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Properties

Label 2-76e2-1.1-c1-0-57
Degree 22
Conductor 57765776
Sign 11
Analytic cond. 46.121546.1215
Root an. cond. 6.791286.79128
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  − 2·3-s − 5-s + 3·7-s + 9-s + 3·11-s + 4·13-s + 2·15-s + 5·17-s − 6·21-s − 4·25-s + 4·27-s − 2·29-s + 8·31-s − 6·33-s − 3·35-s + 10·37-s − 8·39-s − 6·41-s + 7·43-s − 45-s + 9·47-s + 2·49-s − 10·51-s + 8·53-s − 3·55-s + 14·59-s − 5·61-s + ⋯
L(s)  = 1  − 1.15·3-s − 0.447·5-s + 1.13·7-s + 1/3·9-s + 0.904·11-s + 1.10·13-s + 0.516·15-s + 1.21·17-s − 1.30·21-s − 4/5·25-s + 0.769·27-s − 0.371·29-s + 1.43·31-s − 1.04·33-s − 0.507·35-s + 1.64·37-s − 1.28·39-s − 0.937·41-s + 1.06·43-s − 0.149·45-s + 1.31·47-s + 2/7·49-s − 1.40·51-s + 1.09·53-s − 0.404·55-s + 1.82·59-s − 0.640·61-s + ⋯

Functional equation

Λ(s)=(5776s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 5776 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(5776s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 5776 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 57765776    =    24⋅1922^{4} \cdot 19^{2}
Sign: 11
Analytic conductor: 46.121546.1215
Root analytic conductor: 6.791286.79128
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 5776, ( :1/2), 1)(2,\ 5776,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.6727375171.672737517
L(12)L(\frac12) ≈\approx 1.6727375171.672737517
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
19 1 1
good3 1+2T+pT2 1 + 2 T + p T^{2}
5 1+T+pT2 1 + T + p T^{2}
7 1−3T+pT2 1 - 3 T + p T^{2}
11 1−3T+pT2 1 - 3 T + p T^{2}
13 1−4T+pT2 1 - 4 T + p T^{2}
17 1−5T+pT2 1 - 5 T + p T^{2}
23 1+pT2 1 + p T^{2}
29 1+2T+pT2 1 + 2 T + p T^{2}
31 1−8T+pT2 1 - 8 T + p T^{2}
37 1−10T+pT2 1 - 10 T + p T^{2}
41 1+6T+pT2 1 + 6 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 1−8T+pT2 1 - 8 T + p T^{2}
59 1−14T+pT2 1 - 14 T + p T^{2}
61 1+5T+pT2 1 + 5 T + p T^{2}
67 1+pT2 1 + p T^{2}
71 1+6T+pT2 1 + 6 T + p T^{2}
73 1+15T+pT2 1 + 15 T + p T^{2}
79 1+4T+pT2 1 + 4 T + p T^{2}
83 1+4T+pT2 1 + 4 T + p T^{2}
89 1+pT2 1 + p T^{2}
97 1+16T+pT2 1 + 16 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

−8.148060259363710086433322724847, −7.38601845396983265703380778176, −6.58341608756454848978175875806, −5.75508495262059909206416744964, −5.51991869338759908788687173901, −4.33693159410665846542938642673, −4.05153477392287557754730772390, −2.83555716155125599118268588753, −1.44011806260301581515132764673, −0.833864002051176659308197110715, 0.833864002051176659308197110715, 1.44011806260301581515132764673, 2.83555716155125599118268588753, 4.05153477392287557754730772390, 4.33693159410665846542938642673, 5.51991869338759908788687173901, 5.75508495262059909206416744964, 6.58341608756454848978175875806, 7.38601845396983265703380778176, 8.148060259363710086433322724847

Graph of the ZZ-function along the critical line