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Properties

Label 2-197-1.1-c3-0-22
Degree 22
Conductor 197197
Sign 11
Analytic cond. 11.623311.6233
Root an. cond. 3.409303.40930
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5.09·2-s − 7.11·3-s + 17.9·4-s + 8.87·5-s − 36.2·6-s − 1.96·7-s + 50.5·8-s + 23.6·9-s + 45.1·10-s + 42.4·11-s − 127.·12-s + 41.5·13-s − 10.0·14-s − 63.1·15-s + 113.·16-s + 12.7·17-s + 120.·18-s + 52.0·19-s + 159.·20-s + 14.0·21-s + 215.·22-s + 43.8·23-s − 359.·24-s − 46.1·25-s + 211.·26-s + 23.9·27-s − 35.2·28-s + ⋯
L(s)  = 1  + 1.79·2-s − 1.36·3-s + 2.23·4-s + 0.794·5-s − 2.46·6-s − 0.106·7-s + 2.23·8-s + 0.875·9-s + 1.42·10-s + 1.16·11-s − 3.06·12-s + 0.886·13-s − 0.191·14-s − 1.08·15-s + 1.77·16-s + 0.181·17-s + 1.57·18-s + 0.629·19-s + 1.77·20-s + 0.145·21-s + 2.09·22-s + 0.397·23-s − 3.05·24-s − 0.369·25-s + 1.59·26-s + 0.170·27-s − 0.238·28-s + ⋯

Functional equation

Λ(s)=(197s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(197s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 197197
Sign: 11
Analytic conductor: 11.623311.6233
Root analytic conductor: 3.409303.40930
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 197, ( :3/2), 1)(2,\ 197,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 3.9231144893.923114489
L(12)L(\frac12) ≈\approx 3.9231144893.923114489
L(52)L(\frac{5}{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+197T 1 + 197T
good2 1−5.09T+8T2 1 - 5.09T + 8T^{2}
3 1+7.11T+27T2 1 + 7.11T + 27T^{2}
5 1−8.87T+125T2 1 - 8.87T + 125T^{2}
7 1+1.96T+343T2 1 + 1.96T + 343T^{2}
11 1−42.4T+1.33e3T2 1 - 42.4T + 1.33e3T^{2}
13 1−41.5T+2.19e3T2 1 - 41.5T + 2.19e3T^{2}
17 1−12.7T+4.91e3T2 1 - 12.7T + 4.91e3T^{2}
19 1−52.0T+6.85e3T2 1 - 52.0T + 6.85e3T^{2}
23 1−43.8T+1.21e4T2 1 - 43.8T + 1.21e4T^{2}
29 1+233.T+2.43e4T2 1 + 233.T + 2.43e4T^{2}
31 1−15.4T+2.97e4T2 1 - 15.4T + 2.97e4T^{2}
37 1+215.T+5.06e4T2 1 + 215.T + 5.06e4T^{2}
41 1−240.T+6.89e4T2 1 - 240.T + 6.89e4T^{2}
43 1−72.1T+7.95e4T2 1 - 72.1T + 7.95e4T^{2}
47 1−249.T+1.03e5T2 1 - 249.T + 1.03e5T^{2}
53 1+656.T+1.48e5T2 1 + 656.T + 1.48e5T^{2}
59 1−282.T+2.05e5T2 1 - 282.T + 2.05e5T^{2}
61 1+705.T+2.26e5T2 1 + 705.T + 2.26e5T^{2}
67 1+601.T+3.00e5T2 1 + 601.T + 3.00e5T^{2}
71 1+839.T+3.57e5T2 1 + 839.T + 3.57e5T^{2}
73 1−194.T+3.89e5T2 1 - 194.T + 3.89e5T^{2}
79 1−450.T+4.93e5T2 1 - 450.T + 4.93e5T^{2}
83 1+156.T+5.71e5T2 1 + 156.T + 5.71e5T^{2}
89 1−315.T+7.04e5T2 1 - 315.T + 7.04e5T^{2}
97 1−1.20e3T+9.12e5T2 1 - 1.20e3T + 9.12e5T^{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

−12.10316267388592749738123433328, −11.40009914109244724409352672778, −10.70797245811444177713983360544, −9.329456048117356419067369345783, −7.18547363262216600523407791236, −6.10318898452810479430865726286, −5.83033384430894628224667883889, −4.68604179011077023694566456704, −3.46347946582898727283669312996, −1.54993107992247367298299613348, 1.54993107992247367298299613348, 3.46347946582898727283669312996, 4.68604179011077023694566456704, 5.83033384430894628224667883889, 6.10318898452810479430865726286, 7.18547363262216600523407791236, 9.329456048117356419067369345783, 10.70797245811444177713983360544, 11.40009914109244724409352672778, 12.10316267388592749738123433328

Graph of the ZZ-function along the critical line