Properties

Label 2-197-1.1-c13-0-61
Degree 22
Conductor 197197
Sign 11
Analytic cond. 211.244211.244
Root an. cond. 14.534214.5342
Motivic weight 1313
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 133.·2-s + 64.3·3-s + 9.58e3·4-s + 1.95e4·5-s − 8.57e3·6-s + 1.77e5·7-s − 1.85e5·8-s − 1.59e6·9-s − 2.60e6·10-s − 3.36e5·11-s + 6.16e5·12-s + 1.41e7·13-s − 2.36e7·14-s + 1.25e6·15-s − 5.37e7·16-s + 1.10e8·17-s + 2.12e8·18-s − 1.18e8·19-s + 1.87e8·20-s + 1.14e7·21-s + 4.47e7·22-s − 7.03e8·23-s − 1.19e7·24-s − 8.37e8·25-s − 1.89e9·26-s − 2.04e8·27-s + 1.69e9·28-s + ⋯
L(s)  = 1  − 1.47·2-s + 0.0509·3-s + 1.16·4-s + 0.559·5-s − 0.0750·6-s + 0.569·7-s − 0.250·8-s − 0.997·9-s − 0.824·10-s − 0.0571·11-s + 0.0596·12-s + 0.815·13-s − 0.839·14-s + 0.0285·15-s − 0.801·16-s + 1.11·17-s + 1.46·18-s − 0.575·19-s + 0.654·20-s + 0.0290·21-s + 0.0842·22-s − 0.990·23-s − 0.0127·24-s − 0.686·25-s − 1.20·26-s − 0.101·27-s + 0.666·28-s + ⋯

Functional equation

Λ(s)=(197s/2ΓC(s)L(s)=(Λ(14s)\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(14-s) \end{aligned}
Λ(s)=(197s/2ΓC(s+13/2)L(s)=(Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 197197
Sign: 11
Analytic conductor: 211.244211.244
Root analytic conductor: 14.534214.5342
Motivic weight: 1313
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 197, ( :13/2), 1)(2,\ 197,\ (\ :13/2),\ 1)

Particular Values

L(7)L(7) \approx 0.98402626250.9840262625
L(12)L(\frac12) \approx 0.98402626250.9840262625
L(152)L(\frac{15}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad197 15.84e13T 1 - 5.84e13T
good2 1+133.T+8.19e3T2 1 + 133.T + 8.19e3T^{2}
3 164.3T+1.59e6T2 1 - 64.3T + 1.59e6T^{2}
5 11.95e4T+1.22e9T2 1 - 1.95e4T + 1.22e9T^{2}
7 11.77e5T+9.68e10T2 1 - 1.77e5T + 9.68e10T^{2}
11 1+3.36e5T+3.45e13T2 1 + 3.36e5T + 3.45e13T^{2}
13 11.41e7T+3.02e14T2 1 - 1.41e7T + 3.02e14T^{2}
17 11.10e8T+9.90e15T2 1 - 1.10e8T + 9.90e15T^{2}
19 1+1.18e8T+4.20e16T2 1 + 1.18e8T + 4.20e16T^{2}
23 1+7.03e8T+5.04e17T2 1 + 7.03e8T + 5.04e17T^{2}
29 1+3.44e9T+1.02e19T2 1 + 3.44e9T + 1.02e19T^{2}
31 1+5.07e9T+2.44e19T2 1 + 5.07e9T + 2.44e19T^{2}
37 12.95e10T+2.43e20T2 1 - 2.95e10T + 2.43e20T^{2}
41 13.95e10T+9.25e20T2 1 - 3.95e10T + 9.25e20T^{2}
43 1+5.83e10T+1.71e21T2 1 + 5.83e10T + 1.71e21T^{2}
47 18.53e10T+5.46e21T2 1 - 8.53e10T + 5.46e21T^{2}
53 1+2.30e11T+2.60e22T2 1 + 2.30e11T + 2.60e22T^{2}
59 12.95e11T+1.04e23T2 1 - 2.95e11T + 1.04e23T^{2}
61 13.76e11T+1.61e23T2 1 - 3.76e11T + 1.61e23T^{2}
67 15.34e11T+5.48e23T2 1 - 5.34e11T + 5.48e23T^{2}
71 11.73e12T+1.16e24T2 1 - 1.73e12T + 1.16e24T^{2}
73 16.32e11T+1.67e24T2 1 - 6.32e11T + 1.67e24T^{2}
79 1+1.64e12T+4.66e24T2 1 + 1.64e12T + 4.66e24T^{2}
83 1+7.58e11T+8.87e24T2 1 + 7.58e11T + 8.87e24T^{2}
89 11.01e11T+2.19e25T2 1 - 1.01e11T + 2.19e25T^{2}
97 1+1.06e12T+6.73e25T2 1 + 1.06e12T + 6.73e25T^{2}
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   L(s)=p j=12(1αj,pps)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

−9.917110897775969962423120305841, −9.228999269754609107886690237397, −8.212401890454571390395868178109, −7.75244428763011171391723583953, −6.28122600901542719184391858681, −5.45246981919792835667509007961, −3.85512020974786597001394527392, −2.36883831769383456422097539829, −1.57310229908007807017715488488, −0.52716100692339385437479282583, 0.52716100692339385437479282583, 1.57310229908007807017715488488, 2.36883831769383456422097539829, 3.85512020974786597001394527392, 5.45246981919792835667509007961, 6.28122600901542719184391858681, 7.75244428763011171391723583953, 8.212401890454571390395868178109, 9.228999269754609107886690237397, 9.917110897775969962423120305841

Graph of the ZZ-function along the critical line