↑

Properties

Label 2-5220-1.1-c1-0-37
Degree 22
Conductor 52205220
Sign −1-1
Analytic cond. 41.681941.6819
Root an. cond. 6.456156.45615
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 5-s + 1.32·7-s − 5.32·11-s − 5.02·13-s + 6.34·17-s − 4.34·19-s + 1.70·23-s + 25-s + 29-s + 8.34·31-s + 1.32·35-s − 6.93·37-s + 1.02·41-s + 10.7·43-s + 0.679·47-s − 5.25·49-s − 2.38·53-s − 5.32·55-s − 10.4·59-s − 6.38·61-s − 5.02·65-s − 5.70·67-s + 3.61·71-s − 6.73·73-s − 7.02·77-s − 11.3·79-s + 3.96·83-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.499·7-s − 1.60·11-s − 1.39·13-s + 1.53·17-s − 0.997·19-s + 0.356·23-s + 0.200·25-s + 0.185·29-s + 1.49·31-s + 0.223·35-s − 1.14·37-s + 0.160·41-s + 1.63·43-s + 0.0990·47-s − 0.750·49-s − 0.327·53-s − 0.717·55-s − 1.35·59-s − 0.817·61-s − 0.623·65-s − 0.697·67-s + 0.428·71-s − 0.788·73-s − 0.800·77-s − 1.28·79-s + 0.434·83-s + ⋯

Functional equation

Λ(s)=(5220s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 5220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(5220s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 5220 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 52205220    =    22⋅32⋅5⋅292^{2} \cdot 3^{2} \cdot 5 \cdot 29
Sign: −1-1
Analytic conductor: 41.681941.6819
Root analytic conductor: 6.456156.45615
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 5220, ( :1/2), −1)(2,\ 5220,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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)
bad2 1 1
3 1 1
5 1−T 1 - T
29 1−T 1 - T
good7 1−1.32T+7T2 1 - 1.32T + 7T^{2}
11 1+5.32T+11T2 1 + 5.32T + 11T^{2}
13 1+5.02T+13T2 1 + 5.02T + 13T^{2}
17 1−6.34T+17T2 1 - 6.34T + 17T^{2}
19 1+4.34T+19T2 1 + 4.34T + 19T^{2}
23 1−1.70T+23T2 1 - 1.70T + 23T^{2}
31 1−8.34T+31T2 1 - 8.34T + 31T^{2}
37 1+6.93T+37T2 1 + 6.93T + 37T^{2}
41 1−1.02T+41T2 1 - 1.02T + 41T^{2}
43 1−10.7T+43T2 1 - 10.7T + 43T^{2}
47 1−0.679T+47T2 1 - 0.679T + 47T^{2}
53 1+2.38T+53T2 1 + 2.38T + 53T^{2}
59 1+10.4T+59T2 1 + 10.4T + 59T^{2}
61 1+6.38T+61T2 1 + 6.38T + 61T^{2}
67 1+5.70T+67T2 1 + 5.70T + 67T^{2}
71 1−3.61T+71T2 1 - 3.61T + 71T^{2}
73 1+6.73T+73T2 1 + 6.73T + 73T^{2}
79 1+11.3T+79T2 1 + 11.3T + 79T^{2}
83 1−3.96T+83T2 1 - 3.96T + 83T^{2}
89 1+2.58T+89T2 1 + 2.58T + 89T^{2}
97 1+15.3T+97T2 1 + 15.3T + 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

−7.79726546313300968099328794911, −7.35118778127331763399440308823, −6.34785680134332272186749732621, −5.51592119291581419662411843173, −5.01405023583652197521142122963, −4.34484038441904270340928245055, −2.95193618032618234049668460411, −2.54456662855585889951573625539, −1.42096228406856806185233043419, 0, 1.42096228406856806185233043419, 2.54456662855585889951573625539, 2.95193618032618234049668460411, 4.34484038441904270340928245055, 5.01405023583652197521142122963, 5.51592119291581419662411843173, 6.34785680134332272186749732621, 7.35118778127331763399440308823, 7.79726546313300968099328794911

Graph of the ZZ-function along the critical line