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Properties

Label 2-197-1.1-c5-0-24
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  + 9.73·2-s − 22.3·3-s + 62.8·4-s − 58.5·5-s − 217.·6-s − 2.91·7-s + 300.·8-s + 254.·9-s − 570.·10-s − 29.1·11-s − 1.40e3·12-s + 771.·13-s − 28.3·14-s + 1.30e3·15-s + 916.·16-s + 1.05e3·17-s + 2.48e3·18-s + 668.·19-s − 3.68e3·20-s + 65.0·21-s − 283.·22-s + 1.20e3·23-s − 6.70e3·24-s + 302.·25-s + 7.51e3·26-s − 267.·27-s − 183.·28-s + ⋯
L(s)  = 1  + 1.72·2-s − 1.43·3-s + 1.96·4-s − 1.04·5-s − 2.46·6-s − 0.0224·7-s + 1.66·8-s + 1.04·9-s − 1.80·10-s − 0.0725·11-s − 2.81·12-s + 1.26·13-s − 0.0386·14-s + 1.49·15-s + 0.894·16-s + 0.885·17-s + 1.80·18-s + 0.424·19-s − 2.05·20-s + 0.0321·21-s − 0.124·22-s + 0.474·23-s − 2.37·24-s + 0.0967·25-s + 2.18·26-s − 0.0706·27-s − 0.0441·28-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 2.9843545032.984354503
L(12)L(\frac12) ≈\approx 2.9843545032.984354503
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 1−9.73T+32T2 1 - 9.73T + 32T^{2}
3 1+22.3T+243T2 1 + 22.3T + 243T^{2}
5 1+58.5T+3.12e3T2 1 + 58.5T + 3.12e3T^{2}
7 1+2.91T+1.68e4T2 1 + 2.91T + 1.68e4T^{2}
11 1+29.1T+1.61e5T2 1 + 29.1T + 1.61e5T^{2}
13 1−771.T+3.71e5T2 1 - 771.T + 3.71e5T^{2}
17 1−1.05e3T+1.41e6T2 1 - 1.05e3T + 1.41e6T^{2}
19 1−668.T+2.47e6T2 1 - 668.T + 2.47e6T^{2}
23 1−1.20e3T+6.43e6T2 1 - 1.20e3T + 6.43e6T^{2}
29 1−3.77e3T+2.05e7T2 1 - 3.77e3T + 2.05e7T^{2}
31 1−1.06e4T+2.86e7T2 1 - 1.06e4T + 2.86e7T^{2}
37 1+688.T+6.93e7T2 1 + 688.T + 6.93e7T^{2}
41 1−85.4T+1.15e8T2 1 - 85.4T + 1.15e8T^{2}
43 1+1.92e3T+1.47e8T2 1 + 1.92e3T + 1.47e8T^{2}
47 1+190.T+2.29e8T2 1 + 190.T + 2.29e8T^{2}
53 1−1.56e4T+4.18e8T2 1 - 1.56e4T + 4.18e8T^{2}
59 1−6.73e3T+7.14e8T2 1 - 6.73e3T + 7.14e8T^{2}
61 1−3.46e4T+8.44e8T2 1 - 3.46e4T + 8.44e8T^{2}
67 1+2.10e4T+1.35e9T2 1 + 2.10e4T + 1.35e9T^{2}
71 1−5.34e4T+1.80e9T2 1 - 5.34e4T + 1.80e9T^{2}
73 1−6.91e3T+2.07e9T2 1 - 6.91e3T + 2.07e9T^{2}
79 1+3.64e4T+3.07e9T2 1 + 3.64e4T + 3.07e9T^{2}
83 1+1.42e4T+3.93e9T2 1 + 1.42e4T + 3.93e9T^{2}
89 1+2.67e4T+5.58e9T2 1 + 2.67e4T + 5.58e9T^{2}
97 1−5.06e4T+8.58e9T2 1 - 5.06e4T + 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

−11.72207136454237611381735895657, −11.30185599071021561080008673998, −10.28473824431592159581006752647, −8.195425047247561596383396246553, −6.88600583517086732224128143278, −6.08288044721032318197316201221, −5.16196647161861787893410082738, −4.22942047272726962207822934453, −3.18122054267582537735511141592, −0.920127943011961056368696424143, 0.920127943011961056368696424143, 3.18122054267582537735511141592, 4.22942047272726962207822934453, 5.16196647161861787893410082738, 6.08288044721032318197316201221, 6.88600583517086732224128143278, 8.195425047247561596383396246553, 10.28473824431592159581006752647, 11.30185599071021561080008673998, 11.72207136454237611381735895657

Graph of the ZZ-function along the critical line