↑

Properties

Label 2-3800-1.1-c1-0-61
Degree 22
Conductor 38003800
Sign 11
Analytic cond. 30.343130.3431
Root an. cond. 5.508465.50846
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2.77·3-s + 2.31·7-s + 4.67·9-s + 2.16·11-s + 6.25·13-s + 7.10·17-s − 19-s + 6.40·21-s − 8.99·23-s + 4.63·27-s + 3.16·29-s − 9.95·31-s + 5.98·33-s + 9.43·37-s + 17.3·39-s − 10.8·41-s + 1.05·43-s − 6.98·47-s − 1.66·49-s + 19.6·51-s − 2.69·53-s − 2.77·57-s + 11.2·59-s − 4.36·61-s + 10.7·63-s + 6.25·67-s − 24.9·69-s + ⋯
L(s)  = 1  + 1.59·3-s + 0.873·7-s + 1.55·9-s + 0.651·11-s + 1.73·13-s + 1.72·17-s − 0.229·19-s + 1.39·21-s − 1.87·23-s + 0.892·27-s + 0.587·29-s − 1.78·31-s + 1.04·33-s + 1.55·37-s + 2.77·39-s − 1.68·41-s + 0.160·43-s − 1.01·47-s − 0.237·49-s + 2.75·51-s − 0.370·53-s − 0.366·57-s + 1.45·59-s − 0.558·61-s + 1.36·63-s + 0.764·67-s − 3.00·69-s + ⋯

Functional equation

Λ(s)=(3800s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(3800s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 38003800    =    23⋅52⋅192^{3} \cdot 5^{2} \cdot 19
Sign: 11
Analytic conductor: 30.343130.3431
Root analytic conductor: 5.508465.50846
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 3800, ( :1/2), 1)(2,\ 3800,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 4.4583666084.458366608
L(12)L(\frac12) ≈\approx 4.4583666084.458366608
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
5 1 1
19 1+T 1 + T
good3 1−2.77T+3T2 1 - 2.77T + 3T^{2}
7 1−2.31T+7T2 1 - 2.31T + 7T^{2}
11 1−2.16T+11T2 1 - 2.16T + 11T^{2}
13 1−6.25T+13T2 1 - 6.25T + 13T^{2}
17 1−7.10T+17T2 1 - 7.10T + 17T^{2}
23 1+8.99T+23T2 1 + 8.99T + 23T^{2}
29 1−3.16T+29T2 1 - 3.16T + 29T^{2}
31 1+9.95T+31T2 1 + 9.95T + 31T^{2}
37 1−9.43T+37T2 1 - 9.43T + 37T^{2}
41 1+10.8T+41T2 1 + 10.8T + 41T^{2}
43 1−1.05T+43T2 1 - 1.05T + 43T^{2}
47 1+6.98T+47T2 1 + 6.98T + 47T^{2}
53 1+2.69T+53T2 1 + 2.69T + 53T^{2}
59 1−11.2T+59T2 1 - 11.2T + 59T^{2}
61 1+4.36T+61T2 1 + 4.36T + 61T^{2}
67 1−6.25T+67T2 1 - 6.25T + 67T^{2}
71 1+13.9T+71T2 1 + 13.9T + 71T^{2}
73 1+8.64T+73T2 1 + 8.64T + 73T^{2}
79 1+9.93T+79T2 1 + 9.93T + 79T^{2}
83 1+9.82T+83T2 1 + 9.82T + 83T^{2}
89 1+1.63T+89T2 1 + 1.63T + 89T^{2}
97 1−9.27T+97T2 1 - 9.27T + 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

−8.416844845775407031458014684867, −8.003322234917908542258292407072, −7.37882186968181741335174974399, −6.27974157867221583863041133764, −5.58404774455847619878433390470, −4.34622309274093397128652057411, −3.71298704342400990336202592685, −3.15702560461685351166608566606, −1.86132197103629459089971158830, −1.36995978014762498193570884767, 1.36995978014762498193570884767, 1.86132197103629459089971158830, 3.15702560461685351166608566606, 3.71298704342400990336202592685, 4.34622309274093397128652057411, 5.58404774455847619878433390470, 6.27974157867221583863041133764, 7.37882186968181741335174974399, 8.003322234917908542258292407072, 8.416844845775407031458014684867

Graph of the ZZ-function along the critical line