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Properties

Label 2-48e2-1.1-c3-0-99
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  + 4·5-s + 92·13-s − 94·17-s − 109·25-s − 284·29-s + 396·37-s − 230·41-s − 343·49-s − 572·53-s − 468·61-s + 368·65-s + 1.09e3·73-s − 376·85-s + 1.67e3·89-s − 594·97-s + 1.94e3·101-s − 1.46e3·109-s − 2.00e3·113-s + ⋯
L(s)  = 1  + 0.357·5-s + 1.96·13-s − 1.34·17-s − 0.871·25-s − 1.81·29-s + 1.75·37-s − 0.876·41-s − 49-s − 1.48·53-s − 0.982·61-s + 0.702·65-s + 1.76·73-s − 0.479·85-s + 1.98·89-s − 0.621·97-s + 1.91·101-s − 1.28·109-s − 1.66·113-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 1−4T+p3T2 1 - 4 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+94T+p3T2 1 + 94 T + p^{3} T^{2}
19 1+p3T2 1 + p^{3} T^{2}
23 1+p3T2 1 + p^{3} T^{2}
29 1+284T+p3T2 1 + 284 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+230T+p3T2 1 + 230 T + p^{3} T^{2}
43 1+p3T2 1 + p^{3} T^{2}
47 1+p3T2 1 + p^{3} T^{2}
53 1+572T+p3T2 1 + 572 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 1−1670T+p3T2 1 - 1670 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

−8.275820327009108507535643786739, −7.62176511165678453514136771728, −6.36537148568650961056236973393, −6.20429297938648501412898692621, −5.13367420173070648613621483128, −4.11057385268763954228436424500, −3.43583664079371395343193902063, −2.17875190948423196858828206037, −1.35093743101236521253426765698, 0, 1.35093743101236521253426765698, 2.17875190948423196858828206037, 3.43583664079371395343193902063, 4.11057385268763954228436424500, 5.13367420173070648613621483128, 6.20429297938648501412898692621, 6.36537148568650961056236973393, 7.62176511165678453514136771728, 8.275820327009108507535643786739

Graph of the ZZ-function along the critical line