↑

Properties

Label 2-4140-1.1-c1-0-13
Degree 22
Conductor 41404140
Sign 11
Analytic cond. 33.058033.0580
Root an. cond. 5.749615.74961
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  − 5-s + 4.46·7-s + 6.19·11-s − 6.73·13-s + 4.26·17-s − 2.73·19-s + 23-s + 25-s + 3.19·29-s − 31-s − 4.46·35-s + 9.39·37-s + 4.26·41-s − 3.46·43-s + 2.73·47-s + 12.9·49-s + 10.6·53-s − 6.19·55-s − 3.19·59-s − 7.26·61-s + 6.73·65-s + 5·67-s − 10.1·71-s − 14.7·73-s + 27.6·77-s + 4·79-s − 10.6·83-s + ⋯
L(s)  = 1  − 0.447·5-s + 1.68·7-s + 1.86·11-s − 1.86·13-s + 1.03·17-s − 0.626·19-s + 0.208·23-s + 0.200·25-s + 0.593·29-s − 0.179·31-s − 0.754·35-s + 1.54·37-s + 0.666·41-s − 0.528·43-s + 0.398·47-s + 1.84·49-s + 1.46·53-s − 0.835·55-s − 0.416·59-s − 0.930·61-s + 0.835·65-s + 0.610·67-s − 1.20·71-s − 1.72·73-s + 3.15·77-s + 0.450·79-s − 1.17·83-s + ⋯

Functional equation

Λ(s)=(4140s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(4140s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 41404140    =    22⋅32⋅5⋅232^{2} \cdot 3^{2} \cdot 5 \cdot 23
Sign: 11
Analytic conductor: 33.058033.0580
Root analytic conductor: 5.749615.74961
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 4140, ( :1/2), 1)(2,\ 4140,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.3954393972.395439397
L(12)L(\frac12) ≈\approx 2.3954393972.395439397
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
23 1−T 1 - T
good7 1−4.46T+7T2 1 - 4.46T + 7T^{2}
11 1−6.19T+11T2 1 - 6.19T + 11T^{2}
13 1+6.73T+13T2 1 + 6.73T + 13T^{2}
17 1−4.26T+17T2 1 - 4.26T + 17T^{2}
19 1+2.73T+19T2 1 + 2.73T + 19T^{2}
29 1−3.19T+29T2 1 - 3.19T + 29T^{2}
31 1+T+31T2 1 + T + 31T^{2}
37 1−9.39T+37T2 1 - 9.39T + 37T^{2}
41 1−4.26T+41T2 1 - 4.26T + 41T^{2}
43 1+3.46T+43T2 1 + 3.46T + 43T^{2}
47 1−2.73T+47T2 1 - 2.73T + 47T^{2}
53 1−10.6T+53T2 1 - 10.6T + 53T^{2}
59 1+3.19T+59T2 1 + 3.19T + 59T^{2}
61 1+7.26T+61T2 1 + 7.26T + 61T^{2}
67 1−5T+67T2 1 - 5T + 67T^{2}
71 1+10.1T+71T2 1 + 10.1T + 71T^{2}
73 1+14.7T+73T2 1 + 14.7T + 73T^{2}
79 1−4T+79T2 1 - 4T + 79T^{2}
83 1+10.6T+83T2 1 + 10.6T + 83T^{2}
89 1+4.39T+89T2 1 + 4.39T + 89T^{2}
97 1−4T+97T2 1 - 4T + 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

−8.398394972117492602225408327999, −7.55491602449439102140022244576, −7.25851005391246496524177318065, −6.22161442695359963300102650094, −5.31749171100122395914654245978, −4.47178189430518470967501179811, −4.17806601028777823618010976529, −2.87801811757684074313834299640, −1.84670153597132589616586768874, −0.944740134060137521817513579406, 0.944740134060137521817513579406, 1.84670153597132589616586768874, 2.87801811757684074313834299640, 4.17806601028777823618010976529, 4.47178189430518470967501179811, 5.31749171100122395914654245978, 6.22161442695359963300102650094, 7.25851005391246496524177318065, 7.55491602449439102140022244576, 8.398394972117492602225408327999

Graph of the ZZ-function along the critical line