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Properties

Label 2-40e2-1.1-c3-0-16
Degree 22
Conductor 16001600
Sign 11
Analytic cond. 94.403094.4030
Root an. cond. 9.716129.71612
Motivic weight 33
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 8·3-s + 16·7-s + 37·9-s − 40·11-s − 50·13-s + 30·17-s + 40·19-s − 128·21-s + 48·23-s − 80·27-s + 34·29-s − 320·31-s + 320·33-s + 310·37-s + 400·39-s + 410·41-s − 152·43-s − 416·47-s − 87·49-s − 240·51-s − 410·53-s − 320·57-s − 200·59-s − 30·61-s + 592·63-s − 776·67-s − 384·69-s + ⋯
L(s)  = 1  − 1.53·3-s + 0.863·7-s + 1.37·9-s − 1.09·11-s − 1.06·13-s + 0.428·17-s + 0.482·19-s − 1.33·21-s + 0.435·23-s − 0.570·27-s + 0.217·29-s − 1.85·31-s + 1.68·33-s + 1.37·37-s + 1.64·39-s + 1.56·41-s − 0.539·43-s − 1.29·47-s − 0.253·49-s − 0.658·51-s − 1.06·53-s − 0.743·57-s − 0.441·59-s − 0.0629·61-s + 1.18·63-s − 1.41·67-s − 0.669·69-s + ⋯

Functional equation

Λ(s)=(1600s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(1600s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 16001600    =    26⋅522^{6} \cdot 5^{2}
Sign: 11
Analytic conductor: 94.403094.4030
Root analytic conductor: 9.716129.71612
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 1600, ( :3/2), 1)(2,\ 1600,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 0.80831623000.8083162300
L(12)L(\frac12) ≈\approx 0.80831623000.8083162300
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 1
good3 1+8T+p3T2 1 + 8 T + p^{3} T^{2}
7 1−16T+p3T2 1 - 16 T + p^{3} T^{2}
11 1+40T+p3T2 1 + 40 T + p^{3} T^{2}
13 1+50T+p3T2 1 + 50 T + p^{3} T^{2}
17 1−30T+p3T2 1 - 30 T + p^{3} T^{2}
19 1−40T+p3T2 1 - 40 T + p^{3} T^{2}
23 1−48T+p3T2 1 - 48 T + p^{3} T^{2}
29 1−34T+p3T2 1 - 34 T + p^{3} T^{2}
31 1+320T+p3T2 1 + 320 T + p^{3} T^{2}
37 1−310T+p3T2 1 - 310 T + p^{3} T^{2}
41 1−10pT+p3T2 1 - 10 p T + p^{3} T^{2}
43 1+152T+p3T2 1 + 152 T + p^{3} T^{2}
47 1+416T+p3T2 1 + 416 T + p^{3} T^{2}
53 1+410T+p3T2 1 + 410 T + p^{3} T^{2}
59 1+200T+p3T2 1 + 200 T + p^{3} T^{2}
61 1+30T+p3T2 1 + 30 T + p^{3} T^{2}
67 1+776T+p3T2 1 + 776 T + p^{3} T^{2}
71 1+400T+p3T2 1 + 400 T + p^{3} T^{2}
73 1−630T+p3T2 1 - 630 T + p^{3} T^{2}
79 1−1120T+p3T2 1 - 1120 T + p^{3} T^{2}
83 1+552T+p3T2 1 + 552 T + p^{3} T^{2}
89 1+326T+p3T2 1 + 326 T + p^{3} T^{2}
97 1−110T+p3T2 1 - 110 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

−9.246057983553326163200671499598, −7.85307592800567424949871789418, −7.55419681364820494835359746952, −6.50176061984022841871306569955, −5.54358639289057795733329563901, −5.10280849466233146169527442782, −4.43656617653808281793710472000, −2.92455275558375989384975319898, −1.66086619973863982051246998957, −0.47444034053957629981648384660, 0.47444034053957629981648384660, 1.66086619973863982051246998957, 2.92455275558375989384975319898, 4.43656617653808281793710472000, 5.10280849466233146169527442782, 5.54358639289057795733329563901, 6.50176061984022841871306569955, 7.55419681364820494835359746952, 7.85307592800567424949871789418, 9.246057983553326163200671499598

Graph of the ZZ-function along the critical line