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Properties

Label 2-197-1.1-c1-0-3
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  − 0.530·2-s + 0.899·3-s − 1.71·4-s + 3.03·5-s − 0.476·6-s + 0.743·7-s + 1.97·8-s − 2.19·9-s − 1.60·10-s + 3.91·11-s − 1.54·12-s − 1.48·13-s − 0.394·14-s + 2.72·15-s + 2.39·16-s + 5.45·17-s + 1.16·18-s + 1.78·19-s − 5.21·20-s + 0.668·21-s − 2.07·22-s − 4.14·23-s + 1.77·24-s + 4.19·25-s + 0.787·26-s − 4.66·27-s − 1.27·28-s + ⋯
L(s)  = 1  − 0.374·2-s + 0.519·3-s − 0.859·4-s + 1.35·5-s − 0.194·6-s + 0.281·7-s + 0.697·8-s − 0.730·9-s − 0.508·10-s + 1.17·11-s − 0.446·12-s − 0.411·13-s − 0.105·14-s + 0.704·15-s + 0.597·16-s + 1.32·17-s + 0.273·18-s + 0.409·19-s − 1.16·20-s + 0.145·21-s − 0.442·22-s − 0.863·23-s + 0.361·24-s + 0.838·25-s + 0.154·26-s − 0.898·27-s − 0.241·28-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 1.2080918141.208091814
L(12)L(\frac12) ≈\approx 1.2080918141.208091814
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+0.530T+2T2 1 + 0.530T + 2T^{2}
3 1−0.899T+3T2 1 - 0.899T + 3T^{2}
5 1−3.03T+5T2 1 - 3.03T + 5T^{2}
7 1−0.743T+7T2 1 - 0.743T + 7T^{2}
11 1−3.91T+11T2 1 - 3.91T + 11T^{2}
13 1+1.48T+13T2 1 + 1.48T + 13T^{2}
17 1−5.45T+17T2 1 - 5.45T + 17T^{2}
19 1−1.78T+19T2 1 - 1.78T + 19T^{2}
23 1+4.14T+23T2 1 + 4.14T + 23T^{2}
29 1+4.67T+29T2 1 + 4.67T + 29T^{2}
31 1+0.723T+31T2 1 + 0.723T + 31T^{2}
37 1+0.460T+37T2 1 + 0.460T + 37T^{2}
41 1+6.33T+41T2 1 + 6.33T + 41T^{2}
43 1−0.907T+43T2 1 - 0.907T + 43T^{2}
47 1+2.75T+47T2 1 + 2.75T + 47T^{2}
53 1+11.7T+53T2 1 + 11.7T + 53T^{2}
59 1−1.19T+59T2 1 - 1.19T + 59T^{2}
61 1−5.55T+61T2 1 - 5.55T + 61T^{2}
67 1+1.92T+67T2 1 + 1.92T + 67T^{2}
71 1−12.9T+71T2 1 - 12.9T + 71T^{2}
73 1+13.5T+73T2 1 + 13.5T + 73T^{2}
79 1−6.59T+79T2 1 - 6.59T + 79T^{2}
83 1+2.20T+83T2 1 + 2.20T + 83T^{2}
89 1+5.90T+89T2 1 + 5.90T + 89T^{2}
97 1−17.7T+97T2 1 - 17.7T + 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

−12.65236875114677382241127525525, −11.48774094126597515522097754254, −9.995436354886708022464939945429, −9.563797734390071901165869257566, −8.705504178036617020568536648820, −7.70077236295669609096332332748, −6.08267796047963842289649758594, −5.13145926464621747028654331750, −3.50951263376479321404567945896, −1.69597385220128136931077491004, 1.69597385220128136931077491004, 3.50951263376479321404567945896, 5.13145926464621747028654331750, 6.08267796047963842289649758594, 7.70077236295669609096332332748, 8.705504178036617020568536648820, 9.563797734390071901165869257566, 9.995436354886708022464939945429, 11.48774094126597515522097754254, 12.65236875114677382241127525525

Graph of the ZZ-function along the critical line