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Properties

Label 2-19-1.1-c5-0-3
Degree 22
Conductor 1919
Sign 11
Analytic cond. 3.047293.04729
Root an. cond. 1.745641.74564
Motivic weight 55
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 7.37·2-s − 3.41·3-s + 22.4·4-s + 108.·5-s − 25.1·6-s − 80.6·7-s − 70.6·8-s − 231.·9-s + 800.·10-s − 472.·11-s − 76.5·12-s + 639.·13-s − 594.·14-s − 370.·15-s − 1.23e3·16-s + 478.·17-s − 1.70e3·18-s + 361·19-s + 2.43e3·20-s + 275.·21-s − 3.48e3·22-s − 190.·23-s + 241.·24-s + 8.65e3·25-s + 4.71e3·26-s + 1.61e3·27-s − 1.80e3·28-s + ⋯
L(s)  = 1  + 1.30·2-s − 0.219·3-s + 0.700·4-s + 1.94·5-s − 0.285·6-s − 0.622·7-s − 0.390·8-s − 0.952·9-s + 2.53·10-s − 1.17·11-s − 0.153·12-s + 1.04·13-s − 0.811·14-s − 0.425·15-s − 1.20·16-s + 0.401·17-s − 1.24·18-s + 0.229·19-s + 1.35·20-s + 0.136·21-s − 1.53·22-s − 0.0749·23-s + 0.0855·24-s + 2.76·25-s + 1.36·26-s + 0.427·27-s − 0.435·28-s + ⋯

Functional equation

Λ(s)=(19s/2 ΓC(s) L(s)=( Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
Λ(s)=(19s/2 ΓC(s+5/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 1919
Sign: 11
Analytic conductor: 3.047293.04729
Root analytic conductor: 1.745641.74564
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 19, ( :5/2), 1)(2,\ 19,\ (\ :5/2),\ 1)

Particular Values

L(3)L(3) ≈\approx 2.5193192032.519319203
L(12)L(\frac12) ≈\approx 2.5193192032.519319203
L(72)L(\frac{7}{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)
bad19 1−361T 1 - 361T
good2 1−7.37T+32T2 1 - 7.37T + 32T^{2}
3 1+3.41T+243T2 1 + 3.41T + 243T^{2}
5 1−108.T+3.12e3T2 1 - 108.T + 3.12e3T^{2}
7 1+80.6T+1.68e4T2 1 + 80.6T + 1.68e4T^{2}
11 1+472.T+1.61e5T2 1 + 472.T + 1.61e5T^{2}
13 1−639.T+3.71e5T2 1 - 639.T + 3.71e5T^{2}
17 1−478.T+1.41e6T2 1 - 478.T + 1.41e6T^{2}
23 1+190.T+6.43e6T2 1 + 190.T + 6.43e6T^{2}
29 1+1.84e3T+2.05e7T2 1 + 1.84e3T + 2.05e7T^{2}
31 1−381.T+2.86e7T2 1 - 381.T + 2.86e7T^{2}
37 1+7.53e3T+6.93e7T2 1 + 7.53e3T + 6.93e7T^{2}
41 1−1.30e4T+1.15e8T2 1 - 1.30e4T + 1.15e8T^{2}
43 1−2.07e4T+1.47e8T2 1 - 2.07e4T + 1.47e8T^{2}
47 1−1.16e4T+2.29e8T2 1 - 1.16e4T + 2.29e8T^{2}
53 1+6.50e3T+4.18e8T2 1 + 6.50e3T + 4.18e8T^{2}
59 1+2.94e4T+7.14e8T2 1 + 2.94e4T + 7.14e8T^{2}
61 1−1.09e4T+8.44e8T2 1 - 1.09e4T + 8.44e8T^{2}
67 1+5.13e4T+1.35e9T2 1 + 5.13e4T + 1.35e9T^{2}
71 1−4.83e4T+1.80e9T2 1 - 4.83e4T + 1.80e9T^{2}
73 1+829.T+2.07e9T2 1 + 829.T + 2.07e9T^{2}
79 1+8.82e4T+3.07e9T2 1 + 8.82e4T + 3.07e9T^{2}
83 1−1.61e4T+3.93e9T2 1 - 1.61e4T + 3.93e9T^{2}
89 1+2.14e4T+5.58e9T2 1 + 2.14e4T + 5.58e9T^{2}
97 1−1.63e5T+8.58e9T2 1 - 1.63e5T + 8.58e9T^{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

−17.45498751107657677016098092274, −15.99503224744896544461783648807, −14.30125195043826012973489098203, −13.54321105546753378491555252021, −12.67648076189829934273544966978, −10.72770315762460556906986048843, −9.175710294921058131929316016309, −6.12505511149204918988044270822, −5.45686701406109831448983355719, −2.79576152434414397203283737437, 2.79576152434414397203283737437, 5.45686701406109831448983355719, 6.12505511149204918988044270822, 9.175710294921058131929316016309, 10.72770315762460556906986048843, 12.67648076189829934273544966978, 13.54321105546753378491555252021, 14.30125195043826012973489098203, 15.99503224744896544461783648807, 17.45498751107657677016098092274

Graph of the ZZ-function along the critical line