↑

Properties

Label 2-4925-1.1-c1-0-1
Degree 22
Conductor 49254925
Sign 11
Analytic cond. 39.326339.3263
Root an. cond. 6.271076.27107
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 0.183·2-s − 2.64·3-s − 1.96·4-s + 0.484·6-s − 4.36·7-s + 0.727·8-s + 3.97·9-s − 0.259·11-s + 5.19·12-s − 0.888·13-s + 0.800·14-s + 3.79·16-s − 3.65·17-s − 0.728·18-s + 3.95·19-s + 11.5·21-s + 0.0475·22-s + 2.56·23-s − 1.92·24-s + 0.162·26-s − 2.57·27-s + 8.57·28-s − 2.50·29-s − 6.76·31-s − 2.15·32-s + 0.684·33-s + 0.670·34-s + ⋯
L(s)  = 1  − 0.129·2-s − 1.52·3-s − 0.983·4-s + 0.197·6-s − 1.64·7-s + 0.257·8-s + 1.32·9-s − 0.0781·11-s + 1.49·12-s − 0.246·13-s + 0.213·14-s + 0.949·16-s − 0.886·17-s − 0.171·18-s + 0.908·19-s + 2.51·21-s + 0.0101·22-s + 0.535·23-s − 0.392·24-s + 0.0319·26-s − 0.494·27-s + 1.62·28-s − 0.464·29-s − 1.21·31-s − 0.380·32-s + 0.119·33-s + 0.115·34-s + ⋯

Functional equation

Λ(s)=(4925s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(4925s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 49254925    =    52⋅1975^{2} \cdot 197
Sign: 11
Analytic conductor: 39.326339.3263
Root analytic conductor: 6.271076.27107
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 4925, ( :1/2), 1)(2,\ 4925,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.0036682460220.003668246022
L(12)L(\frac12) ≈\approx 0.0036682460220.003668246022
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)
bad5 1 1
197 1+T 1 + T
good2 1+0.183T+2T2 1 + 0.183T + 2T^{2}
3 1+2.64T+3T2 1 + 2.64T + 3T^{2}
7 1+4.36T+7T2 1 + 4.36T + 7T^{2}
11 1+0.259T+11T2 1 + 0.259T + 11T^{2}
13 1+0.888T+13T2 1 + 0.888T + 13T^{2}
17 1+3.65T+17T2 1 + 3.65T + 17T^{2}
19 1−3.95T+19T2 1 - 3.95T + 19T^{2}
23 1−2.56T+23T2 1 - 2.56T + 23T^{2}
29 1+2.50T+29T2 1 + 2.50T + 29T^{2}
31 1+6.76T+31T2 1 + 6.76T + 31T^{2}
37 1+9.87T+37T2 1 + 9.87T + 37T^{2}
41 1+10.6T+41T2 1 + 10.6T + 41T^{2}
43 1+4.66T+43T2 1 + 4.66T + 43T^{2}
47 1+11.0T+47T2 1 + 11.0T + 47T^{2}
53 1+10.0T+53T2 1 + 10.0T + 53T^{2}
59 1+8.02T+59T2 1 + 8.02T + 59T^{2}
61 1−2.13T+61T2 1 - 2.13T + 61T^{2}
67 1+1.00T+67T2 1 + 1.00T + 67T^{2}
71 1+0.407T+71T2 1 + 0.407T + 71T^{2}
73 1−1.66T+73T2 1 - 1.66T + 73T^{2}
79 1−3.08T+79T2 1 - 3.08T + 79T^{2}
83 1−12.2T+83T2 1 - 12.2T + 83T^{2}
89 1+6.10T+89T2 1 + 6.10T + 89T^{2}
97 1+8.15T+97T2 1 + 8.15T + 97T^{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

−8.348811675403205095483634249536, −7.23777232816997929751415706619, −6.71287532696150428558809252081, −6.09167072915374353138101077746, −5.17942443444345990920534752610, −4.95271419842830442191274970541, −3.75094278071744066533552927877, −3.17884264527991712336692651702, −1.50106250291919630617288439099, −0.03780971162271424173118123128, 0.03780971162271424173118123128, 1.50106250291919630617288439099, 3.17884264527991712336692651702, 3.75094278071744066533552927877, 4.95271419842830442191274970541, 5.17942443444345990920534752610, 6.09167072915374353138101077746, 6.71287532696150428558809252081, 7.23777232816997929751415706619, 8.348811675403205095483634249536

Graph of the ZZ-function along the critical line