↑

Properties

Label 2-40e2-1.1-c3-0-36
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

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 9·3-s − 26·7-s + 54·9-s − 59·11-s − 28·13-s + 5·17-s + 109·19-s − 234·21-s + 194·23-s + 243·27-s + 32·29-s − 10·31-s − 531·33-s + 198·37-s − 252·39-s + 117·41-s + 388·43-s + 68·47-s + 333·49-s + 45·51-s + 18·53-s + 981·57-s + 392·59-s + 710·61-s − 1.40e3·63-s − 253·67-s + 1.74e3·69-s + ⋯
L(s)  = 1  + 1.73·3-s − 1.40·7-s + 2·9-s − 1.61·11-s − 0.597·13-s + 0.0713·17-s + 1.31·19-s − 2.43·21-s + 1.75·23-s + 1.73·27-s + 0.204·29-s − 0.0579·31-s − 2.80·33-s + 0.879·37-s − 1.03·39-s + 0.445·41-s + 1.37·43-s + 0.211·47-s + 0.970·49-s + 0.123·51-s + 0.0466·53-s + 2.27·57-s + 0.864·59-s + 1.49·61-s − 2.80·63-s − 0.461·67-s + 3.04·69-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 3.3339504703.333950470
L(12)L(\frac12) ≈\approx 3.3339504703.333950470
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−p2T+p3T2 1 - p^{2} T + p^{3} T^{2}
7 1+26T+p3T2 1 + 26 T + p^{3} T^{2}
11 1+59T+p3T2 1 + 59 T + p^{3} T^{2}
13 1+28T+p3T2 1 + 28 T + p^{3} T^{2}
17 1−5T+p3T2 1 - 5 T + p^{3} T^{2}
19 1−109T+p3T2 1 - 109 T + p^{3} T^{2}
23 1−194T+p3T2 1 - 194 T + p^{3} T^{2}
29 1−32T+p3T2 1 - 32 T + p^{3} T^{2}
31 1+10T+p3T2 1 + 10 T + p^{3} T^{2}
37 1−198T+p3T2 1 - 198 T + p^{3} T^{2}
41 1−117T+p3T2 1 - 117 T + p^{3} T^{2}
43 1−388T+p3T2 1 - 388 T + p^{3} T^{2}
47 1−68T+p3T2 1 - 68 T + p^{3} T^{2}
53 1−18T+p3T2 1 - 18 T + p^{3} T^{2}
59 1−392T+p3T2 1 - 392 T + p^{3} T^{2}
61 1−710T+p3T2 1 - 710 T + p^{3} T^{2}
67 1+253T+p3T2 1 + 253 T + p^{3} T^{2}
71 1−612T+p3T2 1 - 612 T + p^{3} T^{2}
73 1+549T+p3T2 1 + 549 T + p^{3} T^{2}
79 1+414T+p3T2 1 + 414 T + p^{3} T^{2}
83 1+121T+p3T2 1 + 121 T + p^{3} T^{2}
89 1+81T+p3T2 1 + 81 T + p^{3} T^{2}
97 1+1502T+p3T2 1 + 1502 T + p^{3} T^{2}
show more
show less
   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

−9.105921820664199125260757361906, −8.300637610774630132117018879082, −7.37795196084154701844332503743, −7.14668462352730111396516721517, −5.75502762664416276467004496662, −4.76882678009111725199752431059, −3.57393278607174061446424571583, −2.85469636941045360769011860044, −2.48468896344146973630823661749, −0.78880587193246888501100179150, 0.78880587193246888501100179150, 2.48468896344146973630823661749, 2.85469636941045360769011860044, 3.57393278607174061446424571583, 4.76882678009111725199752431059, 5.75502762664416276467004496662, 7.14668462352730111396516721517, 7.37795196084154701844332503743, 8.300637610774630132117018879082, 9.105921820664199125260757361906

Graph of the ZZ-function along the critical line