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Properties

Label 2-2e8-1.1-c3-0-21
Degree 22
Conductor 256256
Sign −1-1
Analytic cond. 15.104415.1044
Root an. cond. 3.886443.88644
Motivic weight 33
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 8·3-s − 12·5-s − 32·7-s + 37·9-s − 8·11-s + 20·13-s − 96·15-s − 98·17-s − 88·19-s − 256·21-s + 32·23-s + 19·25-s + 80·27-s − 172·29-s + 256·31-s − 64·33-s + 384·35-s − 92·37-s + 160·39-s + 102·41-s − 296·43-s − 444·45-s + 320·47-s + 681·49-s − 784·51-s − 76·53-s + 96·55-s + ⋯
L(s)  = 1  + 1.53·3-s − 1.07·5-s − 1.72·7-s + 1.37·9-s − 0.219·11-s + 0.426·13-s − 1.65·15-s − 1.39·17-s − 1.06·19-s − 2.66·21-s + 0.290·23-s + 0.151·25-s + 0.570·27-s − 1.10·29-s + 1.48·31-s − 0.337·33-s + 1.85·35-s − 0.408·37-s + 0.656·39-s + 0.388·41-s − 1.04·43-s − 1.47·45-s + 0.993·47-s + 1.98·49-s − 2.15·51-s − 0.196·53-s + 0.235·55-s + ⋯

Functional equation

Λ(s)=(256s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(256s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 256256    =    282^{8}
Sign: −1-1
Analytic conductor: 15.104415.1044
Root analytic conductor: 3.886443.88644
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 256, ( :3/2), −1)(2,\ 256,\ (\ :3/2),\ -1)

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
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
good3 1−8T+p3T2 1 - 8 T + p^{3} T^{2}
5 1+12T+p3T2 1 + 12 T + p^{3} T^{2}
7 1+32T+p3T2 1 + 32 T + p^{3} T^{2}
11 1+8T+p3T2 1 + 8 T + p^{3} T^{2}
13 1−20T+p3T2 1 - 20 T + p^{3} T^{2}
17 1+98T+p3T2 1 + 98 T + p^{3} T^{2}
19 1+88T+p3T2 1 + 88 T + p^{3} T^{2}
23 1−32T+p3T2 1 - 32 T + p^{3} T^{2}
29 1+172T+p3T2 1 + 172 T + p^{3} T^{2}
31 1−256T+p3T2 1 - 256 T + p^{3} T^{2}
37 1+92T+p3T2 1 + 92 T + p^{3} T^{2}
41 1−102T+p3T2 1 - 102 T + p^{3} T^{2}
43 1+296T+p3T2 1 + 296 T + p^{3} T^{2}
47 1−320T+p3T2 1 - 320 T + p^{3} T^{2}
53 1+76T+p3T2 1 + 76 T + p^{3} T^{2}
59 1−408T+p3T2 1 - 408 T + p^{3} T^{2}
61 1+636T+p3T2 1 + 636 T + p^{3} T^{2}
67 1−552T+p3T2 1 - 552 T + p^{3} T^{2}
71 1+416T+p3T2 1 + 416 T + p^{3} T^{2}
73 1−138T+p3T2 1 - 138 T + p^{3} T^{2}
79 1−64T+p3T2 1 - 64 T + p^{3} T^{2}
83 1−392T+p3T2 1 - 392 T + p^{3} T^{2}
89 1+582T+p3T2 1 + 582 T + p^{3} T^{2}
97 1−238T+p3T2 1 - 238 T + p^{3} T^{2}
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   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

−11.00424551040880696546651053123, −9.889264219654524422736722071666, −8.978913783491527809837262997417, −8.347715054493995942484846769909, −7.25699648671309243965857937439, −6.35957109580888645021585828402, −4.21738485745119910644235562172, −3.46965481684877858922540241781, −2.45364195143905733044109761993, 0, 2.45364195143905733044109761993, 3.46965481684877858922540241781, 4.21738485745119910644235562172, 6.35957109580888645021585828402, 7.25699648671309243965857937439, 8.347715054493995942484846769909, 8.978913783491527809837262997417, 9.889264219654524422736722071666, 11.00424551040880696546651053123

Graph of the ZZ-function along the critical line