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Properties

Label 2-197-1.1-c1-0-9
Degree 22
Conductor 197197
Sign 11
Analytic cond. 1.573051.57305
Root an. cond. 1.254211.25421
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.896·2-s + 3.41·3-s − 1.19·4-s − 0.448·5-s + 3.05·6-s − 2.39·7-s − 2.86·8-s + 8.64·9-s − 0.401·10-s + 3.36·11-s − 4.08·12-s − 3.92·13-s − 2.14·14-s − 1.53·15-s − 0.175·16-s + 0.536·17-s + 7.74·18-s − 3.95·19-s + 0.536·20-s − 8.16·21-s + 3.01·22-s − 5.71·23-s − 9.77·24-s − 4.79·25-s − 3.52·26-s + 19.2·27-s + 2.86·28-s + ⋯
L(s)  = 1  + 0.633·2-s + 1.97·3-s − 0.598·4-s − 0.200·5-s + 1.24·6-s − 0.903·7-s − 1.01·8-s + 2.88·9-s − 0.127·10-s + 1.01·11-s − 1.17·12-s − 1.08·13-s − 0.572·14-s − 0.395·15-s − 0.0438·16-s + 0.130·17-s + 1.82·18-s − 0.908·19-s + 0.119·20-s − 1.78·21-s + 0.642·22-s − 1.19·23-s − 1.99·24-s − 0.959·25-s − 0.690·26-s + 3.70·27-s + 0.540·28-s + ⋯

Functional equation

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

Invariants

Degree: 22
Conductor: 197197
Sign: 11
Analytic conductor: 1.573051.57305
Root analytic conductor: 1.254211.25421
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 197, ( :1/2), 1)(2,\ 197,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.1711491652.171149165
L(12)L(\frac12) ≈\approx 2.1711491652.171149165
L(32)L(\frac{3}{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−T 1 - T
good2 1−0.896T+2T2 1 - 0.896T + 2T^{2}
3 1−3.41T+3T2 1 - 3.41T + 3T^{2}
5 1+0.448T+5T2 1 + 0.448T + 5T^{2}
7 1+2.39T+7T2 1 + 2.39T + 7T^{2}
11 1−3.36T+11T2 1 - 3.36T + 11T^{2}
13 1+3.92T+13T2 1 + 3.92T + 13T^{2}
17 1−0.536T+17T2 1 - 0.536T + 17T^{2}
19 1+3.95T+19T2 1 + 3.95T + 19T^{2}
23 1+5.71T+23T2 1 + 5.71T + 23T^{2}
29 1+6.05T+29T2 1 + 6.05T + 29T^{2}
31 1−6.74T+31T2 1 - 6.74T + 31T^{2}
37 1−8.96T+37T2 1 - 8.96T + 37T^{2}
41 1−4.65T+41T2 1 - 4.65T + 41T^{2}
43 1+2.58T+43T2 1 + 2.58T + 43T^{2}
47 1−1.84T+47T2 1 - 1.84T + 47T^{2}
53 1−9.84T+53T2 1 - 9.84T + 53T^{2}
59 1+4.25T+59T2 1 + 4.25T + 59T^{2}
61 1+0.391T+61T2 1 + 0.391T + 61T^{2}
67 1−7.54T+67T2 1 - 7.54T + 67T^{2}
71 1+8.07T+71T2 1 + 8.07T + 71T^{2}
73 1−11.2T+73T2 1 - 11.2T + 73T^{2}
79 1+2.38T+79T2 1 + 2.38T + 79T^{2}
83 1+1.38T+83T2 1 + 1.38T + 83T^{2}
89 1+1.54T+89T2 1 + 1.54T + 89T^{2}
97 1+11.1T+97T2 1 + 11.1T + 97T^{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

−12.82957803084216534502116201369, −12.06378931200110123501258261716, −9.848923205750769912872849113672, −9.569164464581891717710623859414, −8.568977710575794230809824739828, −7.60873228653269623285234981291, −6.32527951321528156948281656717, −4.32346730235717361093642779739, −3.71933729323563848878574036472, −2.47236704046632529570457532303, 2.47236704046632529570457532303, 3.71933729323563848878574036472, 4.32346730235717361093642779739, 6.32527951321528156948281656717, 7.60873228653269623285234981291, 8.568977710575794230809824739828, 9.569164464581891717710623859414, 9.848923205750769912872849113672, 12.06378931200110123501258261716, 12.82957803084216534502116201369

Graph of the ZZ-function along the critical line