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Properties

Label 2-197-1.1-c5-0-28
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  − 10.7·2-s + 19.4·3-s + 82.7·4-s + 86.8·5-s − 208.·6-s − 107.·7-s − 543.·8-s + 136.·9-s − 930.·10-s + 73.1·11-s + 1.61e3·12-s − 497.·13-s + 1.15e3·14-s + 1.69e3·15-s + 3.17e3·16-s + 368.·17-s − 1.46e3·18-s + 976.·19-s + 7.18e3·20-s − 2.10e3·21-s − 783.·22-s + 4.09e3·23-s − 1.05e4·24-s + 4.41e3·25-s + 5.33e3·26-s − 2.06e3·27-s − 8.92e3·28-s + ⋯
L(s)  = 1  − 1.89·2-s + 1.25·3-s + 2.58·4-s + 1.55·5-s − 2.36·6-s − 0.832·7-s − 3.00·8-s + 0.563·9-s − 2.94·10-s + 0.182·11-s + 3.23·12-s − 0.816·13-s + 1.57·14-s + 1.94·15-s + 3.10·16-s + 0.309·17-s − 1.06·18-s + 0.620·19-s + 4.01·20-s − 1.04·21-s − 0.345·22-s + 1.61·23-s − 3.75·24-s + 1.41·25-s + 1.54·26-s − 0.546·27-s − 2.15·28-s + ⋯

Functional equation

Λ(s)=(197s/2 ΓC(s) L(s)=( Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
Λ(s)=(197s/2 ΓC(s+5/2) L(s)=( Λ(1−s)\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 1.6284590321.628459032
L(12)L(\frac12) ≈\approx 1.6284590321.628459032
L(72)L(\frac{7}{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 1−3.88e4T 1 - 3.88e4T
good2 1+10.7T+32T2 1 + 10.7T + 32T^{2}
3 1−19.4T+243T2 1 - 19.4T + 243T^{2}
5 1−86.8T+3.12e3T2 1 - 86.8T + 3.12e3T^{2}
7 1+107.T+1.68e4T2 1 + 107.T + 1.68e4T^{2}
11 1−73.1T+1.61e5T2 1 - 73.1T + 1.61e5T^{2}
13 1+497.T+3.71e5T2 1 + 497.T + 3.71e5T^{2}
17 1−368.T+1.41e6T2 1 - 368.T + 1.41e6T^{2}
19 1−976.T+2.47e6T2 1 - 976.T + 2.47e6T^{2}
23 1−4.09e3T+6.43e6T2 1 - 4.09e3T + 6.43e6T^{2}
29 1+3.14e3T+2.05e7T2 1 + 3.14e3T + 2.05e7T^{2}
31 1−4.04e3T+2.86e7T2 1 - 4.04e3T + 2.86e7T^{2}
37 1−1.44e4T+6.93e7T2 1 - 1.44e4T + 6.93e7T^{2}
41 1−1.00e4T+1.15e8T2 1 - 1.00e4T + 1.15e8T^{2}
43 1−8.39e3T+1.47e8T2 1 - 8.39e3T + 1.47e8T^{2}
47 1−7.53e3T+2.29e8T2 1 - 7.53e3T + 2.29e8T^{2}
53 1−3.18e4T+4.18e8T2 1 - 3.18e4T + 4.18e8T^{2}
59 1+3.71e4T+7.14e8T2 1 + 3.71e4T + 7.14e8T^{2}
61 1−4.58e3T+8.44e8T2 1 - 4.58e3T + 8.44e8T^{2}
67 1+1.66e4T+1.35e9T2 1 + 1.66e4T + 1.35e9T^{2}
71 1−1.97e4T+1.80e9T2 1 - 1.97e4T + 1.80e9T^{2}
73 1−8.33e4T+2.07e9T2 1 - 8.33e4T + 2.07e9T^{2}
79 1−7.32e4T+3.07e9T2 1 - 7.32e4T + 3.07e9T^{2}
83 1+9.00e4T+3.93e9T2 1 + 9.00e4T + 3.93e9T^{2}
89 1+1.03e5T+5.58e9T2 1 + 1.03e5T + 5.58e9T^{2}
97 1−1.36e5T+8.58e9T2 1 - 1.36e5T + 8.58e9T^{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

−11.07506173113465435809700907094, −9.933051274225912322068738873480, −9.456392466739912960429343915045, −9.052068512871726627414031971541, −7.81245969078994898086970680693, −6.89635790226067647775830606773, −5.80349582151469204699240657679, −2.93753304344969174342694365815, −2.33241885369118099625058241868, −1.01066128037433157390832138485, 1.01066128037433157390832138485, 2.33241885369118099625058241868, 2.93753304344969174342694365815, 5.80349582151469204699240657679, 6.89635790226067647775830606773, 7.81245969078994898086970680693, 9.052068512871726627414031971541, 9.456392466739912960429343915045, 9.933051274225912322068738873480, 11.07506173113465435809700907094

Graph of the ZZ-function along the critical line