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Properties

Label 2-320-1.1-c3-0-23
Degree 22
Conductor 320320
Sign −1-1
Analytic cond. 18.880618.8806
Root an. cond. 4.345184.34518
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  + 6·3-s + 5·5-s − 34·7-s + 9·9-s − 16·11-s − 58·13-s + 30·15-s − 70·17-s − 4·19-s − 204·21-s − 134·23-s + 25·25-s − 108·27-s + 242·29-s + 100·31-s − 96·33-s − 170·35-s + 438·37-s − 348·39-s − 138·41-s − 178·43-s + 45·45-s + 22·47-s + 813·49-s − 420·51-s − 162·53-s − 80·55-s + ⋯
L(s)  = 1  + 1.15·3-s + 0.447·5-s − 1.83·7-s + 1/3·9-s − 0.438·11-s − 1.23·13-s + 0.516·15-s − 0.998·17-s − 0.0482·19-s − 2.11·21-s − 1.21·23-s + 1/5·25-s − 0.769·27-s + 1.54·29-s + 0.579·31-s − 0.506·33-s − 0.821·35-s + 1.94·37-s − 1.42·39-s − 0.525·41-s − 0.631·43-s + 0.149·45-s + 0.0682·47-s + 2.37·49-s − 1.15·51-s − 0.419·53-s − 0.196·55-s + ⋯

Functional equation

Λ(s)=(320s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(320s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 320320    =    26⋅52^{6} \cdot 5
Sign: −1-1
Analytic conductor: 18.880618.8806
Root analytic conductor: 4.345184.34518
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 320, ( :3/2), −1)(2,\ 320,\ (\ :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
5 1−pT 1 - p T
good3 1−2pT+p3T2 1 - 2 p T + p^{3} T^{2}
7 1+34T+p3T2 1 + 34 T + p^{3} T^{2}
11 1+16T+p3T2 1 + 16 T + p^{3} T^{2}
13 1+58T+p3T2 1 + 58 T + p^{3} T^{2}
17 1+70T+p3T2 1 + 70 T + p^{3} T^{2}
19 1+4T+p3T2 1 + 4 T + p^{3} T^{2}
23 1+134T+p3T2 1 + 134 T + p^{3} T^{2}
29 1−242T+p3T2 1 - 242 T + p^{3} T^{2}
31 1−100T+p3T2 1 - 100 T + p^{3} T^{2}
37 1−438T+p3T2 1 - 438 T + p^{3} T^{2}
41 1+138T+p3T2 1 + 138 T + p^{3} T^{2}
43 1+178T+p3T2 1 + 178 T + p^{3} T^{2}
47 1−22T+p3T2 1 - 22 T + p^{3} T^{2}
53 1+162T+p3T2 1 + 162 T + p^{3} T^{2}
59 1−268T+p3T2 1 - 268 T + p^{3} T^{2}
61 1+250T+p3T2 1 + 250 T + p^{3} T^{2}
67 1+422T+p3T2 1 + 422 T + p^{3} T^{2}
71 1+12pT+p3T2 1 + 12 p T + p^{3} T^{2}
73 1−306T+p3T2 1 - 306 T + p^{3} T^{2}
79 1+456T+p3T2 1 + 456 T + p^{3} T^{2}
83 1+434T+p3T2 1 + 434 T + p^{3} T^{2}
89 1+726T+p3T2 1 + 726 T + p^{3} T^{2}
97 1−1378T+p3T2 1 - 1378 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

−10.23014463176491978369381799521, −9.781885963717609691940987832204, −8.992234361736871636430047539745, −7.984125124870024250470059182788, −6.84662389215761778858782885596, −5.98525973062276473797780306416, −4.38194918512963248962026338711, −2.99401994143364764477955915129, −2.42446982631790162680635176093, 0, 2.42446982631790162680635176093, 2.99401994143364764477955915129, 4.38194918512963248962026338711, 5.98525973062276473797780306416, 6.84662389215761778858782885596, 7.984125124870024250470059182788, 8.992234361736871636430047539745, 9.781885963717609691940987832204, 10.23014463176491978369381799521

Graph of the ZZ-function along the critical line