↑

Properties

Label 2-197-1.1-c13-0-13
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

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 145.·2-s − 2.02e3·3-s + 1.28e4·4-s − 1.01e4·5-s + 2.93e5·6-s + 3.45e5·7-s − 6.80e5·8-s + 2.50e6·9-s + 1.47e6·10-s + 1.29e6·11-s − 2.60e7·12-s − 2.93e7·13-s − 5.02e7·14-s + 2.05e7·15-s − 6.76e6·16-s − 7.66e7·17-s − 3.63e8·18-s + 1.13e8·19-s − 1.30e8·20-s − 7.00e8·21-s − 1.87e8·22-s − 9.76e8·23-s + 1.37e9·24-s − 1.11e9·25-s + 4.25e9·26-s − 1.84e9·27-s + 4.45e9·28-s + ⋯
L(s)  = 1  − 1.60·2-s − 1.60·3-s + 1.57·4-s − 0.290·5-s + 2.57·6-s + 1.11·7-s − 0.917·8-s + 1.57·9-s + 0.466·10-s + 0.220·11-s − 2.52·12-s − 1.68·13-s − 1.78·14-s + 0.466·15-s − 0.100·16-s − 0.770·17-s − 2.52·18-s + 0.555·19-s − 0.457·20-s − 1.78·21-s − 0.353·22-s − 1.37·23-s + 1.47·24-s − 0.915·25-s + 2.70·26-s − 0.917·27-s + 1.74·28-s + ⋯

Functional equation

Λ(s)=(197s/2 ΓC(s) L(s)=( Λ(14−s)\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(14-s) \end{aligned}
Λ(s)=(197s/2 ΓC(s+13/2) L(s)=( Λ(1−s)\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 0.076688381940.07668838194
L(12)L(\frac12) ≈\approx 0.076688381940.07668838194
L(152)L(\frac{15}{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 1−5.84e13T 1 - 5.84e13T
good2 1+145.T+8.19e3T2 1 + 145.T + 8.19e3T^{2}
3 1+2.02e3T+1.59e6T2 1 + 2.02e3T + 1.59e6T^{2}
5 1+1.01e4T+1.22e9T2 1 + 1.01e4T + 1.22e9T^{2}
7 1−3.45e5T+9.68e10T2 1 - 3.45e5T + 9.68e10T^{2}
11 1−1.29e6T+3.45e13T2 1 - 1.29e6T + 3.45e13T^{2}
13 1+2.93e7T+3.02e14T2 1 + 2.93e7T + 3.02e14T^{2}
17 1+7.66e7T+9.90e15T2 1 + 7.66e7T + 9.90e15T^{2}
19 1−1.13e8T+4.20e16T2 1 - 1.13e8T + 4.20e16T^{2}
23 1+9.76e8T+5.04e17T2 1 + 9.76e8T + 5.04e17T^{2}
29 1−4.14e9T+1.02e19T2 1 - 4.14e9T + 1.02e19T^{2}
31 1+5.83e9T+2.44e19T2 1 + 5.83e9T + 2.44e19T^{2}
37 1−3.87e9T+2.43e20T2 1 - 3.87e9T + 2.43e20T^{2}
41 1+3.31e10T+9.25e20T2 1 + 3.31e10T + 9.25e20T^{2}
43 1+4.34e10T+1.71e21T2 1 + 4.34e10T + 1.71e21T^{2}
47 1−1.16e11T+5.46e21T2 1 - 1.16e11T + 5.46e21T^{2}
53 1−3.23e10T+2.60e22T2 1 - 3.23e10T + 2.60e22T^{2}
59 1+4.58e11T+1.04e23T2 1 + 4.58e11T + 1.04e23T^{2}
61 1+7.04e11T+1.61e23T2 1 + 7.04e11T + 1.61e23T^{2}
67 1−7.66e11T+5.48e23T2 1 - 7.66e11T + 5.48e23T^{2}
71 1+1.10e12T+1.16e24T2 1 + 1.10e12T + 1.16e24T^{2}
73 1−2.36e12T+1.67e24T2 1 - 2.36e12T + 1.67e24T^{2}
79 1−7.79e11T+4.66e24T2 1 - 7.79e11T + 4.66e24T^{2}
83 1+6.47e11T+8.87e24T2 1 + 6.47e11T + 8.87e24T^{2}
89 1−1.00e12T+2.19e25T2 1 - 1.00e12T + 2.19e25T^{2}
97 1−1.52e13T+6.73e25T2 1 - 1.52e13T + 6.73e25T^{2}
show more
show less
   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

−10.26185144741292670980411419279, −9.387721139966086537224948162031, −8.083485074107388433836465787506, −7.39351013308023232175737017808, −6.47336620009903149360724138120, −5.21557589687898099310241384254, −4.40417537251106014500315838701, −2.17992291487395160363956825499, −1.32949828037724494854108753722, −0.17334210609360390726615418409, 0.17334210609360390726615418409, 1.32949828037724494854108753722, 2.17992291487395160363956825499, 4.40417537251106014500315838701, 5.21557589687898099310241384254, 6.47336620009903149360724138120, 7.39351013308023232175737017808, 8.083485074107388433836465787506, 9.387721139966086537224948162031, 10.26185144741292670980411419279

Graph of the ZZ-function along the critical line