395331212686291950472319307, −3.82844694626200613701514838124, −3.16039042154847503047624020435, −2.00139176724287747848757418899, −0.52054713672409666411604741362, 0.52054713672409666411604741362, 2.00139176724287747848757418899, 3.16039042154847503047624020435, 3.82844694626200613701514838124, 4.92395331212686291950472319307, 5.55058267506529268713362678327, 6.31376271112230576625519630596, 7.01608618441703671864583600001, 7.82448368747913391276430948485, 8.608740568697848080864230626915

Graph of the ZZ-function along the critical line