↑

Properties

Label 2-735-1.1-c3-0-42
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  + 4.70·2-s − 3·3-s + 14.1·4-s + 5·5-s − 14.1·6-s + 28.7·8-s + 9·9-s + 23.5·10-s + 24.5·11-s − 42.3·12-s + 35.0·13-s − 15·15-s + 22.1·16-s + 18.4·17-s + 42.3·18-s + 67.4·19-s + 70.5·20-s + 115.·22-s − 145.·23-s − 86.1·24-s + 25·25-s + 164.·26-s − 27·27-s + 214.·29-s − 70.5·30-s + 88.6·31-s − 125.·32-s + ⋯
L(s)  = 1  + 1.66·2-s − 0.577·3-s + 1.76·4-s + 0.447·5-s − 0.959·6-s + 1.26·8-s + 0.333·9-s + 0.743·10-s + 0.674·11-s − 1.01·12-s + 0.747·13-s − 0.258·15-s + 0.345·16-s + 0.262·17-s + 0.554·18-s + 0.813·19-s + 0.788·20-s + 1.12·22-s − 1.32·23-s − 0.732·24-s + 0.200·25-s + 1.24·26-s − 0.192·27-s + 1.37·29-s − 0.429·30-s + 0.513·31-s − 0.694·32-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 5.4860270285.486027028
L(12)L(\frac12) ≈\approx 5.4860270285.486027028
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 1−4.70T+8T2 1 - 4.70T + 8T^{2}
11 1−24.5T+1.33e3T2 1 - 24.5T + 1.33e3T^{2}
13 1−35.0T+2.19e3T2 1 - 35.0T + 2.19e3T^{2}
17 1−18.4T+4.91e3T2 1 - 18.4T + 4.91e3T^{2}
19 1−67.4T+6.85e3T2 1 - 67.4T + 6.85e3T^{2}
23 1+145.T+1.21e4T2 1 + 145.T + 1.21e4T^{2}
29 1−214.T+2.43e4T2 1 - 214.T + 2.43e4T^{2}
31 1−88.6T+2.97e4T2 1 - 88.6T + 2.97e4T^{2}
37 1−162.T+5.06e4T2 1 - 162.T + 5.06e4T^{2}
41 1−337.T+6.89e4T2 1 - 337.T + 6.89e4T^{2}
43 1−122.T+7.95e4T2 1 - 122.T + 7.95e4T^{2}
47 1+354.T+1.03e5T2 1 + 354.T + 1.03e5T^{2}
53 1−676.T+1.48e5T2 1 - 676.T + 1.48e5T^{2}
59 1+501.T+2.05e5T2 1 + 501.T + 2.05e5T^{2}
61 1−708.T+2.26e5T2 1 - 708.T + 2.26e5T^{2}
67 1+907.T+3.00e5T2 1 + 907.T + 3.00e5T^{2}
71 1−430.T+3.57e5T2 1 - 430.T + 3.57e5T^{2}
73 1+41.3T+3.89e5T2 1 + 41.3T + 3.89e5T^{2}
79 1−890.T+4.93e5T2 1 - 890.T + 4.93e5T^{2}
83 1−1.05e3T+5.71e5T2 1 - 1.05e3T + 5.71e5T^{2}
89 1+1.47e3T+7.04e5T2 1 + 1.47e3T + 7.04e5T^{2}
97 1+555.T+9.12e5T2 1 + 555.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

−10.26243306453489764673215649701, −9.312958750931259854264478827614, −8.014715513445701045886512751649, −6.80558046091901650361741660254, −6.15139409839427904295959889246, −5.53996684522321551081043933933, −4.51515373973666636588781518725, −3.73929958869782030011687353693, −2.55989918564782867568502447585, −1.17920045757108547304782338798, 1.17920045757108547304782338798, 2.55989918564782867568502447585, 3.73929958869782030011687353693, 4.51515373973666636588781518725, 5.53996684522321551081043933933, 6.15139409839427904295959889246, 6.80558046091901650361741660254, 8.014715513445701045886512751649, 9.312958750931259854264478827614, 10.26243306453489764673215649701

Graph of the ZZ-function along the critical line