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Properties

Label 2-6552-1.1-c1-0-55
Degree 22
Conductor 65526552
Sign −1-1
Analytic cond. 52.317952.3179
Root an. cond. 7.233117.23311
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  − 2·5-s + 7-s − 4·11-s − 13-s − 6·17-s + 4·19-s + 8·23-s − 25-s + 6·29-s + 8·31-s − 2·35-s + 2·37-s − 2·41-s − 4·43-s + 12·47-s + 49-s + 6·53-s + 8·55-s − 12·59-s − 6·61-s + 2·65-s − 16·71-s − 2·73-s − 4·77-s + 4·83-s + 12·85-s − 2·89-s + ⋯
L(s)  = 1  − 0.894·5-s + 0.377·7-s − 1.20·11-s − 0.277·13-s − 1.45·17-s + 0.917·19-s + 1.66·23-s − 1/5·25-s + 1.11·29-s + 1.43·31-s − 0.338·35-s + 0.328·37-s − 0.312·41-s − 0.609·43-s + 1.75·47-s + 1/7·49-s + 0.824·53-s + 1.07·55-s − 1.56·59-s − 0.768·61-s + 0.248·65-s − 1.89·71-s − 0.234·73-s − 0.455·77-s + 0.439·83-s + 1.30·85-s − 0.211·89-s + ⋯

Functional equation

Λ(s)=(6552s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 6552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(6552s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 6552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 65526552    =    23⋅32⋅7⋅132^{3} \cdot 3^{2} \cdot 7 \cdot 13
Sign: −1-1
Analytic conductor: 52.317952.3179
Root analytic conductor: 7.233117.23311
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 6552, ( :1/2), −1)(2,\ 6552,\ (\ :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
7 1−T 1 - T
13 1+T 1 + T
good5 1+2T+pT2 1 + 2 T + p T^{2}
11 1+4T+pT2 1 + 4 T + p T^{2}
17 1+6T+pT2 1 + 6 T + p T^{2}
19 1−4T+pT2 1 - 4 T + p T^{2}
23 1−8T+pT2 1 - 8 T + p T^{2}
29 1−6T+pT2 1 - 6 T + 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+2T+pT2 1 + 2 T + 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+12T+pT2 1 + 12 T + p T^{2}
61 1+6T+pT2 1 + 6 T + p T^{2}
67 1+pT2 1 + p T^{2}
71 1+16T+pT2 1 + 16 T + p T^{2}
73 1+2T+pT2 1 + 2 T + p T^{2}
79 1+pT2 1 + 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

−7.61138227150661299886474120026, −7.15058479585012122876172412971, −6.33746865755064908823612227695, −5.33311206879602121953924418003, −4.72639825567634000418862949259, −4.19237928482593613286833254108, −2.98810415480492252830043880919, −2.56814021450321899428515125912, −1.15840199401338845116090360872, 0, 1.15840199401338845116090360872, 2.56814021450321899428515125912, 2.98810415480492252830043880919, 4.19237928482593613286833254108, 4.72639825567634000418862949259, 5.33311206879602121953924418003, 6.33746865755064908823612227695, 7.15058479585012122876172412971, 7.61138227150661299886474120026

Graph of the ZZ-function along the critical line