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Properties

Label 2-4788-1.1-c1-0-11
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  + 5-s + 7-s − 6.09·11-s − 2.23·13-s + 7.61·17-s − 19-s + 8.70·23-s − 4·25-s − 0.854·29-s − 2.38·31-s + 35-s − 6.70·37-s − 0.0901·41-s + 8.47·43-s − 4.70·47-s + 49-s + 14.3·53-s − 6.09·55-s + 10.7·59-s + 5.47·61-s − 2.23·65-s − 2.09·67-s + 11.1·71-s − 0.909·73-s − 6.09·77-s − 6.94·79-s − 14.6·83-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.377·7-s − 1.83·11-s − 0.620·13-s + 1.84·17-s − 0.229·19-s + 1.81·23-s − 0.800·25-s − 0.158·29-s − 0.427·31-s + 0.169·35-s − 1.10·37-s − 0.0140·41-s + 1.29·43-s − 0.686·47-s + 0.142·49-s + 1.96·53-s − 0.821·55-s + 1.39·59-s + 0.700·61-s − 0.277·65-s − 0.255·67-s + 1.32·71-s − 0.106·73-s − 0.694·77-s − 0.781·79-s − 1.60·83-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 1.9387210161.938721016
L(12)L(\frac12) ≈\approx 1.9387210161.938721016
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−T+5T2 1 - T + 5T^{2}
11 1+6.09T+11T2 1 + 6.09T + 11T^{2}
13 1+2.23T+13T2 1 + 2.23T + 13T^{2}
17 1−7.61T+17T2 1 - 7.61T + 17T^{2}
23 1−8.70T+23T2 1 - 8.70T + 23T^{2}
29 1+0.854T+29T2 1 + 0.854T + 29T^{2}
31 1+2.38T+31T2 1 + 2.38T + 31T^{2}
37 1+6.70T+37T2 1 + 6.70T + 37T^{2}
41 1+0.0901T+41T2 1 + 0.0901T + 41T^{2}
43 1−8.47T+43T2 1 - 8.47T + 43T^{2}
47 1+4.70T+47T2 1 + 4.70T + 47T^{2}
53 1−14.3T+53T2 1 - 14.3T + 53T^{2}
59 1−10.7T+59T2 1 - 10.7T + 59T^{2}
61 1−5.47T+61T2 1 - 5.47T + 61T^{2}
67 1+2.09T+67T2 1 + 2.09T + 67T^{2}
71 1−11.1T+71T2 1 - 11.1T + 71T^{2}
73 1+0.909T+73T2 1 + 0.909T + 73T^{2}
79 1+6.94T+79T2 1 + 6.94T + 79T^{2}
83 1+14.6T+83T2 1 + 14.6T + 83T^{2}
89 1−12.1T+89T2 1 - 12.1T + 89T^{2}
97 1−8.70T+97T2 1 - 8.70T + 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

−8.236199094675516919659078092017, −7.44898231417672297571748422030, −7.15686561583390945993862983723, −5.79575393009660039737125421828, −5.37421634891878140555700028969, −4.86840392221287782521119728974, −3.63009913458997148789985714402, −2.78958210214579634829878740168, −2.03847630787601743863151891662, −0.76054146354421515469178553735, 0.76054146354421515469178553735, 2.03847630787601743863151891662, 2.78958210214579634829878740168, 3.63009913458997148789985714402, 4.86840392221287782521119728974, 5.37421634891878140555700028969, 5.79575393009660039737125421828, 7.15686561583390945993862983723, 7.44898231417672297571748422030, 8.236199094675516919659078092017

Graph of the ZZ-function along the critical line