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Properties

Label 2-197-1.1-c5-0-17
Degree 22
Conductor 197197
Sign 11
Analytic cond. 31.595631.5956
Root an. cond. 5.620995.62099
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  − 5.80·2-s + 12.9·3-s + 1.67·4-s + 23.4·5-s − 74.9·6-s − 167.·7-s + 175.·8-s − 76.3·9-s − 136.·10-s − 355.·11-s + 21.6·12-s + 481.·13-s + 970.·14-s + 303.·15-s − 1.07e3·16-s − 535.·17-s + 443.·18-s + 1.41e3·19-s + 39.3·20-s − 2.15e3·21-s + 2.06e3·22-s + 4.28e3·23-s + 2.27e3·24-s − 2.57e3·25-s − 2.79e3·26-s − 4.12e3·27-s − 280.·28-s + ⋯
L(s)  = 1  − 1.02·2-s + 0.828·3-s + 0.0523·4-s + 0.420·5-s − 0.849·6-s − 1.29·7-s + 0.972·8-s − 0.314·9-s − 0.431·10-s − 0.885·11-s + 0.0433·12-s + 0.789·13-s + 1.32·14-s + 0.348·15-s − 1.04·16-s − 0.449·17-s + 0.322·18-s + 0.897·19-s + 0.0220·20-s − 1.06·21-s + 0.908·22-s + 1.68·23-s + 0.805·24-s − 0.823·25-s − 0.810·26-s − 1.08·27-s − 0.0675·28-s + ⋯

Functional equation

Λ(s)=(197s/2 ΓC(s) L(s)=( Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
Λ(s)=(197s/2 ΓC(s+5/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 197197
Sign: 11
Analytic conductor: 31.595631.5956
Root analytic conductor: 5.620995.62099
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 197, ( :5/2), 1)(2,\ 197,\ (\ :5/2),\ 1)

Particular Values

L(3)L(3) ≈\approx 1.0753728081.075372808
L(12)L(\frac12) ≈\approx 1.0753728081.075372808
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)
bad197 1−3.88e4T 1 - 3.88e4T
good2 1+5.80T+32T2 1 + 5.80T + 32T^{2}
3 1−12.9T+243T2 1 - 12.9T + 243T^{2}
5 1−23.4T+3.12e3T2 1 - 23.4T + 3.12e3T^{2}
7 1+167.T+1.68e4T2 1 + 167.T + 1.68e4T^{2}
11 1+355.T+1.61e5T2 1 + 355.T + 1.61e5T^{2}
13 1−481.T+3.71e5T2 1 - 481.T + 3.71e5T^{2}
17 1+535.T+1.41e6T2 1 + 535.T + 1.41e6T^{2}
19 1−1.41e3T+2.47e6T2 1 - 1.41e3T + 2.47e6T^{2}
23 1−4.28e3T+6.43e6T2 1 - 4.28e3T + 6.43e6T^{2}
29 1−7.41e3T+2.05e7T2 1 - 7.41e3T + 2.05e7T^{2}
31 1+6.67e3T+2.86e7T2 1 + 6.67e3T + 2.86e7T^{2}
37 1−3.74e3T+6.93e7T2 1 - 3.74e3T + 6.93e7T^{2}
41 1−2.17e3T+1.15e8T2 1 - 2.17e3T + 1.15e8T^{2}
43 1+1.76e3T+1.47e8T2 1 + 1.76e3T + 1.47e8T^{2}
47 1−8.64e3T+2.29e8T2 1 - 8.64e3T + 2.29e8T^{2}
53 1−1.67e3T+4.18e8T2 1 - 1.67e3T + 4.18e8T^{2}
59 1−2.29e4T+7.14e8T2 1 - 2.29e4T + 7.14e8T^{2}
61 1−4.47e4T+8.44e8T2 1 - 4.47e4T + 8.44e8T^{2}
67 1−4.58e4T+1.35e9T2 1 - 4.58e4T + 1.35e9T^{2}
71 1−7.48e4T+1.80e9T2 1 - 7.48e4T + 1.80e9T^{2}
73 1+2.96e4T+2.07e9T2 1 + 2.96e4T + 2.07e9T^{2}
79 1−2.09e4T+3.07e9T2 1 - 2.09e4T + 3.07e9T^{2}
83 1−2.75e4T+3.93e9T2 1 - 2.75e4T + 3.93e9T^{2}
89 1−7.41e4T+5.58e9T2 1 - 7.41e4T + 5.58e9T^{2}
97 1−1.04e4T+8.58e9T2 1 - 1.04e4T + 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

−11.24074646506951728124456638109, −10.23147882985160439998132336584, −9.420846907912336575588336290417, −8.819490346614092040378190775726, −7.87139663901265104017585479280, −6.73688005487510798961327257291, −5.34480076198986330860008276516, −3.55656077281658554682419897809, −2.43776094813388608696629441808, −0.70819644205491908138051422173, 0.70819644205491908138051422173, 2.43776094813388608696629441808, 3.55656077281658554682419897809, 5.34480076198986330860008276516, 6.73688005487510798961327257291, 7.87139663901265104017585479280, 8.819490346614092040378190775726, 9.420846907912336575588336290417, 10.23147882985160439998132336584, 11.24074646506951728124456638109

Graph of the ZZ-function along the critical line