↑

Properties

Label 2-735-1.1-c3-0-5
Degree 22
Conductor 735735
Sign 11
Analytic cond. 43.366443.3664
Root an. cond. 6.585316.58531
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2.32·2-s − 3·3-s − 2.57·4-s − 5·5-s + 6.98·6-s + 24.6·8-s + 9·9-s + 11.6·10-s + 52.6·11-s + 7.73·12-s − 78.1·13-s + 15·15-s − 36.7·16-s + 76.1·17-s − 20.9·18-s − 73.8·19-s + 12.8·20-s − 122.·22-s + 137.·23-s − 73.8·24-s + 25·25-s + 182.·26-s − 27·27-s − 289.·29-s − 34.9·30-s + 83.9·31-s − 111.·32-s + ⋯
L(s)  = 1  − 0.823·2-s − 0.577·3-s − 0.322·4-s − 0.447·5-s + 0.475·6-s + 1.08·8-s + 0.333·9-s + 0.368·10-s + 1.44·11-s + 0.186·12-s − 1.66·13-s + 0.258·15-s − 0.574·16-s + 1.08·17-s − 0.274·18-s − 0.892·19-s + 0.144·20-s − 1.18·22-s + 1.25·23-s − 0.628·24-s + 0.200·25-s + 1.37·26-s − 0.192·27-s − 1.85·29-s − 0.212·30-s + 0.486·31-s − 0.615·32-s + ⋯

Functional equation

Λ(s)=(735s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(735s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 735735    =    3⋅5⋅723 \cdot 5 \cdot 7^{2}
Sign: 11
Analytic conductor: 43.366443.3664
Root analytic conductor: 6.585316.58531
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 735, ( :3/2), 1)(2,\ 735,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 0.61428216070.6142821607
L(12)L(\frac12) ≈\approx 0.61428216070.6142821607
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)
bad3 1+3T 1 + 3T
5 1+5T 1 + 5T
7 1 1
good2 1+2.32T+8T2 1 + 2.32T + 8T^{2}
11 1−52.6T+1.33e3T2 1 - 52.6T + 1.33e3T^{2}
13 1+78.1T+2.19e3T2 1 + 78.1T + 2.19e3T^{2}
17 1−76.1T+4.91e3T2 1 - 76.1T + 4.91e3T^{2}
19 1+73.8T+6.85e3T2 1 + 73.8T + 6.85e3T^{2}
23 1−137.T+1.21e4T2 1 - 137.T + 1.21e4T^{2}
29 1+289.T+2.43e4T2 1 + 289.T + 2.43e4T^{2}
31 1−83.9T+2.97e4T2 1 - 83.9T + 2.97e4T^{2}
37 1+324.T+5.06e4T2 1 + 324.T + 5.06e4T^{2}
41 1+392.T+6.89e4T2 1 + 392.T + 6.89e4T^{2}
43 1−459.T+7.95e4T2 1 - 459.T + 7.95e4T^{2}
47 1+138.T+1.03e5T2 1 + 138.T + 1.03e5T^{2}
53 1+494.T+1.48e5T2 1 + 494.T + 1.48e5T^{2}
59 1−174.T+2.05e5T2 1 - 174.T + 2.05e5T^{2}
61 1−88.6T+2.26e5T2 1 - 88.6T + 2.26e5T^{2}
67 1−641.T+3.00e5T2 1 - 641.T + 3.00e5T^{2}
71 1−672.T+3.57e5T2 1 - 672.T + 3.57e5T^{2}
73 1−456.T+3.89e5T2 1 - 456.T + 3.89e5T^{2}
79 1+604.T+4.93e5T2 1 + 604.T + 4.93e5T^{2}
83 1+868.T+5.71e5T2 1 + 868.T + 5.71e5T^{2}
89 1−126.T+7.04e5T2 1 - 126.T + 7.04e5T^{2}
97 1+289.T+9.12e5T2 1 + 289.T + 9.12e5T^{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

−9.791629285685427396301272924939, −9.299025697396090363861341555212, −8.356499894665962983358668168581, −7.36198856867049039050792492127, −6.82363541675593018287473658891, −5.39574348618124850871360967461, −4.57863605538812904309095350011, −3.57689704710759379144958574584, −1.74148627855439960545428395030, −0.53712849838396446595494277940, 0.53712849838396446595494277940, 1.74148627855439960545428395030, 3.57689704710759379144958574584, 4.57863605538812904309095350011, 5.39574348618124850871360967461, 6.82363541675593018287473658891, 7.36198856867049039050792492127, 8.356499894665962983358668168581, 9.299025697396090363861341555212, 9.791629285685427396301272924939

Graph of the ZZ-function along the critical line