Properties

Label 2-197-1.1-c5-0-27
Degree 22
Conductor 197197
Sign 11
Analytic cond. 31.595631.5956
Root an. cond. 5.620995.62099
Motivic weight 55
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4.71·2-s + 27.6·3-s − 9.75·4-s − 79.0·5-s − 130.·6-s + 189.·7-s + 196.·8-s + 524.·9-s + 372.·10-s − 396.·11-s − 270.·12-s + 722.·13-s − 891.·14-s − 2.18e3·15-s − 616.·16-s − 684.·17-s − 2.47e3·18-s − 2.52e3·19-s + 770.·20-s + 5.23e3·21-s + 1.87e3·22-s + 3.87e3·23-s + 5.45e3·24-s + 3.11e3·25-s − 3.40e3·26-s + 7.79e3·27-s − 1.84e3·28-s + ⋯
L(s)  = 1  − 0.833·2-s + 1.77·3-s − 0.304·4-s − 1.41·5-s − 1.48·6-s + 1.45·7-s + 1.08·8-s + 2.15·9-s + 1.17·10-s − 0.989·11-s − 0.541·12-s + 1.18·13-s − 1.21·14-s − 2.51·15-s − 0.602·16-s − 0.574·17-s − 1.79·18-s − 1.60·19-s + 0.430·20-s + 2.59·21-s + 0.824·22-s + 1.52·23-s + 1.93·24-s + 0.998·25-s − 0.988·26-s + 2.05·27-s − 0.444·28-s + ⋯

Functional equation

Λ(s)=(197s/2ΓC(s)L(s)=(Λ(6s)\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
Λ(s)=(197s/2ΓC(s+5/2)L(s)=(Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 197197
Sign: 11
Analytic conductor: 31.595631.5956
Root analytic conductor: 5.620995.62099
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 197, ( :5/2), 1)(2,\ 197,\ (\ :5/2),\ 1)

Particular Values

L(3)L(3) \approx 1.9856631631.985663163
L(12)L(\frac12) \approx 1.9856631631.985663163
L(72)L(\frac{7}{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 13.88e4T 1 - 3.88e4T
good2 1+4.71T+32T2 1 + 4.71T + 32T^{2}
3 127.6T+243T2 1 - 27.6T + 243T^{2}
5 1+79.0T+3.12e3T2 1 + 79.0T + 3.12e3T^{2}
7 1189.T+1.68e4T2 1 - 189.T + 1.68e4T^{2}
11 1+396.T+1.61e5T2 1 + 396.T + 1.61e5T^{2}
13 1722.T+3.71e5T2 1 - 722.T + 3.71e5T^{2}
17 1+684.T+1.41e6T2 1 + 684.T + 1.41e6T^{2}
19 1+2.52e3T+2.47e6T2 1 + 2.52e3T + 2.47e6T^{2}
23 13.87e3T+6.43e6T2 1 - 3.87e3T + 6.43e6T^{2}
29 1+2.25e3T+2.05e7T2 1 + 2.25e3T + 2.05e7T^{2}
31 15.83e3T+2.86e7T2 1 - 5.83e3T + 2.86e7T^{2}
37 11.58e4T+6.93e7T2 1 - 1.58e4T + 6.93e7T^{2}
41 1+5.10e3T+1.15e8T2 1 + 5.10e3T + 1.15e8T^{2}
43 11.36e4T+1.47e8T2 1 - 1.36e4T + 1.47e8T^{2}
47 12.55e4T+2.29e8T2 1 - 2.55e4T + 2.29e8T^{2}
53 12.89e3T+4.18e8T2 1 - 2.89e3T + 4.18e8T^{2}
59 1+698.T+7.14e8T2 1 + 698.T + 7.14e8T^{2}
61 11.46e4T+8.44e8T2 1 - 1.46e4T + 8.44e8T^{2}
67 16.47e4T+1.35e9T2 1 - 6.47e4T + 1.35e9T^{2}
71 1+4.14e4T+1.80e9T2 1 + 4.14e4T + 1.80e9T^{2}
73 1+4.97e3T+2.07e9T2 1 + 4.97e3T + 2.07e9T^{2}
79 1+3.49e4T+3.07e9T2 1 + 3.49e4T + 3.07e9T^{2}
83 12.79e4T+3.93e9T2 1 - 2.79e4T + 3.93e9T^{2}
89 11.24e5T+5.58e9T2 1 - 1.24e5T + 5.58e9T^{2}
97 1+1.35e5T+8.58e9T2 1 + 1.35e5T + 8.58e9T^{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

−11.17486610235758670720280296413, −10.62176192545037162551305041742, −9.077294852098335710042586748660, −8.387424369754199961657178971456, −8.083177695634992083728651414618, −7.30251388707580509119347301313, −4.58901825809281318883919487563, −3.98419717346803907051562394322, −2.39885886187902438878229633830, −0.965217301508788902871111023886, 0.965217301508788902871111023886, 2.39885886187902438878229633830, 3.98419717346803907051562394322, 4.58901825809281318883919487563, 7.30251388707580509119347301313, 8.083177695634992083728651414618, 8.387424369754199961657178971456, 9.077294852098335710042586748660, 10.62176192545037162551305041742, 11.17486610235758670720280296413

Graph of the ZZ-function along the critical line