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Properties

Label 2-3840-1.1-c1-0-54
Degree 22
Conductor 38403840
Sign −1-1
Analytic cond. 30.662530.6625
Root an. cond. 5.537375.53737
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 5-s − 4.82·7-s + 9-s + 0.828·11-s + 2·13-s + 15-s + 5.65·17-s − 6.82·19-s − 4.82·21-s − 1.17·23-s + 25-s + 27-s − 6·29-s − 4·31-s + 0.828·33-s − 4.82·35-s − 7.65·37-s + 2·39-s − 3.65·41-s − 9.65·43-s + 45-s − 5.17·47-s + 16.3·49-s + 5.65·51-s + 9.31·53-s + 0.828·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s − 1.82·7-s + 0.333·9-s + 0.249·11-s + 0.554·13-s + 0.258·15-s + 1.37·17-s − 1.56·19-s − 1.05·21-s − 0.244·23-s + 0.200·25-s + 0.192·27-s − 1.11·29-s − 0.718·31-s + 0.144·33-s − 0.816·35-s − 1.25·37-s + 0.320·39-s − 0.571·41-s − 1.47·43-s + 0.149·45-s − 0.754·47-s + 2.33·49-s + 0.792·51-s + 1.27·53-s + 0.111·55-s + ⋯

Functional equation

Λ(s)=(3840s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(3840s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 38403840    =    28⋅3⋅52^{8} \cdot 3 \cdot 5
Sign: −1-1
Analytic conductor: 30.662530.6625
Root analytic conductor: 5.537375.53737
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 3840, ( :1/2), −1)(2,\ 3840,\ (\ :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−T 1 - T
5 1−T 1 - T
good7 1+4.82T+7T2 1 + 4.82T + 7T^{2}
11 1−0.828T+11T2 1 - 0.828T + 11T^{2}
13 1−2T+13T2 1 - 2T + 13T^{2}
17 1−5.65T+17T2 1 - 5.65T + 17T^{2}
19 1+6.82T+19T2 1 + 6.82T + 19T^{2}
23 1+1.17T+23T2 1 + 1.17T + 23T^{2}
29 1+6T+29T2 1 + 6T + 29T^{2}
31 1+4T+31T2 1 + 4T + 31T^{2}
37 1+7.65T+37T2 1 + 7.65T + 37T^{2}
41 1+3.65T+41T2 1 + 3.65T + 41T^{2}
43 1+9.65T+43T2 1 + 9.65T + 43T^{2}
47 1+5.17T+47T2 1 + 5.17T + 47T^{2}
53 1−9.31T+53T2 1 - 9.31T + 53T^{2}
59 1−14.4T+59T2 1 - 14.4T + 59T^{2}
61 1+4T+61T2 1 + 4T + 61T^{2}
67 1−4T+67T2 1 - 4T + 67T^{2}
71 1−5.65T+71T2 1 - 5.65T + 71T^{2}
73 1+13.3T+73T2 1 + 13.3T + 73T^{2}
79 1+4T+79T2 1 + 4T + 79T^{2}
83 1−2.34T+83T2 1 - 2.34T + 83T^{2}
89 1−0.343T+89T2 1 - 0.343T + 89T^{2}
97 1+14T+97T2 1 + 14T + 97T^{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

−8.353928403589865262204718743251, −7.20645523713973161029720978273, −6.68967407576349151280366566906, −5.99077263816598197128166679878, −5.28678265527901992328333751890, −3.80506209521218478110385258156, −3.58721319899676426299426398163, −2.58981933385541535918782042168, −1.56778293428891148188984382172, 0, 1.56778293428891148188984382172, 2.58981933385541535918782042168, 3.58721319899676426299426398163, 3.80506209521218478110385258156, 5.28678265527901992328333751890, 5.99077263816598197128166679878, 6.68967407576349151280366566906, 7.20645523713973161029720978273, 8.353928403589865262204718743251

Graph of the ZZ-function along the critical line