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Properties

Label 2-4788-1.1-c1-0-13
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  + 3.64·5-s − 7-s − 5.29·11-s − 0.354·17-s + 19-s − 5.29·23-s + 8.29·25-s + 9.64·29-s − 6·31-s − 3.64·35-s + 5.29·37-s + 3.29·41-s − 1.29·43-s + 4.35·47-s + 49-s + 13.6·53-s − 19.2·55-s + 8·59-s + 9.29·61-s + 7.29·67-s + 8.93·71-s − 2·73-s + 5.29·77-s − 3.29·79-s + 14.9·83-s − 1.29·85-s + 3.29·89-s + ⋯
L(s)  = 1  + 1.63·5-s − 0.377·7-s − 1.59·11-s − 0.0859·17-s + 0.229·19-s − 1.10·23-s + 1.65·25-s + 1.79·29-s − 1.07·31-s − 0.616·35-s + 0.869·37-s + 0.514·41-s − 0.196·43-s + 0.635·47-s + 0.142·49-s + 1.87·53-s − 2.60·55-s + 1.04·59-s + 1.18·61-s + 0.890·67-s + 1.06·71-s − 0.234·73-s + 0.603·77-s − 0.370·79-s + 1.63·83-s − 0.140·85-s + 0.348·89-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 2.3028931852.302893185
L(12)L(\frac12) ≈\approx 2.3028931852.302893185
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 1−3.64T+5T2 1 - 3.64T + 5T^{2}
11 1+5.29T+11T2 1 + 5.29T + 11T^{2}
13 1+13T2 1 + 13T^{2}
17 1+0.354T+17T2 1 + 0.354T + 17T^{2}
23 1+5.29T+23T2 1 + 5.29T + 23T^{2}
29 1−9.64T+29T2 1 - 9.64T + 29T^{2}
31 1+6T+31T2 1 + 6T + 31T^{2}
37 1−5.29T+37T2 1 - 5.29T + 37T^{2}
41 1−3.29T+41T2 1 - 3.29T + 41T^{2}
43 1+1.29T+43T2 1 + 1.29T + 43T^{2}
47 1−4.35T+47T2 1 - 4.35T + 47T^{2}
53 1−13.6T+53T2 1 - 13.6T + 53T^{2}
59 1−8T+59T2 1 - 8T + 59T^{2}
61 1−9.29T+61T2 1 - 9.29T + 61T^{2}
67 1−7.29T+67T2 1 - 7.29T + 67T^{2}
71 1−8.93T+71T2 1 - 8.93T + 71T^{2}
73 1+2T+73T2 1 + 2T + 73T^{2}
79 1+3.29T+79T2 1 + 3.29T + 79T^{2}
83 1−14.9T+83T2 1 - 14.9T + 83T^{2}
89 1−3.29T+89T2 1 - 3.29T + 89T^{2}
97 1−4.58T+97T2 1 - 4.58T + 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

−8.342498511489194493128300538292, −7.55514568041951029706274017288, −6.70885970927139041592557610147, −5.99358146865372199238851536794, −5.46523503775127360361047023021, −4.83611213004660297604455525197, −3.66354092059792735001229663505, −2.45024797473173932349400087792, −2.29569875313131728817694321872, −0.822697844479424147022668890964, 0.822697844479424147022668890964, 2.29569875313131728817694321872, 2.45024797473173932349400087792, 3.66354092059792735001229663505, 4.83611213004660297604455525197, 5.46523503775127360361047023021, 5.99358146865372199238851536794, 6.70885970927139041592557610147, 7.55514568041951029706274017288, 8.342498511489194493128300538292

Graph of the ZZ-function along the critical line