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Properties

Label 2-48e2-1.1-c1-0-17
Degree 22
Conductor 23042304
Sign 11
Analytic cond. 18.397518.3975
Root an. cond. 4.289234.28923
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  + 4·7-s + 4·11-s + 4·13-s + 2·17-s + 4·19-s − 8·23-s − 5·25-s + 8·29-s + 4·31-s − 4·37-s − 6·41-s − 4·43-s − 8·47-s + 9·49-s + 8·53-s − 12·59-s + 12·61-s − 12·67-s + 8·71-s − 6·73-s + 16·77-s + 4·79-s − 4·83-s + 6·89-s + 16·91-s − 2·97-s + 8·101-s + ⋯
L(s)  = 1  + 1.51·7-s + 1.20·11-s + 1.10·13-s + 0.485·17-s + 0.917·19-s − 1.66·23-s − 25-s + 1.48·29-s + 0.718·31-s − 0.657·37-s − 0.937·41-s − 0.609·43-s − 1.16·47-s + 9/7·49-s + 1.09·53-s − 1.56·59-s + 1.53·61-s − 1.46·67-s + 0.949·71-s − 0.702·73-s + 1.82·77-s + 0.450·79-s − 0.439·83-s + 0.635·89-s + 1.67·91-s − 0.203·97-s + 0.796·101-s + ⋯

Functional equation

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

Invariants

Degree: 22
Conductor: 23042304    =    28⋅322^{8} \cdot 3^{2}
Sign: 11
Analytic conductor: 18.397518.3975
Root analytic conductor: 4.289234.28923
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 2304, ( :1/2), 1)(2,\ 2304,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.5205234712.520523471
L(12)L(\frac12) ≈\approx 2.5205234712.520523471
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
good5 1+pT2 1 + p T^{2}
7 1−4T+pT2 1 - 4 T + p T^{2}
11 1−4T+pT2 1 - 4 T + p T^{2}
13 1−4T+pT2 1 - 4 T + p T^{2}
17 1−2T+pT2 1 - 2 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 1−8T+pT2 1 - 8 T + p T^{2}
31 1−4T+pT2 1 - 4 T + p T^{2}
37 1+4T+pT2 1 + 4 T + p T^{2}
41 1+6T+pT2 1 + 6 T + p T^{2}
43 1+4T+pT2 1 + 4 T + p T^{2}
47 1+8T+pT2 1 + 8 T + p T^{2}
53 1−8T+pT2 1 - 8 T + p T^{2}
59 1+12T+pT2 1 + 12 T + p T^{2}
61 1−12T+pT2 1 - 12 T + p T^{2}
67 1+12T+pT2 1 + 12 T + p T^{2}
71 1−8T+pT2 1 - 8 T + 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 1−6T+pT2 1 - 6 T + 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

−8.770780213851605268751258605897, −8.273401806148982335189253098083, −7.66603558879594006866408095630, −6.59361365188310900976865652323, −5.90734355349400077940850184592, −4.98531549119289558181541630311, −4.16197372918625835514829553019, −3.38591647618518318996390357483, −1.87178167433974199275141976633, −1.18014352853545560604310847207, 1.18014352853545560604310847207, 1.87178167433974199275141976633, 3.38591647618518318996390357483, 4.16197372918625835514829553019, 4.98531549119289558181541630311, 5.90734355349400077940850184592, 6.59361365188310900976865652323, 7.66603558879594006866408095630, 8.273401806148982335189253098083, 8.770780213851605268751258605897

Graph of the ZZ-function along the critical line