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Properties

Label 2-48e2-1.1-c3-0-112
Degree 22
Conductor 23042304
Sign −1-1
Analytic cond. 135.940135.940
Root an. cond. 11.659311.6593
Motivic weight 33
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 22·5-s − 92·13-s − 104·17-s + 359·25-s + 130·29-s + 396·37-s − 472·41-s − 343·49-s − 518·53-s − 468·61-s − 2.02e3·65-s − 1.09e3·73-s − 2.28e3·85-s − 176·89-s + 594·97-s − 598·101-s + 1.46e3·109-s − 1.32e3·113-s + ⋯
L(s)  = 1  + 1.96·5-s − 1.96·13-s − 1.48·17-s + 2.87·25-s + 0.832·29-s + 1.75·37-s − 1.79·41-s − 49-s − 1.34·53-s − 0.982·61-s − 3.86·65-s − 1.76·73-s − 2.91·85-s − 0.209·89-s + 0.621·97-s − 0.589·101-s + 1.28·109-s − 1.10·113-s + ⋯

Functional equation

Λ(s)=(2304s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(2304s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 23042304    =    28⋅322^{8} \cdot 3^{2}
Sign: −1-1
Analytic conductor: 135.940135.940
Root analytic conductor: 11.659311.6593
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 2304, ( :3/2), −1)(2,\ 2304,\ (\ :3/2),\ -1)

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
L(52)L(\frac{5}{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
good5 1−22T+p3T2 1 - 22 T + p^{3} T^{2}
7 1+p3T2 1 + p^{3} T^{2}
11 1+p3T2 1 + p^{3} T^{2}
13 1+92T+p3T2 1 + 92 T + p^{3} T^{2}
17 1+104T+p3T2 1 + 104 T + p^{3} T^{2}
19 1+p3T2 1 + p^{3} T^{2}
23 1+p3T2 1 + p^{3} T^{2}
29 1−130T+p3T2 1 - 130 T + p^{3} T^{2}
31 1+p3T2 1 + p^{3} T^{2}
37 1−396T+p3T2 1 - 396 T + p^{3} T^{2}
41 1+472T+p3T2 1 + 472 T + p^{3} T^{2}
43 1+p3T2 1 + p^{3} T^{2}
47 1+p3T2 1 + p^{3} T^{2}
53 1+518T+p3T2 1 + 518 T + p^{3} T^{2}
59 1+p3T2 1 + p^{3} T^{2}
61 1+468T+p3T2 1 + 468 T + p^{3} T^{2}
67 1+p3T2 1 + p^{3} T^{2}
71 1+p3T2 1 + p^{3} T^{2}
73 1+1098T+p3T2 1 + 1098 T + p^{3} T^{2}
79 1+p3T2 1 + p^{3} T^{2}
83 1+p3T2 1 + p^{3} T^{2}
89 1+176T+p3T2 1 + 176 T + p^{3} T^{2}
97 1−594T+p3T2 1 - 594 T + p^{3} 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

−8.442721614606649735742770949180, −7.32253373224090007499932330360, −6.56858279412549644069088810088, −6.03891654579303787676946986678, −4.94290952356971411996007797812, −4.67721544481997312806899500589, −2.88205342209647591616129487103, −2.33222271120770206960248713590, −1.49537304557110902716672127340, 0, 1.49537304557110902716672127340, 2.33222271120770206960248713590, 2.88205342209647591616129487103, 4.67721544481997312806899500589, 4.94290952356971411996007797812, 6.03891654579303787676946986678, 6.56858279412549644069088810088, 7.32253373224090007499932330360, 8.442721614606649735742770949180

Graph of the ZZ-function along the critical line