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Properties

Label 2-3888-1.1-c1-0-39
Degree 22
Conductor 38883888
Sign 11
Analytic cond. 31.045831.0458
Root an. cond. 5.571875.57187
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.46·5-s + 7-s + 3.46·11-s + 5·13-s + 19-s + 6.92·23-s + 6.99·25-s − 3.46·29-s − 5·31-s + 3.46·35-s − 37-s + 3.46·41-s + 43-s + 3.46·47-s − 6·49-s − 10.3·53-s + 11.9·55-s − 3.46·59-s + 2·61-s + 17.3·65-s − 8·67-s − 10.3·71-s + 2·73-s + 3.46·77-s + 79-s − 6.92·83-s + 10.3·89-s + ⋯
L(s)  = 1  + 1.54·5-s + 0.377·7-s + 1.04·11-s + 1.38·13-s + 0.229·19-s + 1.44·23-s + 1.39·25-s − 0.643·29-s − 0.898·31-s + 0.585·35-s − 0.164·37-s + 0.541·41-s + 0.152·43-s + 0.505·47-s − 0.857·49-s − 1.42·53-s + 1.61·55-s − 0.450·59-s + 0.256·61-s + 2.14·65-s − 0.977·67-s − 1.23·71-s + 0.234·73-s + 0.394·77-s + 0.112·79-s − 0.760·83-s + 1.10·89-s + ⋯

Functional equation

Λ(s)=(3888s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(3888s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3888 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 38883888    =    24⋅352^{4} \cdot 3^{5}
Sign: 11
Analytic conductor: 31.045831.0458
Root analytic conductor: 5.571875.57187
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 3888, ( :1/2), 1)(2,\ 3888,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 3.3023877583.302387758
L(12)L(\frac12) ≈\approx 3.3023877583.302387758
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
good5 1−3.46T+5T2 1 - 3.46T + 5T^{2}
7 1−T+7T2 1 - T + 7T^{2}
11 1−3.46T+11T2 1 - 3.46T + 11T^{2}
13 1−5T+13T2 1 - 5T + 13T^{2}
17 1+17T2 1 + 17T^{2}
19 1−T+19T2 1 - T + 19T^{2}
23 1−6.92T+23T2 1 - 6.92T + 23T^{2}
29 1+3.46T+29T2 1 + 3.46T + 29T^{2}
31 1+5T+31T2 1 + 5T + 31T^{2}
37 1+T+37T2 1 + T + 37T^{2}
41 1−3.46T+41T2 1 - 3.46T + 41T^{2}
43 1−T+43T2 1 - T + 43T^{2}
47 1−3.46T+47T2 1 - 3.46T + 47T^{2}
53 1+10.3T+53T2 1 + 10.3T + 53T^{2}
59 1+3.46T+59T2 1 + 3.46T + 59T^{2}
61 1−2T+61T2 1 - 2T + 61T^{2}
67 1+8T+67T2 1 + 8T + 67T^{2}
71 1+10.3T+71T2 1 + 10.3T + 71T^{2}
73 1−2T+73T2 1 - 2T + 73T^{2}
79 1−T+79T2 1 - T + 79T^{2}
83 1+6.92T+83T2 1 + 6.92T + 83T^{2}
89 1−10.3T+89T2 1 - 10.3T + 89T^{2}
97 1−17T+97T2 1 - 17T + 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

−8.823148884472770583686001033090, −7.72253562023634107256833095808, −6.81574589779543637368539971185, −6.19601898208036417515526512565, −5.63429678017844926628142228106, −4.83706904053187060669113718097, −3.81573367725641187139774016798, −2.92573440025065869437037121154, −1.72304635608116323224123903823, −1.23579631315049212244451551519, 1.23579631315049212244451551519, 1.72304635608116323224123903823, 2.92573440025065869437037121154, 3.81573367725641187139774016798, 4.83706904053187060669113718097, 5.63429678017844926628142228106, 6.19601898208036417515526512565, 6.81574589779543637368539971185, 7.72253562023634107256833095808, 8.823148884472770583686001033090

Graph of the ZZ-function along the critical line