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Properties

Label 2-1472-1.1-c1-0-14
Degree 22
Conductor 14721472
Sign 11
Analytic cond. 11.753911.7539
Root an. cond. 3.428403.42840
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  − 3·3-s + 2·5-s + 4·7-s + 6·9-s + 2·11-s + 5·13-s − 6·15-s + 4·17-s − 2·19-s − 12·21-s − 23-s − 25-s − 9·27-s + 7·29-s + 3·31-s − 6·33-s + 8·35-s − 2·37-s − 15·39-s − 9·41-s − 8·43-s + 12·45-s − 9·47-s + 9·49-s − 12·51-s − 2·53-s + 4·55-s + ⋯
L(s)  = 1  − 1.73·3-s + 0.894·5-s + 1.51·7-s + 2·9-s + 0.603·11-s + 1.38·13-s − 1.54·15-s + 0.970·17-s − 0.458·19-s − 2.61·21-s − 0.208·23-s − 1/5·25-s − 1.73·27-s + 1.29·29-s + 0.538·31-s − 1.04·33-s + 1.35·35-s − 0.328·37-s − 2.40·39-s − 1.40·41-s − 1.21·43-s + 1.78·45-s − 1.31·47-s + 9/7·49-s − 1.68·51-s − 0.274·53-s + 0.539·55-s + ⋯

Functional equation

Λ(s)=(1472s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(1472s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 14721472    =    26⋅232^{6} \cdot 23
Sign: 11
Analytic conductor: 11.753911.7539
Root analytic conductor: 3.428403.42840
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 1472, ( :1/2), 1)(2,\ 1472,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.5532187701.553218770
L(12)L(\frac12) ≈\approx 1.5532187701.553218770
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
23 1+T 1 + T
good3 1+pT+pT2 1 + p T + p T^{2}
5 1−2T+pT2 1 - 2 T + p T^{2}
7 1−4T+pT2 1 - 4 T + p T^{2}
11 1−2T+pT2 1 - 2 T + p T^{2}
13 1−5T+pT2 1 - 5 T + p T^{2}
17 1−4T+pT2 1 - 4 T + p T^{2}
19 1+2T+pT2 1 + 2 T + p T^{2}
29 1−7T+pT2 1 - 7 T + p T^{2}
31 1−3T+pT2 1 - 3 T + p T^{2}
37 1+2T+pT2 1 + 2 T + p T^{2}
41 1+9T+pT2 1 + 9 T + p T^{2}
43 1+8T+pT2 1 + 8 T + p T^{2}
47 1+9T+pT2 1 + 9 T + p T^{2}
53 1+2T+pT2 1 + 2 T + p T^{2}
59 1+pT2 1 + p T^{2}
61 1−2T+pT2 1 - 2 T + p T^{2}
67 1−14T+pT2 1 - 14 T + p T^{2}
71 1−3T+pT2 1 - 3 T + p T^{2}
73 1+3T+pT2 1 + 3 T + p T^{2}
79 1−6T+pT2 1 - 6 T + p T^{2}
83 1−8T+pT2 1 - 8 T + p T^{2}
89 1−12T+pT2 1 - 12 T + p T^{2}
97 1+pT2 1 + 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

−9.866992454756712207798623866216, −8.608238956261300097073169258146, −7.940995723732562677345333914110, −6.56889089382194248049775428165, −6.29658206116634366505284597932, −5.28615499787164158880149394506, −4.88899916461674802073833414510, −3.75737483622489943039302637500, −1.75965596937335757332008532480, −1.11834268237837652771743629830, 1.11834268237837652771743629830, 1.75965596937335757332008532480, 3.75737483622489943039302637500, 4.88899916461674802073833414510, 5.28615499787164158880149394506, 6.29658206116634366505284597932, 6.56889089382194248049775428165, 7.940995723732562677345333914110, 8.608238956261300097073169258146, 9.866992454756712207798623866216

Graph of the ZZ-function along the critical line