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Properties

Label 2-51520-1.1-c1-0-20
Degree 22
Conductor 5152051520
Sign 11
Analytic cond. 411.389411.389
Root an. cond. 20.282720.2827
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 + 3·11-s − 3·13-s − 15-s + 3·17-s − 21-s + 23-s + 25-s + 5·27-s + 9·29-s + 2·31-s − 3·33-s + 35-s + 4·37-s + 3·39-s + 4·41-s + 2·43-s − 2·45-s + 13·47-s + 49-s − 3·51-s + 2·53-s + 3·55-s + 6·59-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s + 0.377·7-s − 2/3·9-s + 0.904·11-s − 0.832·13-s − 0.258·15-s + 0.727·17-s − 0.218·21-s + 0.208·23-s + 1/5·25-s + 0.962·27-s + 1.67·29-s + 0.359·31-s − 0.522·33-s + 0.169·35-s + 0.657·37-s + 0.480·39-s + 0.624·41-s + 0.304·43-s − 0.298·45-s + 1.89·47-s + 1/7·49-s − 0.420·51-s + 0.274·53-s + 0.404·55-s + 0.781·59-s + ⋯

Functional equation

Λ(s)=(51520s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 51520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(51520s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 51520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 5152051520    =    26⋅5⋅7⋅232^{6} \cdot 5 \cdot 7 \cdot 23
Sign: 11
Analytic conductor: 411.389411.389
Root analytic conductor: 20.282720.2827
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 51520, ( :1/2), 1)(2,\ 51520,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.6690646952.669064695
L(12)L(\frac12) ≈\approx 2.6690646952.669064695
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
7 1−T 1 - T
23 1−T 1 - T
good3 1+T+pT2 1 + T + p T^{2}
11 1−3T+pT2 1 - 3 T + p T^{2}
13 1+3T+pT2 1 + 3 T + p T^{2}
17 1−3T+pT2 1 - 3 T + p T^{2}
19 1+pT2 1 + p T^{2}
29 1−9T+pT2 1 - 9 T + p T^{2}
31 1−2T+pT2 1 - 2 T + p T^{2}
37 1−4T+pT2 1 - 4 T + p T^{2}
41 1−4T+pT2 1 - 4 T + p T^{2}
43 1−2T+pT2 1 - 2 T + p T^{2}
47 1−13T+pT2 1 - 13 T + p T^{2}
53 1−2T+pT2 1 - 2 T + p T^{2}
59 1−6T+pT2 1 - 6 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+8T+pT2 1 + 8 T + p T^{2}
73 1+6T+pT2 1 + 6 T + p T^{2}
79 1+13T+pT2 1 + 13 T + p T^{2}
83 1+12T+pT2 1 + 12 T + p T^{2}
89 1−6T+pT2 1 - 6 T + p T^{2}
97 1−7T+pT2 1 - 7 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

−14.44041912190354, −14.19147192115801, −13.55599424599438, −12.83935494792298, −12.28113881402940, −11.95764558304751, −11.45640693234486, −11.00191052251672, −10.22005916173901, −10.01521625306642, −9.321732654418923, −8.610969600203587, −8.447570009843260, −7.440445639328562, −7.137483633226479, −6.353694590938309, −5.902347907505197, −5.464660566601361, −4.718025476606102, −4.356437242292180, −3.442691280735218, −2.697112501464693, −2.225002416506506, −1.114858728788691, −0.7018454719089312, 0.7018454719089312, 1.114858728788691, 2.225002416506506, 2.697112501464693, 3.442691280735218, 4.356437242292180, 4.718025476606102, 5.464660566601361, 5.902347907505197, 6.353694590938309, 7.137483633226479, 7.440445639328562, 8.447570009843260, 8.610969600203587, 9.321732654418923, 10.01521625306642, 10.22005916173901, 11.00191052251672, 11.45640693234486, 11.95764558304751, 12.28113881402940, 12.83935494792298, 13.55599424599438, 14.19147192115801, 14.44041912190354

Graph of the ZZ-function along the critical line