↑

Properties

Label 2-240-1.1-c5-0-0
Degree 22
Conductor 240240
Sign 11
Analytic cond. 38.492138.4921
Root an. cond. 6.204206.20420
Motivic weight 55
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 − 25·5-s − 164·7-s + 81·9-s − 720·11-s + 698·13-s + 225·15-s − 2.22e3·17-s − 356·19-s + 1.47e3·21-s + 1.80e3·23-s + 625·25-s − 729·27-s + 714·29-s − 848·31-s + 6.48e3·33-s + 4.10e3·35-s − 1.13e4·37-s − 6.28e3·39-s + 9.35e3·41-s + 5.95e3·43-s − 2.02e3·45-s + 1.11e4·47-s + 1.00e4·49-s + 2.00e4·51-s + 1.41e4·53-s + 1.80e4·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s − 1.26·7-s + 1/3·9-s − 1.79·11-s + 1.14·13-s + 0.258·15-s − 1.86·17-s − 0.226·19-s + 0.730·21-s + 0.709·23-s + 1/5·25-s − 0.192·27-s + 0.157·29-s − 0.158·31-s + 1.03·33-s + 0.565·35-s − 1.35·37-s − 0.661·39-s + 0.869·41-s + 0.491·43-s − 0.149·45-s + 0.736·47-s + 0.600·49-s + 1.07·51-s + 0.689·53-s + 0.802·55-s + ⋯

Functional equation

Λ(s)=(240s/2 ΓC(s) L(s)=( Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
Λ(s)=(240s/2 ΓC(s+5/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 240240    =    24⋅3⋅52^{4} \cdot 3 \cdot 5
Sign: 11
Analytic conductor: 38.492138.4921
Root analytic conductor: 6.204206.20420
Motivic weight: 55
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 240, ( :5/2), 1)(2,\ 240,\ (\ :5/2),\ 1)

Particular Values

L(3)L(3) ≈\approx 0.61189800920.6118980092
L(12)L(\frac12) ≈\approx 0.61189800920.6118980092
L(72)L(\frac{7}{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
3 1+p2T 1 + p^{2} T
5 1+p2T 1 + p^{2} T
good7 1+164T+p5T2 1 + 164 T + p^{5} T^{2}
11 1+720T+p5T2 1 + 720 T + p^{5} T^{2}
13 1−698T+p5T2 1 - 698 T + p^{5} T^{2}
17 1+2226T+p5T2 1 + 2226 T + p^{5} T^{2}
19 1+356T+p5T2 1 + 356 T + p^{5} T^{2}
23 1−1800T+p5T2 1 - 1800 T + p^{5} T^{2}
29 1−714T+p5T2 1 - 714 T + p^{5} T^{2}
31 1+848T+p5T2 1 + 848 T + p^{5} T^{2}
37 1+11302T+p5T2 1 + 11302 T + p^{5} T^{2}
41 1−9354T+p5T2 1 - 9354 T + p^{5} T^{2}
43 1−5956T+p5T2 1 - 5956 T + p^{5} T^{2}
47 1−11160T+p5T2 1 - 11160 T + p^{5} T^{2}
53 1−14106T+p5T2 1 - 14106 T + p^{5} T^{2}
59 1+7920T+p5T2 1 + 7920 T + p^{5} T^{2}
61 1+13450T+p5T2 1 + 13450 T + p^{5} T^{2}
67 1−65476T+p5T2 1 - 65476 T + p^{5} T^{2}
71 1+34560T+p5T2 1 + 34560 T + p^{5} T^{2}
73 1−86258T+p5T2 1 - 86258 T + p^{5} T^{2}
79 1−108832T+p5T2 1 - 108832 T + p^{5} T^{2}
83 1+10668T+p5T2 1 + 10668 T + p^{5} T^{2}
89 1−10818T+p5T2 1 - 10818 T + p^{5} T^{2}
97 1−4418T+p5T2 1 - 4418 T + p^{5} 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

−10.89581353712377024745827803222, −10.72022391522388610886261329095, −9.344872854140805077518835704760, −8.348341694552421965485012134524, −7.08399523225577673860474131480, −6.25263771345026341743253032696, −5.10529778034645029102703644458, −3.80552022498707371377988913896, −2.51953314733531064468403921413, −0.44981945464797214779218231291, 0.44981945464797214779218231291, 2.51953314733531064468403921413, 3.80552022498707371377988913896, 5.10529778034645029102703644458, 6.25263771345026341743253032696, 7.08399523225577673860474131480, 8.348341694552421965485012134524, 9.344872854140805077518835704760, 10.72022391522388610886261329095, 10.89581353712377024745827803222

Graph of the ZZ-function along the critical line