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Properties

Label 2-4788-1.1-c1-0-23
Degree 22
Conductor 47884788
Sign 11
Analytic cond. 38.232338.2323
Root an. cond. 6.183236.18323
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  + 3.23·5-s + 7-s + 2·11-s + 4.47·13-s − 3.23·17-s − 19-s + 2·23-s + 5.47·25-s + 2.76·29-s + 4·31-s + 3.23·35-s − 4.47·37-s + 2.47·41-s − 1.52·43-s + 4.76·47-s + 49-s + 10.1·53-s + 6.47·55-s − 4.94·59-s − 8.47·61-s + 14.4·65-s − 12.9·67-s + 2.76·71-s + 6·73-s + 2·77-s + 0.944·79-s + 11.2·83-s + ⋯
L(s)  = 1  + 1.44·5-s + 0.377·7-s + 0.603·11-s + 1.24·13-s − 0.784·17-s − 0.229·19-s + 0.417·23-s + 1.09·25-s + 0.513·29-s + 0.718·31-s + 0.546·35-s − 0.735·37-s + 0.386·41-s − 0.232·43-s + 0.694·47-s + 0.142·49-s + 1.39·53-s + 0.872·55-s − 0.643·59-s − 1.08·61-s + 1.79·65-s − 1.58·67-s + 0.328·71-s + 0.702·73-s + 0.227·77-s + 0.106·79-s + 1.23·83-s + ⋯

Functional equation

Λ(s)=(4788s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4788 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(4788s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4788 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 47884788    =    22⋅32⋅7⋅192^{2} \cdot 3^{2} \cdot 7 \cdot 19
Sign: 11
Analytic conductor: 38.232338.2323
Root analytic conductor: 6.183236.18323
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 4788, ( :1/2), 1)(2,\ 4788,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 3.1486640973.148664097
L(12)L(\frac12) ≈\approx 3.1486640973.148664097
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
7 1−T 1 - T
19 1+T 1 + T
good5 1−3.23T+5T2 1 - 3.23T + 5T^{2}
11 1−2T+11T2 1 - 2T + 11T^{2}
13 1−4.47T+13T2 1 - 4.47T + 13T^{2}
17 1+3.23T+17T2 1 + 3.23T + 17T^{2}
23 1−2T+23T2 1 - 2T + 23T^{2}
29 1−2.76T+29T2 1 - 2.76T + 29T^{2}
31 1−4T+31T2 1 - 4T + 31T^{2}
37 1+4.47T+37T2 1 + 4.47T + 37T^{2}
41 1−2.47T+41T2 1 - 2.47T + 41T^{2}
43 1+1.52T+43T2 1 + 1.52T + 43T^{2}
47 1−4.76T+47T2 1 - 4.76T + 47T^{2}
53 1−10.1T+53T2 1 - 10.1T + 53T^{2}
59 1+4.94T+59T2 1 + 4.94T + 59T^{2}
61 1+8.47T+61T2 1 + 8.47T + 61T^{2}
67 1+12.9T+67T2 1 + 12.9T + 67T^{2}
71 1−2.76T+71T2 1 - 2.76T + 71T^{2}
73 1−6T+73T2 1 - 6T + 73T^{2}
79 1−0.944T+79T2 1 - 0.944T + 79T^{2}
83 1−11.2T+83T2 1 - 11.2T + 83T^{2}
89 1−10.4T+89T2 1 - 10.4T + 89T^{2}
97 1−7.52T+97T2 1 - 7.52T + 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

−8.561266748619750062654643145899, −7.52568362109191971729194594285, −6.49965704325248316658661055143, −6.27630192873524620341954861513, −5.44321374509681900649684391464, −4.65161518070291075287410499799, −3.78878996874181176894084618008, −2.71119375745888847995330347808, −1.84769743631374320526850239478, −1.07111329255288240712707202781, 1.07111329255288240712707202781, 1.84769743631374320526850239478, 2.71119375745888847995330347808, 3.78878996874181176894084618008, 4.65161518070291075287410499799, 5.44321374509681900649684391464, 6.27630192873524620341954861513, 6.49965704325248316658661055143, 7.52568362109191971729194594285, 8.561266748619750062654643145899

Graph of the ZZ-function along the critical line