↑

Properties

Label 2-464-1.1-c3-0-8
Degree 22
Conductor 464464
Sign 11
Analytic cond. 27.376827.3768
Root an. cond. 5.232295.23229
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 6.13·3-s + 2.36·5-s + 18.5·7-s + 10.6·9-s + 15.3·11-s + 27.8·13-s − 14.4·15-s − 62.4·17-s − 55.3·19-s − 113.·21-s − 44.3·23-s − 119.·25-s + 100.·27-s + 29·29-s + 207.·31-s − 94.3·33-s + 43.6·35-s + 303.·37-s − 170.·39-s + 125.·41-s − 101.·43-s + 25.0·45-s − 50.8·47-s − 0.583·49-s + 382.·51-s + 692.·53-s + 36.3·55-s + ⋯
L(s)  = 1  − 1.18·3-s + 0.211·5-s + 0.999·7-s + 0.393·9-s + 0.421·11-s + 0.593·13-s − 0.249·15-s − 0.890·17-s − 0.668·19-s − 1.17·21-s − 0.401·23-s − 0.955·25-s + 0.716·27-s + 0.185·29-s + 1.19·31-s − 0.497·33-s + 0.210·35-s + 1.34·37-s − 0.700·39-s + 0.478·41-s − 0.359·43-s + 0.0829·45-s − 0.157·47-s − 0.00170·49-s + 1.05·51-s + 1.79·53-s + 0.0890·55-s + ⋯

Functional equation

Λ(s)=(464s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(464s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 464 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 464464    =    24⋅292^{4} \cdot 29
Sign: 11
Analytic conductor: 27.376827.3768
Root analytic conductor: 5.232295.23229
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 464, ( :3/2), 1)(2,\ 464,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 1.3735653931.373565393
L(12)L(\frac12) ≈\approx 1.3735653931.373565393
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)
bad2 1 1
29 1−29T 1 - 29T
good3 1+6.13T+27T2 1 + 6.13T + 27T^{2}
5 1−2.36T+125T2 1 - 2.36T + 125T^{2}
7 1−18.5T+343T2 1 - 18.5T + 343T^{2}
11 1−15.3T+1.33e3T2 1 - 15.3T + 1.33e3T^{2}
13 1−27.8T+2.19e3T2 1 - 27.8T + 2.19e3T^{2}
17 1+62.4T+4.91e3T2 1 + 62.4T + 4.91e3T^{2}
19 1+55.3T+6.85e3T2 1 + 55.3T + 6.85e3T^{2}
23 1+44.3T+1.21e4T2 1 + 44.3T + 1.21e4T^{2}
31 1−207.T+2.97e4T2 1 - 207.T + 2.97e4T^{2}
37 1−303.T+5.06e4T2 1 - 303.T + 5.06e4T^{2}
41 1−125.T+6.89e4T2 1 - 125.T + 6.89e4T^{2}
43 1+101.T+7.95e4T2 1 + 101.T + 7.95e4T^{2}
47 1+50.8T+1.03e5T2 1 + 50.8T + 1.03e5T^{2}
53 1−692.T+1.48e5T2 1 - 692.T + 1.48e5T^{2}
59 1+557.T+2.05e5T2 1 + 557.T + 2.05e5T^{2}
61 1−809.T+2.26e5T2 1 - 809.T + 2.26e5T^{2}
67 1−749.T+3.00e5T2 1 - 749.T + 3.00e5T^{2}
71 1−54.7T+3.57e5T2 1 - 54.7T + 3.57e5T^{2}
73 1+184.T+3.89e5T2 1 + 184.T + 3.89e5T^{2}
79 1−752.T+4.93e5T2 1 - 752.T + 4.93e5T^{2}
83 1−902.T+5.71e5T2 1 - 902.T + 5.71e5T^{2}
89 1−953.T+7.04e5T2 1 - 953.T + 7.04e5T^{2}
97 1+1.11e3T+9.12e5T2 1 + 1.11e3T + 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

−10.89413929168215486096150024725, −9.954568727924472185916418233730, −8.753313449214129755316438224850, −7.946121515845367842464592195337, −6.59246006138717735587416733633, −5.99944551614285643960280578844, −4.94430085745551361012097073739, −4.08721955136850121331935798714, −2.17127880065591908178462290516, −0.799316347517982804200599575553, 0.799316347517982804200599575553, 2.17127880065591908178462290516, 4.08721955136850121331935798714, 4.94430085745551361012097073739, 5.99944551614285643960280578844, 6.59246006138717735587416733633, 7.946121515845367842464592195337, 8.753313449214129755316438224850, 9.954568727924472185916418233730, 10.89413929168215486096150024725

Graph of the ZZ-function along the critical line