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Properties

Label 2-4840-1.1-c1-0-99
Degree 22
Conductor 48404840
Sign −1-1
Analytic cond. 38.647538.6475
Root an. cond. 6.216716.21671
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  + 0.294·3-s + 5-s + 4.39·7-s − 2.91·9-s − 5.96·13-s + 0.294·15-s + 1.66·17-s − 7.69·19-s + 1.29·21-s − 0.904·23-s + 25-s − 1.74·27-s + 4.73·29-s − 5.12·31-s + 4.39·35-s − 0.184·37-s − 1.75·39-s + 2.62·41-s − 3.59·43-s − 2.91·45-s − 0.776·47-s + 12.2·49-s + 0.491·51-s − 9.59·53-s − 2.26·57-s − 11.2·59-s − 13.1·61-s + ⋯
L(s)  = 1  + 0.170·3-s + 0.447·5-s + 1.65·7-s − 0.970·9-s − 1.65·13-s + 0.0761·15-s + 0.404·17-s − 1.76·19-s + 0.282·21-s − 0.188·23-s + 0.200·25-s − 0.335·27-s + 0.880·29-s − 0.920·31-s + 0.742·35-s − 0.0304·37-s − 0.281·39-s + 0.409·41-s − 0.548·43-s − 0.434·45-s − 0.113·47-s + 1.75·49-s + 0.0688·51-s − 1.31·53-s − 0.300·57-s − 1.46·59-s − 1.68·61-s + ⋯

Functional equation

Λ(s)=(4840s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(4840s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 48404840    =    23⋅5⋅1122^{3} \cdot 5 \cdot 11^{2}
Sign: −1-1
Analytic conductor: 38.647538.6475
Root analytic conductor: 6.216716.21671
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 4840, ( :1/2), −1)(2,\ 4840,\ (\ :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
5 1−T 1 - T
11 1 1
good3 1−0.294T+3T2 1 - 0.294T + 3T^{2}
7 1−4.39T+7T2 1 - 4.39T + 7T^{2}
13 1+5.96T+13T2 1 + 5.96T + 13T^{2}
17 1−1.66T+17T2 1 - 1.66T + 17T^{2}
19 1+7.69T+19T2 1 + 7.69T + 19T^{2}
23 1+0.904T+23T2 1 + 0.904T + 23T^{2}
29 1−4.73T+29T2 1 - 4.73T + 29T^{2}
31 1+5.12T+31T2 1 + 5.12T + 31T^{2}
37 1+0.184T+37T2 1 + 0.184T + 37T^{2}
41 1−2.62T+41T2 1 - 2.62T + 41T^{2}
43 1+3.59T+43T2 1 + 3.59T + 43T^{2}
47 1+0.776T+47T2 1 + 0.776T + 47T^{2}
53 1+9.59T+53T2 1 + 9.59T + 53T^{2}
59 1+11.2T+59T2 1 + 11.2T + 59T^{2}
61 1+13.1T+61T2 1 + 13.1T + 61T^{2}
67 1+7.79T+67T2 1 + 7.79T + 67T^{2}
71 1+6.97T+71T2 1 + 6.97T + 71T^{2}
73 1+12.7T+73T2 1 + 12.7T + 73T^{2}
79 1−10.0T+79T2 1 - 10.0T + 79T^{2}
83 1+4.09T+83T2 1 + 4.09T + 83T^{2}
89 1+0.466T+89T2 1 + 0.466T + 89T^{2}
97 1−7.09T+97T2 1 - 7.09T + 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

−7.84566801402113528265155240969, −7.48199636503386561344045474801, −6.34847206348759981062453655963, −5.69741222197338606129086820326, −4.74665164591068857611148800468, −4.57516014815393567223352252220, −3.12815055525495291451958033175, −2.27158304820024163044189897424, −1.65009302910992829969571358442, 0, 1.65009302910992829969571358442, 2.27158304820024163044189897424, 3.12815055525495291451958033175, 4.57516014815393567223352252220, 4.74665164591068857611148800468, 5.69741222197338606129086820326, 6.34847206348759981062453655963, 7.48199636503386561344045474801, 7.84566801402113528265155240969

Graph of the ZZ-function along the critical line