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Properties

Label 2-546-1.1-c1-0-9
Degree 22
Conductor 546546
Sign 11
Analytic cond. 4.359834.35983
Root an. cond. 2.088022.08802
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  + 2-s + 3-s + 4-s − 5-s + 6-s + 7-s + 8-s + 9-s − 10-s + 5·11-s + 12-s − 13-s + 14-s − 15-s + 16-s − 3·17-s + 18-s − 19-s − 20-s + 21-s + 5·22-s + 3·23-s + 24-s − 4·25-s − 26-s + 27-s + 28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s + 1.50·11-s + 0.288·12-s − 0.277·13-s + 0.267·14-s − 0.258·15-s + 1/4·16-s − 0.727·17-s + 0.235·18-s − 0.229·19-s − 0.223·20-s + 0.218·21-s + 1.06·22-s + 0.625·23-s + 0.204·24-s − 4/5·25-s − 0.196·26-s + 0.192·27-s + 0.188·28-s + ⋯

Functional equation

Λ(s)=(546s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(546s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 546546    =    2⋅3⋅7⋅132 \cdot 3 \cdot 7 \cdot 13
Sign: 11
Analytic conductor: 4.359834.35983
Root analytic conductor: 2.088022.08802
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 546, ( :1/2), 1)(2,\ 546,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.6715677592.671567759
L(12)L(\frac12) ≈\approx 2.6715677592.671567759
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−T 1 - T
3 1−T 1 - T
7 1−T 1 - T
13 1+T 1 + T
good5 1+T+pT2 1 + T + p T^{2}
11 1−5T+pT2 1 - 5 T + p T^{2}
17 1+3T+pT2 1 + 3 T + p T^{2}
19 1+T+pT2 1 + T + p T^{2}
23 1−3T+pT2 1 - 3 T + p T^{2}
29 1−9T+pT2 1 - 9 T + p T^{2}
31 1−4T+pT2 1 - 4 T + p T^{2}
37 1+11T+pT2 1 + 11 T + p T^{2}
41 1+pT2 1 + p T^{2}
43 1+5T+pT2 1 + 5 T + p T^{2}
47 1+8T+pT2 1 + 8 T + p T^{2}
53 1+2T+pT2 1 + 2 T + p T^{2}
59 1−4T+pT2 1 - 4 T + p T^{2}
61 1+15T+pT2 1 + 15 T + p T^{2}
67 1+2T+pT2 1 + 2 T + p T^{2}
71 1+12T+pT2 1 + 12 T + p T^{2}
73 1−11T+pT2 1 - 11 T + p T^{2}
79 1−10T+pT2 1 - 10 T + p T^{2}
83 1+14T+pT2 1 + 14 T + p T^{2}
89 1−6T+pT2 1 - 6 T + p T^{2}
97 1+14T+pT2 1 + 14 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

−10.99185846587921421514805776622, −9.946457360732899024095075692007, −8.892862878325232361751976646852, −8.169592135054023904644454836931, −7.01673151448316654720589442436, −6.38424510864517867781756363293, −4.87504488106939908766189839095, −4.12151923019463718852682975283, −3.08986072016271957515480040072, −1.65008882553677401575623098637, 1.65008882553677401575623098637, 3.08986072016271957515480040072, 4.12151923019463718852682975283, 4.87504488106939908766189839095, 6.38424510864517867781756363293, 7.01673151448316654720589442436, 8.169592135054023904644454836931, 8.892862878325232361751976646852, 9.946457360732899024095075692007, 10.99185846587921421514805776622

Graph of the ZZ-function along the critical line