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Properties

Label 2-51-1.1-c3-0-4
Degree 22
Conductor 5151
Sign 11
Analytic cond. 3.009093.00909
Root an. cond. 1.734671.73467
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4.24·2-s − 3·3-s + 9.99·4-s + 19.9·5-s − 12.7·6-s − 20.9·7-s + 8.48·8-s + 9·9-s + 84.7·10-s + 16.0·11-s − 29.9·12-s − 34.9·13-s − 88.9·14-s − 59.9·15-s − 44.0·16-s − 17·17-s + 38.1·18-s − 80.8·19-s + 199.·20-s + 62.9·21-s + 68.0·22-s − 115.·23-s − 25.4·24-s + 273.·25-s − 148.·26-s − 27·27-s − 209.·28-s + ⋯
L(s)  = 1  + 1.49·2-s − 0.577·3-s + 1.24·4-s + 1.78·5-s − 0.866·6-s − 1.13·7-s + 0.374·8-s + 0.333·9-s + 2.67·10-s + 0.439·11-s − 0.721·12-s − 0.745·13-s − 1.69·14-s − 1.03·15-s − 0.687·16-s − 0.242·17-s + 0.500·18-s − 0.976·19-s + 2.23·20-s + 0.653·21-s + 0.659·22-s − 1.05·23-s − 0.216·24-s + 2.19·25-s − 1.11·26-s − 0.192·27-s − 1.41·28-s + ⋯

Functional equation

Λ(s)=(51s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(51s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 5151    =    3⋅173 \cdot 17
Sign: 11
Analytic conductor: 3.009093.00909
Root analytic conductor: 1.734671.73467
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 51, ( :3/2), 1)(2,\ 51,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 2.6367582092.636758209
L(12)L(\frac12) ≈\approx 2.6367582092.636758209
L(52)L(\frac{5}{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)
bad3 1+3T 1 + 3T
17 1+17T 1 + 17T
good2 1−4.24T+8T2 1 - 4.24T + 8T^{2}
5 1−19.9T+125T2 1 - 19.9T + 125T^{2}
7 1+20.9T+343T2 1 + 20.9T + 343T^{2}
11 1−16.0T+1.33e3T2 1 - 16.0T + 1.33e3T^{2}
13 1+34.9T+2.19e3T2 1 + 34.9T + 2.19e3T^{2}
19 1+80.8T+6.85e3T2 1 + 80.8T + 6.85e3T^{2}
23 1+115.T+1.21e4T2 1 + 115.T + 1.21e4T^{2}
29 1−154.T+2.43e4T2 1 - 154.T + 2.43e4T^{2}
31 1−299.T+2.97e4T2 1 - 299.T + 2.97e4T^{2}
37 1−315.T+5.06e4T2 1 - 315.T + 5.06e4T^{2}
41 1−132.T+6.89e4T2 1 - 132.T + 6.89e4T^{2}
43 1+23.1T+7.95e4T2 1 + 23.1T + 7.95e4T^{2}
47 1−260.T+1.03e5T2 1 - 260.T + 1.03e5T^{2}
53 1+676.T+1.48e5T2 1 + 676.T + 1.48e5T^{2}
59 1−629.T+2.05e5T2 1 - 629.T + 2.05e5T^{2}
61 1+461.T+2.26e5T2 1 + 461.T + 2.26e5T^{2}
67 1+789.T+3.00e5T2 1 + 789.T + 3.00e5T^{2}
71 1+686.T+3.57e5T2 1 + 686.T + 3.57e5T^{2}
73 1−484.T+3.89e5T2 1 - 484.T + 3.89e5T^{2}
79 1−254T+4.93e5T2 1 - 254T + 4.93e5T^{2}
83 1−548.T+5.71e5T2 1 - 548.T + 5.71e5T^{2}
89 1−925.T+7.04e5T2 1 - 925.T + 7.04e5T^{2}
97 1−732.T+9.12e5T2 1 - 732.T + 9.12e5T^{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

−14.62362983344960776055261691856, −13.68222288785837112621247610811, −12.93585579326445167328565546576, −12.06647203562279580501175738999, −10.35686539836264492321707011189, −9.408194808190971945735719944340, −6.42113878209371864504508109495, −6.11885099356846956181751603264, −4.61604693073622875217041840227, −2.55702699647835084381266876097, 2.55702699647835084381266876097, 4.61604693073622875217041840227, 6.11885099356846956181751603264, 6.42113878209371864504508109495, 9.408194808190971945735719944340, 10.35686539836264492321707011189, 12.06647203562279580501175738999, 12.93585579326445167328565546576, 13.68222288785837112621247610811, 14.62362983344960776055261691856

Graph of the ZZ-function along the critical line