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Properties

Label 2-2475-1.1-c3-0-3
Degree 22
Conductor 24752475
Sign 11
Analytic cond. 146.029146.029
Root an. cond. 12.084212.0842
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  − 0.732·2-s − 7.46·4-s − 16.9·7-s + 11.3·8-s + 11·11-s − 74.6·13-s + 12.3·14-s + 51.4·16-s − 82.7·17-s − 67.9·19-s − 8.05·22-s + 13.3·23-s + 54.6·26-s + 126.·28-s − 168.·29-s − 65.4·31-s − 128.·32-s + 60.6·34-s − 40.8·37-s + 49.7·38-s − 274.·41-s + 2.28·43-s − 82.1·44-s − 9.77·46-s + 71.8·47-s − 56.4·49-s + 557.·52-s + ⋯
L(s)  = 1  − 0.258·2-s − 0.933·4-s − 0.914·7-s + 0.500·8-s + 0.301·11-s − 1.59·13-s + 0.236·14-s + 0.803·16-s − 1.18·17-s − 0.820·19-s − 0.0780·22-s + 0.121·23-s + 0.412·26-s + 0.852·28-s − 1.08·29-s − 0.379·31-s − 0.708·32-s + 0.305·34-s − 0.181·37-s + 0.212·38-s − 1.04·41-s + 0.00811·43-s − 0.281·44-s − 0.0313·46-s + 0.222·47-s − 0.164·49-s + 1.48·52-s + ⋯

Functional equation

Λ(s)=(2475s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(2475s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 24752475    =    32⋅52⋅113^{2} \cdot 5^{2} \cdot 11
Sign: 11
Analytic conductor: 146.029146.029
Root analytic conductor: 12.084212.0842
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 2475, ( :3/2), 1)(2,\ 2475,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 0.037107894370.03710789437
L(12)L(\frac12) ≈\approx 0.037107894370.03710789437
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 1
5 1 1
11 1−11T 1 - 11T
good2 1+0.732T+8T2 1 + 0.732T + 8T^{2}
7 1+16.9T+343T2 1 + 16.9T + 343T^{2}
13 1+74.6T+2.19e3T2 1 + 74.6T + 2.19e3T^{2}
17 1+82.7T+4.91e3T2 1 + 82.7T + 4.91e3T^{2}
19 1+67.9T+6.85e3T2 1 + 67.9T + 6.85e3T^{2}
23 1−13.3T+1.21e4T2 1 - 13.3T + 1.21e4T^{2}
29 1+168.T+2.43e4T2 1 + 168.T + 2.43e4T^{2}
31 1+65.4T+2.97e4T2 1 + 65.4T + 2.97e4T^{2}
37 1+40.8T+5.06e4T2 1 + 40.8T + 5.06e4T^{2}
41 1+274.T+6.89e4T2 1 + 274.T + 6.89e4T^{2}
43 1−2.28T+7.95e4T2 1 - 2.28T + 7.95e4T^{2}
47 1−71.8T+1.03e5T2 1 - 71.8T + 1.03e5T^{2}
53 1+149.T+1.48e5T2 1 + 149.T + 1.48e5T^{2}
59 1+545.T+2.05e5T2 1 + 545.T + 2.05e5T^{2}
61 1−101.T+2.26e5T2 1 - 101.T + 2.26e5T^{2}
67 1+411.T+3.00e5T2 1 + 411.T + 3.00e5T^{2}
71 1−470.T+3.57e5T2 1 - 470.T + 3.57e5T^{2}
73 1+610.T+3.89e5T2 1 + 610.T + 3.89e5T^{2}
79 1+978.T+4.93e5T2 1 + 978.T + 4.93e5T^{2}
83 1−26.1T+5.71e5T2 1 - 26.1T + 5.71e5T^{2}
89 1−352.T+7.04e5T2 1 - 352.T + 7.04e5T^{2}
97 1+847.T+9.12e5T2 1 + 847.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

−8.845453733810344542760511320816, −7.84847147449465822319118412791, −7.08923631205891256304624426717, −6.36484366068527356967868924771, −5.34067623115730663760787883670, −4.57765227154553697146636426464, −3.85317271792736817146173714985, −2.80463247367345617896671556252, −1.71834942272120858071101274549, −0.085896833210309871970054469928, 0.085896833210309871970054469928, 1.71834942272120858071101274549, 2.80463247367345617896671556252, 3.85317271792736817146173714985, 4.57765227154553697146636426464, 5.34067623115730663760787883670, 6.36484366068527356967868924771, 7.08923631205891256304624426717, 7.84847147449465822319118412791, 8.845453733810344542760511320816

Graph of the ZZ-function along the critical line