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Properties

Label 2-2940-1.1-c1-0-27
Degree 22
Conductor 29402940
Sign −1-1
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 11

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 − 6·17-s − 6·19-s − 8·23-s + 25-s + 27-s − 2·29-s − 10·31-s + 2·33-s + 2·37-s − 4·39-s − 10·41-s − 4·43-s + 45-s + 8·47-s − 6·51-s + 4·53-s + 2·55-s − 6·57-s + 8·59-s − 6·61-s − 4·65-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 − 1.45·17-s − 1.37·19-s − 1.66·23-s + 1/5·25-s + 0.192·27-s − 0.371·29-s − 1.79·31-s + 0.348·33-s + 0.328·37-s − 0.640·39-s − 1.56·41-s − 0.609·43-s + 0.149·45-s + 1.16·47-s − 0.840·51-s + 0.549·53-s + 0.269·55-s − 0.794·57-s + 1.04·59-s − 0.768·61-s − 0.496·65-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: −1-1
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: 11
Selberg data: (2, 2940, ( :1/2), −1)(2,\ 2940,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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+6T+pT2 1 + 6 T + p T^{2}
19 1+6T+pT2 1 + 6 T + p T^{2}
23 1+8T+pT2 1 + 8 T + p T^{2}
29 1+2T+pT2 1 + 2 T + p T^{2}
31 1+10T+pT2 1 + 10 T + p T^{2}
37 1−2T+pT2 1 - 2 T + p T^{2}
41 1+10T+pT2 1 + 10 T + p T^{2}
43 1+4T+pT2 1 + 4 T + p T^{2}
47 1−8T+pT2 1 - 8 T + p T^{2}
53 1−4T+pT2 1 - 4 T + p T^{2}
59 1−8T+pT2 1 - 8 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+6T+pT2 1 + 6 T + p T^{2}
73 1−12T+pT2 1 - 12 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 1−10T+pT2 1 - 10 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

−8.576753063635997780354724816607, −7.61991010314532395293141674162, −6.85500220249878455819000825708, −6.24416431678863774019927421815, −5.23406477936034179422601656737, −4.31129161226649985989005461361, −3.67509976173477623917923137546, −2.25371126242634563634491015097, −1.97163031330732203509706193529, 0, 1.97163031330732203509706193529, 2.25371126242634563634491015097, 3.67509976173477623917923137546, 4.31129161226649985989005461361, 5.23406477936034179422601656737, 6.24416431678863774019927421815, 6.85500220249878455819000825708, 7.61991010314532395293141674162, 8.576753063635997780354724816607

Graph of the ZZ-function along the critical line