↑

Properties

Label 2-197-1.1-c3-0-48
Degree 22
Conductor 197197
Sign −1-1
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 11

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2.32·2-s + 6.85·3-s − 2.61·4-s − 20.6·5-s + 15.9·6-s − 15.9·7-s − 24.6·8-s + 20.0·9-s − 47.9·10-s − 0.680·11-s − 17.9·12-s + 45.4·13-s − 37.0·14-s − 141.·15-s − 36.2·16-s − 51.0·17-s + 46.5·18-s − 104.·19-s + 54.0·20-s − 109.·21-s − 1.57·22-s + 41.2·23-s − 168.·24-s + 301.·25-s + 105.·26-s − 47.6·27-s + 41.7·28-s + ⋯
L(s)  = 1  + 0.820·2-s + 1.32·3-s − 0.326·4-s − 1.84·5-s + 1.08·6-s − 0.861·7-s − 1.08·8-s + 0.742·9-s − 1.51·10-s − 0.0186·11-s − 0.431·12-s + 0.968·13-s − 0.706·14-s − 2.43·15-s − 0.566·16-s − 0.728·17-s + 0.609·18-s − 1.25·19-s + 0.604·20-s − 1.13·21-s − 0.0153·22-s + 0.374·23-s − 1.43·24-s + 2.41·25-s + 0.794·26-s − 0.339·27-s + 0.281·28-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: −1-1
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: 11
Selberg data: (2, 197, ( :3/2), −1)(2,\ 197,\ (\ :3/2),\ -1)

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
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 1−2.32T+8T2 1 - 2.32T + 8T^{2}
3 1−6.85T+27T2 1 - 6.85T + 27T^{2}
5 1+20.6T+125T2 1 + 20.6T + 125T^{2}
7 1+15.9T+343T2 1 + 15.9T + 343T^{2}
11 1+0.680T+1.33e3T2 1 + 0.680T + 1.33e3T^{2}
13 1−45.4T+2.19e3T2 1 - 45.4T + 2.19e3T^{2}
17 1+51.0T+4.91e3T2 1 + 51.0T + 4.91e3T^{2}
19 1+104.T+6.85e3T2 1 + 104.T + 6.85e3T^{2}
23 1−41.2T+1.21e4T2 1 - 41.2T + 1.21e4T^{2}
29 1−168.T+2.43e4T2 1 - 168.T + 2.43e4T^{2}
31 1+336.T+2.97e4T2 1 + 336.T + 2.97e4T^{2}
37 1−292.T+5.06e4T2 1 - 292.T + 5.06e4T^{2}
41 1+186.T+6.89e4T2 1 + 186.T + 6.89e4T^{2}
43 1+34.6T+7.95e4T2 1 + 34.6T + 7.95e4T^{2}
47 1−189.T+1.03e5T2 1 - 189.T + 1.03e5T^{2}
53 1−416.T+1.48e5T2 1 - 416.T + 1.48e5T^{2}
59 1+426.T+2.05e5T2 1 + 426.T + 2.05e5T^{2}
61 1+752.T+2.26e5T2 1 + 752.T + 2.26e5T^{2}
67 1+690.T+3.00e5T2 1 + 690.T + 3.00e5T^{2}
71 1+281.T+3.57e5T2 1 + 281.T + 3.57e5T^{2}
73 1+525.T+3.89e5T2 1 + 525.T + 3.89e5T^{2}
79 1+24.0T+4.93e5T2 1 + 24.0T + 4.93e5T^{2}
83 1−50.4T+5.71e5T2 1 - 50.4T + 5.71e5T^{2}
89 1−708.T+7.04e5T2 1 - 708.T + 7.04e5T^{2}
97 1−786.T+9.12e5T2 1 - 786.T + 9.12e5T^{2}
show more
show less
   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

−11.83870095610078472984003202804, −10.72676311288651598733153512628, −9.006859814760997169539851311158, −8.683616216892366521900576888502, −7.60500414113086035863205162704, −6.34327794211513031411646756482, −4.43787876290347514567084479807, −3.72834790174633590375312635623, −2.96311363415539507575319957422, 0, 2.96311363415539507575319957422, 3.72834790174633590375312635623, 4.43787876290347514567084479807, 6.34327794211513031411646756482, 7.60500414113086035863205162704, 8.683616216892366521900576888502, 9.006859814760997169539851311158, 10.72676311288651598733153512628, 11.83870095610078472984003202804

Graph of the ZZ-function along the critical line