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Properties

Label 2-4032-1.1-c1-0-30
Degree 22
Conductor 40324032
Sign 11
Analytic cond. 32.195632.1956
Root an. cond. 5.674125.67412
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  + 4·5-s + 7-s − 2·11-s + 6·13-s + 4·17-s − 4·19-s + 2·23-s + 11·25-s − 2·29-s + 4·35-s − 2·37-s − 4·43-s + 12·47-s + 49-s − 6·53-s − 8·55-s + 8·59-s − 6·61-s + 24·65-s − 8·67-s + 14·71-s − 2·73-s − 2·77-s − 12·79-s + 4·83-s + 16·85-s + 6·91-s + ⋯
L(s)  = 1  + 1.78·5-s + 0.377·7-s − 0.603·11-s + 1.66·13-s + 0.970·17-s − 0.917·19-s + 0.417·23-s + 11/5·25-s − 0.371·29-s + 0.676·35-s − 0.328·37-s − 0.609·43-s + 1.75·47-s + 1/7·49-s − 0.824·53-s − 1.07·55-s + 1.04·59-s − 0.768·61-s + 2.97·65-s − 0.977·67-s + 1.66·71-s − 0.234·73-s − 0.227·77-s − 1.35·79-s + 0.439·83-s + 1.73·85-s + 0.628·91-s + ⋯

Functional equation

Λ(s)=(4032s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(4032s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 40324032    =    26⋅32⋅72^{6} \cdot 3^{2} \cdot 7
Sign: 11
Analytic conductor: 32.195632.1956
Root analytic conductor: 5.674125.67412
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 4032, ( :1/2), 1)(2,\ 4032,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 3.1760475113.176047511
L(12)L(\frac12) ≈\approx 3.1760475113.176047511
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
good5 1−4T+pT2 1 - 4 T + p T^{2}
11 1+2T+pT2 1 + 2 T + p T^{2}
13 1−6T+pT2 1 - 6 T + p T^{2}
17 1−4T+pT2 1 - 4 T + p T^{2}
19 1+4T+pT2 1 + 4 T + p T^{2}
23 1−2T+pT2 1 - 2 T + p T^{2}
29 1+2T+pT2 1 + 2 T + p T^{2}
31 1+pT2 1 + p T^{2}
37 1+2T+pT2 1 + 2 T + p T^{2}
41 1+pT2 1 + p T^{2}
43 1+4T+pT2 1 + 4 T + p T^{2}
47 1−12T+pT2 1 - 12 T + p T^{2}
53 1+6T+pT2 1 + 6 T + p T^{2}
59 1−8T+pT2 1 - 8 T + p T^{2}
61 1+6T+pT2 1 + 6 T + p T^{2}
67 1+8T+pT2 1 + 8 T + p T^{2}
71 1−14T+pT2 1 - 14 T + p T^{2}
73 1+2T+pT2 1 + 2 T + p T^{2}
79 1+12T+pT2 1 + 12 T + p T^{2}
83 1−4T+pT2 1 - 4 T + p T^{2}
89 1+pT2 1 + p T^{2}
97 1+2T+pT2 1 + 2 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

−8.621234568783701478950617886590, −7.79195451575718587485184031739, −6.78902576466258518849814546939, −6.06564976659535992293108524760, −5.62370507273544228914734043068, −4.91059247592313575518339819369, −3.78443484852293199292674683103, −2.79378519992244683258213909080, −1.88896973398279373978653204444, −1.12644399783657185408249656998, 1.12644399783657185408249656998, 1.88896973398279373978653204444, 2.79378519992244683258213909080, 3.78443484852293199292674683103, 4.91059247592313575518339819369, 5.62370507273544228914734043068, 6.06564976659535992293108524760, 6.78902576466258518849814546939, 7.79195451575718587485184031739, 8.621234568783701478950617886590

Graph of the ZZ-function along the critical line