↑

Properties

Label 2-1960-1.1-c3-0-33
Degree 22
Conductor 19601960
Sign 11
Analytic cond. 115.643115.643
Root an. cond. 10.753710.7537
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  − 4·3-s − 5·5-s − 11·9-s + 36·11-s + 42·13-s + 20·15-s + 110·17-s + 116·19-s + 16·23-s + 25·25-s + 152·27-s + 198·29-s − 240·31-s − 144·33-s − 258·37-s − 168·39-s − 442·41-s − 292·43-s + 55·45-s − 392·47-s − 440·51-s + 142·53-s − 180·55-s − 464·57-s + 348·59-s + 570·61-s − 210·65-s + ⋯
L(s)  = 1  − 0.769·3-s − 0.447·5-s − 0.407·9-s + 0.986·11-s + 0.896·13-s + 0.344·15-s + 1.56·17-s + 1.40·19-s + 0.145·23-s + 1/5·25-s + 1.08·27-s + 1.26·29-s − 1.39·31-s − 0.759·33-s − 1.14·37-s − 0.689·39-s − 1.68·41-s − 1.03·43-s + 0.182·45-s − 1.21·47-s − 1.20·51-s + 0.368·53-s − 0.441·55-s − 1.07·57-s + 0.767·59-s + 1.19·61-s − 0.400·65-s + ⋯

Functional equation

Λ(s)=(1960s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(1960s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 19601960    =    23⋅5⋅722^{3} \cdot 5 \cdot 7^{2}
Sign: 11
Analytic conductor: 115.643115.643
Root analytic conductor: 10.753710.7537
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 1960, ( :3/2), 1)(2,\ 1960,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 1.6655073981.665507398
L(12)L(\frac12) ≈\approx 1.6655073981.665507398
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
7 1 1
good3 1+4T+p3T2 1 + 4 T + p^{3} T^{2}
11 1−36T+p3T2 1 - 36 T + p^{3} T^{2}
13 1−42T+p3T2 1 - 42 T + p^{3} T^{2}
17 1−110T+p3T2 1 - 110 T + p^{3} T^{2}
19 1−116T+p3T2 1 - 116 T + p^{3} T^{2}
23 1−16T+p3T2 1 - 16 T + p^{3} T^{2}
29 1−198T+p3T2 1 - 198 T + p^{3} T^{2}
31 1+240T+p3T2 1 + 240 T + p^{3} T^{2}
37 1+258T+p3T2 1 + 258 T + p^{3} T^{2}
41 1+442T+p3T2 1 + 442 T + p^{3} T^{2}
43 1+292T+p3T2 1 + 292 T + p^{3} T^{2}
47 1+392T+p3T2 1 + 392 T + p^{3} T^{2}
53 1−142T+p3T2 1 - 142 T + p^{3} T^{2}
59 1−348T+p3T2 1 - 348 T + p^{3} T^{2}
61 1−570T+p3T2 1 - 570 T + p^{3} T^{2}
67 1−692T+p3T2 1 - 692 T + p^{3} T^{2}
71 1−168T+p3T2 1 - 168 T + p^{3} T^{2}
73 1−134T+p3T2 1 - 134 T + p^{3} T^{2}
79 1−784T+p3T2 1 - 784 T + p^{3} T^{2}
83 1+564T+p3T2 1 + 564 T + p^{3} T^{2}
89 1+1034T+p3T2 1 + 1034 T + p^{3} T^{2}
97 1−382T+p3T2 1 - 382 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

−8.677008027035987045813432930499, −8.165182801105838533052664435839, −7.06225859411952504838778602299, −6.50105804295316399635856378201, −5.47998847378264623883021332601, −5.07535682573438079351141878036, −3.64360051930141708120540245663, −3.26956951942213813275790513872, −1.49697776397294957042214310589, −0.67879977302095928445962358631, 0.67879977302095928445962358631, 1.49697776397294957042214310589, 3.26956951942213813275790513872, 3.64360051930141708120540245663, 5.07535682573438079351141878036, 5.47998847378264623883021332601, 6.50105804295316399635856378201, 7.06225859411952504838778602299, 8.165182801105838533052664435839, 8.677008027035987045813432930499

Graph of the ZZ-function along the critical line