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Properties

Label 2-145728-1.1-c1-0-175
Degree 22
Conductor 145728145728
Sign 11
Analytic cond. 1163.641163.64
Root an. cond. 34.112234.1122
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 22

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·5-s + 11-s + 2·13-s − 2·17-s − 4·19-s + 23-s + 11·25-s − 8·29-s + 4·31-s − 2·37-s − 4·41-s − 8·43-s − 8·47-s − 7·49-s − 4·53-s − 4·55-s − 4·59-s − 2·61-s − 8·65-s + 10·67-s + 6·73-s − 4·79-s − 4·83-s + 8·85-s + 16·95-s − 2·97-s + 101-s + ⋯
L(s)  = 1  − 1.78·5-s + 0.301·11-s + 0.554·13-s − 0.485·17-s − 0.917·19-s + 0.208·23-s + 11/5·25-s − 1.48·29-s + 0.718·31-s − 0.328·37-s − 0.624·41-s − 1.21·43-s − 1.16·47-s − 49-s − 0.549·53-s − 0.539·55-s − 0.520·59-s − 0.256·61-s − 0.992·65-s + 1.22·67-s + 0.702·73-s − 0.450·79-s − 0.439·83-s + 0.867·85-s + 1.64·95-s − 0.203·97-s + 0.0995·101-s + ⋯

Functional equation

Λ(s)=(145728s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 145728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(145728s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 145728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 145728145728    =    26⋅32⋅11⋅232^{6} \cdot 3^{2} \cdot 11 \cdot 23
Sign: 11
Analytic conductor: 1163.641163.64
Root analytic conductor: 34.112234.1122
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 22
Selberg data: (2, 145728, ( :1/2), 1)(2,\ 145728,\ (\ :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
11 1−T 1 - T
23 1−T 1 - T
good5 1+4T+pT2 1 + 4 T + p T^{2}
7 1+pT2 1 + p T^{2}
13 1−2T+pT2 1 - 2 T + p T^{2}
17 1+2T+pT2 1 + 2 T + p T^{2}
19 1+4T+pT2 1 + 4 T + p T^{2}
29 1+8T+pT2 1 + 8 T + p T^{2}
31 1−4T+pT2 1 - 4 T + p T^{2}
37 1+2T+pT2 1 + 2 T + p T^{2}
41 1+4T+pT2 1 + 4 T + p T^{2}
43 1+8T+pT2 1 + 8 T + p T^{2}
47 1+8T+pT2 1 + 8 T + p T^{2}
53 1+4T+pT2 1 + 4 T + p T^{2}
59 1+4T+pT2 1 + 4 T + p T^{2}
61 1+2T+pT2 1 + 2 T + p T^{2}
67 1−10T+pT2 1 - 10 T + p T^{2}
71 1+pT2 1 + p T^{2}
73 1−6T+pT2 1 - 6 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+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

−13.77755334370273, −13.30228687473297, −12.78080450099184, −12.42869125382595, −11.86983560061137, −11.37845006579406, −11.10141885678790, −10.78747135864125, −9.972012390191640, −9.535062112642312, −8.782469054577014, −8.450445456843434, −8.111680544349361, −7.568450905613502, −6.986146284077573, −6.564216325807749, −6.139624555134935, −5.158866060721922, −4.819442809302897, −4.194960756867093, −3.699416846190283, −3.388828913040949, −2.664213679089494, −1.785712251880946, −1.173261483381862, 0, 0, 1.173261483381862, 1.785712251880946, 2.664213679089494, 3.388828913040949, 3.699416846190283, 4.194960756867093, 4.819442809302897, 5.158866060721922, 6.139624555134935, 6.564216325807749, 6.986146284077573, 7.568450905613502, 8.111680544349361, 8.450445456843434, 8.782469054577014, 9.535062112642312, 9.972012390191640, 10.78747135864125, 11.10141885678790, 11.37845006579406, 11.86983560061137, 12.42869125382595, 12.78080450099184, 13.30228687473297, 13.77755334370273

Graph of the ZZ-function along the critical line