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Properties

Label 2-7488-1.1-c1-0-48
Degree 22
Conductor 74887488
Sign 11
Analytic cond. 59.791959.7919
Root an. cond. 7.732527.73252
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s + 3·7-s + 2·11-s − 13-s + 3·17-s − 2·19-s − 4·23-s − 4·25-s + 2·29-s + 4·31-s + 3·35-s − 5·37-s + 12·41-s − 7·43-s + 9·47-s + 2·49-s + 4·53-s + 2·55-s + 6·59-s + 4·61-s − 65-s + 10·67-s + 15·71-s − 2·73-s + 6·77-s − 8·79-s − 4·83-s + ⋯
L(s)  = 1  + 0.447·5-s + 1.13·7-s + 0.603·11-s − 0.277·13-s + 0.727·17-s − 0.458·19-s − 0.834·23-s − 4/5·25-s + 0.371·29-s + 0.718·31-s + 0.507·35-s − 0.821·37-s + 1.87·41-s − 1.06·43-s + 1.31·47-s + 2/7·49-s + 0.549·53-s + 0.269·55-s + 0.781·59-s + 0.512·61-s − 0.124·65-s + 1.22·67-s + 1.78·71-s − 0.234·73-s + 0.683·77-s − 0.900·79-s − 0.439·83-s + ⋯

Functional equation

Λ(s)=(7488s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 7488 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(7488s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 7488 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 74887488    =    26⋅32⋅132^{6} \cdot 3^{2} \cdot 13
Sign: 11
Analytic conductor: 59.791959.7919
Root analytic conductor: 7.732527.73252
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 7488, ( :1/2), 1)(2,\ 7488,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.7721933022.772193302
L(12)L(\frac12) ≈\approx 2.7721933022.772193302
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
13 1+T 1 + T
good5 1−T+pT2 1 - T + p T^{2}
7 1−3T+pT2 1 - 3 T + p T^{2}
11 1−2T+pT2 1 - 2 T + p T^{2}
17 1−3T+pT2 1 - 3 T + p T^{2}
19 1+2T+pT2 1 + 2 T + p T^{2}
23 1+4T+pT2 1 + 4 T + p T^{2}
29 1−2T+pT2 1 - 2 T + p T^{2}
31 1−4T+pT2 1 - 4 T + p T^{2}
37 1+5T+pT2 1 + 5 T + p T^{2}
41 1−12T+pT2 1 - 12 T + p T^{2}
43 1+7T+pT2 1 + 7 T + p T^{2}
47 1−9T+pT2 1 - 9 T + p T^{2}
53 1−4T+pT2 1 - 4 T + p T^{2}
59 1−6T+pT2 1 - 6 T + p T^{2}
61 1−4T+pT2 1 - 4 T + p T^{2}
67 1−10T+pT2 1 - 10 T + p T^{2}
71 1−15T+pT2 1 - 15 T + p T^{2}
73 1+2T+pT2 1 + 2 T + p T^{2}
79 1+8T+pT2 1 + 8 T + p T^{2}
83 1+4T+pT2 1 + 4 T + p T^{2}
89 1+2T+pT2 1 + 2 T + p T^{2}
97 1−10T+pT2 1 - 10 T + p T^{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

−7.964703245594772929984299285530, −7.26452379598760388338469331118, −6.44784880927663012596041055664, −5.75101064463899872805706654704, −5.13453009003348958858406129103, −4.31169037721870521245593585722, −3.69882720223680610508104183144, −2.47414386983859427575410557935, −1.83509402100962992787287050734, −0.870411349685264859192266368730, 0.870411349685264859192266368730, 1.83509402100962992787287050734, 2.47414386983859427575410557935, 3.69882720223680610508104183144, 4.31169037721870521245593585722, 5.13453009003348958858406129103, 5.75101064463899872805706654704, 6.44784880927663012596041055664, 7.26452379598760388338469331118, 7.964703245594772929984299285530

Graph of the ZZ-function along the critical line