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Properties

Label 2-40e2-1.1-c3-0-18
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  + 3-s − 26·7-s − 26·9-s + 45·11-s − 44·13-s + 117·17-s − 91·19-s − 26·21-s + 18·23-s − 53·27-s − 144·29-s − 26·31-s + 45·33-s + 214·37-s − 44·39-s − 459·41-s − 460·43-s + 468·47-s + 333·49-s + 117·51-s − 558·53-s − 91·57-s − 72·59-s + 118·61-s + 676·63-s + 251·67-s + 18·69-s + ⋯
L(s)  = 1  + 0.192·3-s − 1.40·7-s − 0.962·9-s + 1.23·11-s − 0.938·13-s + 1.66·17-s − 1.09·19-s − 0.270·21-s + 0.163·23-s − 0.377·27-s − 0.922·29-s − 0.150·31-s + 0.237·33-s + 0.950·37-s − 0.180·39-s − 1.74·41-s − 1.63·43-s + 1.45·47-s + 0.970·49-s + 0.321·51-s − 1.44·53-s − 0.211·57-s − 0.158·59-s + 0.247·61-s + 1.35·63-s + 0.457·67-s + 0.0314·69-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 1.2785758391.278575839
L(12)L(\frac12) ≈\approx 1.2785758391.278575839
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−T+p3T2 1 - T + p^{3} T^{2}
7 1+26T+p3T2 1 + 26 T + p^{3} T^{2}
11 1−45T+p3T2 1 - 45 T + p^{3} T^{2}
13 1+44T+p3T2 1 + 44 T + p^{3} T^{2}
17 1−117T+p3T2 1 - 117 T + p^{3} T^{2}
19 1+91T+p3T2 1 + 91 T + p^{3} T^{2}
23 1−18T+p3T2 1 - 18 T + p^{3} T^{2}
29 1+144T+p3T2 1 + 144 T + p^{3} T^{2}
31 1+26T+p3T2 1 + 26 T + p^{3} T^{2}
37 1−214T+p3T2 1 - 214 T + p^{3} T^{2}
41 1+459T+p3T2 1 + 459 T + p^{3} T^{2}
43 1+460T+p3T2 1 + 460 T + p^{3} T^{2}
47 1−468T+p3T2 1 - 468 T + p^{3} T^{2}
53 1+558T+p3T2 1 + 558 T + p^{3} T^{2}
59 1+72T+p3T2 1 + 72 T + p^{3} T^{2}
61 1−118T+p3T2 1 - 118 T + p^{3} T^{2}
67 1−251T+p3T2 1 - 251 T + p^{3} T^{2}
71 1+108T+p3T2 1 + 108 T + p^{3} T^{2}
73 1−299T+p3T2 1 - 299 T + p^{3} T^{2}
79 1−898T+p3T2 1 - 898 T + p^{3} T^{2}
83 1−927T+p3T2 1 - 927 T + p^{3} T^{2}
89 1−351T+p3T2 1 - 351 T + p^{3} T^{2}
97 1−386T+p3T2 1 - 386 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

−9.196529589833981831689860789104, −8.331319455160018615253547295503, −7.40643258767182009746217685718, −6.51672166572814897357218412812, −5.97465969846708911558173623648, −4.96434879447422325919440724358, −3.64147636310899855322869601522, −3.20972271136542460911399217432, −2.01887438152307613027036935458, −0.51783118127359750653814429717, 0.51783118127359750653814429717, 2.01887438152307613027036935458, 3.20972271136542460911399217432, 3.64147636310899855322869601522, 4.96434879447422325919440724358, 5.97465969846708911558173623648, 6.51672166572814897357218412812, 7.40643258767182009746217685718, 8.331319455160018615253547295503, 9.196529589833981831689860789104

Graph of the ZZ-function along the critical line