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Properties

Label 2-7920-1.1-c1-0-77
Degree 22
Conductor 79207920
Sign −1-1
Analytic cond. 63.241563.2415
Root an. cond. 7.952457.95245
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s − 2·7-s + 11-s + 2·13-s − 2·19-s + 25-s − 8·31-s − 2·35-s + 2·37-s − 2·43-s − 3·49-s − 6·53-s + 55-s − 12·59-s + 2·61-s + 2·65-s + 4·67-s + 2·73-s − 2·77-s + 10·79-s − 12·83-s + 6·89-s − 4·91-s − 2·95-s + 14·97-s + 4·103-s − 12·107-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.755·7-s + 0.301·11-s + 0.554·13-s − 0.458·19-s + 1/5·25-s − 1.43·31-s − 0.338·35-s + 0.328·37-s − 0.304·43-s − 3/7·49-s − 0.824·53-s + 0.134·55-s − 1.56·59-s + 0.256·61-s + 0.248·65-s + 0.488·67-s + 0.234·73-s − 0.227·77-s + 1.12·79-s − 1.31·83-s + 0.635·89-s − 0.419·91-s − 0.205·95-s + 1.42·97-s + 0.394·103-s − 1.16·107-s + ⋯

Functional equation

Λ(s)=(7920s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 7920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(7920s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 7920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 79207920    =    24⋅32⋅5⋅112^{4} \cdot 3^{2} \cdot 5 \cdot 11
Sign: −1-1
Analytic conductor: 63.241563.2415
Root analytic conductor: 7.952457.95245
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 7920, ( :1/2), −1)(2,\ 7920,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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
5 1−T 1 - T
11 1−T 1 - T
good7 1+2T+pT2 1 + 2 T + p T^{2}
13 1−2T+pT2 1 - 2 T + p T^{2}
17 1+pT2 1 + p T^{2}
19 1+2T+pT2 1 + 2 T + p T^{2}
23 1+pT2 1 + p T^{2}
29 1+pT2 1 + p T^{2}
31 1+8T+pT2 1 + 8 T + p T^{2}
37 1−2T+pT2 1 - 2 T + p T^{2}
41 1+pT2 1 + p T^{2}
43 1+2T+pT2 1 + 2 T + p T^{2}
47 1+pT2 1 + p T^{2}
53 1+6T+pT2 1 + 6 T + p T^{2}
59 1+12T+pT2 1 + 12 T + p T^{2}
61 1−2T+pT2 1 - 2 T + p T^{2}
67 1−4T+pT2 1 - 4 T + p T^{2}
71 1+pT2 1 + p T^{2}
73 1−2T+pT2 1 - 2 T + p T^{2}
79 1−10T+pT2 1 - 10 T + p T^{2}
83 1+12T+pT2 1 + 12 T + p T^{2}
89 1−6T+pT2 1 - 6 T + p T^{2}
97 1−14T+pT2 1 - 14 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

−7.45950523429408724078642817430, −6.62809575887132393562329754634, −6.23014779417671950131080268515, −5.51350916544264899972205145523, −4.68675354902407805865003313060, −3.78604634402167547094494153595, −3.19237694755342407493957762628, −2.20902138146244912815630447797, −1.32116731583023775044904348023, 0, 1.32116731583023775044904348023, 2.20902138146244912815630447797, 3.19237694755342407493957762628, 3.78604634402167547094494153595, 4.68675354902407805865003313060, 5.51350916544264899972205145523, 6.23014779417671950131080268515, 6.62809575887132393562329754634, 7.45950523429408724078642817430

Graph of the ZZ-function along the critical line