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Properties

Label 2-40e2-1.1-c3-0-110
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 22

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5·3-s + 2·7-s − 2·9-s − 39·11-s − 84·13-s − 61·17-s − 151·19-s − 10·21-s − 58·23-s + 145·27-s − 192·29-s − 18·31-s + 195·33-s + 138·37-s + 420·39-s + 229·41-s + 164·43-s − 212·47-s − 339·49-s + 305·51-s − 578·53-s + 755·57-s + 336·59-s − 858·61-s − 4·63-s + 209·67-s + 290·69-s + ⋯
L(s)  = 1  − 0.962·3-s + 0.107·7-s − 0.0740·9-s − 1.06·11-s − 1.79·13-s − 0.870·17-s − 1.82·19-s − 0.103·21-s − 0.525·23-s + 1.03·27-s − 1.22·29-s − 0.104·31-s + 1.02·33-s + 0.613·37-s + 1.72·39-s + 0.872·41-s + 0.581·43-s − 0.657·47-s − 0.988·49-s + 0.837·51-s − 1.49·53-s + 1.75·57-s + 0.741·59-s − 1.80·61-s − 0.00799·63-s + 0.381·67-s + 0.505·69-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: 22
Selberg data: (2, 1600, ( :3/2), 1)(2,\ 1600,\ (\ :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 1
good3 1+5T+p3T2 1 + 5 T + p^{3} T^{2}
7 1−2T+p3T2 1 - 2 T + p^{3} T^{2}
11 1+39T+p3T2 1 + 39 T + p^{3} T^{2}
13 1+84T+p3T2 1 + 84 T + p^{3} T^{2}
17 1+61T+p3T2 1 + 61 T + p^{3} T^{2}
19 1+151T+p3T2 1 + 151 T + p^{3} T^{2}
23 1+58T+p3T2 1 + 58 T + p^{3} T^{2}
29 1+192T+p3T2 1 + 192 T + p^{3} T^{2}
31 1+18T+p3T2 1 + 18 T + p^{3} T^{2}
37 1−138T+p3T2 1 - 138 T + p^{3} T^{2}
41 1−229T+p3T2 1 - 229 T + p^{3} T^{2}
43 1−164T+p3T2 1 - 164 T + p^{3} T^{2}
47 1+212T+p3T2 1 + 212 T + p^{3} T^{2}
53 1+578T+p3T2 1 + 578 T + p^{3} T^{2}
59 1−336T+p3T2 1 - 336 T + p^{3} T^{2}
61 1+858T+p3T2 1 + 858 T + p^{3} T^{2}
67 1−209T+p3T2 1 - 209 T + p^{3} T^{2}
71 1+780T+p3T2 1 + 780 T + p^{3} T^{2}
73 1+403T+p3T2 1 + 403 T + p^{3} T^{2}
79 1+230T+p3T2 1 + 230 T + p^{3} T^{2}
83 1−1293T+p3T2 1 - 1293 T + p^{3} T^{2}
89 1+1369T+p3T2 1 + 1369 T + p^{3} T^{2}
97 1−382T+p3T2 1 - 382 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

−8.117302036654077042452324726487, −7.44040983733797533097619005938, −6.48001203918006730239959205115, −5.79302437289555283891312692279, −4.89263208113654345457682660750, −4.37095742375389981651409657518, −2.77759834763033318546225377102, −2.00496523756250707031149561131, 0, 0, 2.00496523756250707031149561131, 2.77759834763033318546225377102, 4.37095742375389981651409657518, 4.89263208113654345457682660750, 5.79302437289555283891312692279, 6.48001203918006730239959205115, 7.44040983733797533097619005938, 8.117302036654077042452324726487

Graph of the ZZ-function along the critical line