↑

Properties

Label 2-693-1.1-c3-0-30
Degree 22
Conductor 693693
Sign −1-1
Analytic cond. 40.888340.8883
Root an. cond. 6.394396.39439
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2.21·2-s − 3.08·4-s − 22.0·5-s − 7·7-s + 24.5·8-s + 48.8·10-s + 11·11-s + 2.76·13-s + 15.5·14-s − 29.7·16-s − 50.6·17-s + 56.5·19-s + 68.0·20-s − 24.3·22-s − 172.·23-s + 360.·25-s − 6.13·26-s + 21.6·28-s + 282.·29-s + 237.·31-s − 130.·32-s + 112.·34-s + 154.·35-s + 207.·37-s − 125.·38-s − 541.·40-s + 386.·41-s + ⋯
L(s)  = 1  − 0.783·2-s − 0.386·4-s − 1.97·5-s − 0.377·7-s + 1.08·8-s + 1.54·10-s + 0.301·11-s + 0.0590·13-s + 0.296·14-s − 0.464·16-s − 0.722·17-s + 0.682·19-s + 0.761·20-s − 0.236·22-s − 1.56·23-s + 2.88·25-s − 0.0462·26-s + 0.145·28-s + 1.80·29-s + 1.37·31-s − 0.721·32-s + 0.566·34-s + 0.745·35-s + 0.920·37-s − 0.534·38-s − 2.14·40-s + 1.47·41-s + ⋯

Functional equation

Λ(s)=(693s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(693s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 693693    =    32⋅7⋅113^{2} \cdot 7 \cdot 11
Sign: −1-1
Analytic conductor: 40.888340.8883
Root analytic conductor: 6.394396.39439
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 693, ( :3/2), −1)(2,\ 693,\ (\ :3/2),\ -1)

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
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 1
7 1+7T 1 + 7T
11 1−11T 1 - 11T
good2 1+2.21T+8T2 1 + 2.21T + 8T^{2}
5 1+22.0T+125T2 1 + 22.0T + 125T^{2}
13 1−2.76T+2.19e3T2 1 - 2.76T + 2.19e3T^{2}
17 1+50.6T+4.91e3T2 1 + 50.6T + 4.91e3T^{2}
19 1−56.5T+6.85e3T2 1 - 56.5T + 6.85e3T^{2}
23 1+172.T+1.21e4T2 1 + 172.T + 1.21e4T^{2}
29 1−282.T+2.43e4T2 1 - 282.T + 2.43e4T^{2}
31 1−237.T+2.97e4T2 1 - 237.T + 2.97e4T^{2}
37 1−207.T+5.06e4T2 1 - 207.T + 5.06e4T^{2}
41 1−386.T+6.89e4T2 1 - 386.T + 6.89e4T^{2}
43 1+18.3T+7.95e4T2 1 + 18.3T + 7.95e4T^{2}
47 1+309.T+1.03e5T2 1 + 309.T + 1.03e5T^{2}
53 1+480.T+1.48e5T2 1 + 480.T + 1.48e5T^{2}
59 1−114.T+2.05e5T2 1 - 114.T + 2.05e5T^{2}
61 1−109.T+2.26e5T2 1 - 109.T + 2.26e5T^{2}
67 1+567.T+3.00e5T2 1 + 567.T + 3.00e5T^{2}
71 1+780.T+3.57e5T2 1 + 780.T + 3.57e5T^{2}
73 1+605.T+3.89e5T2 1 + 605.T + 3.89e5T^{2}
79 1−686.T+4.93e5T2 1 - 686.T + 4.93e5T^{2}
83 1−316.T+5.71e5T2 1 - 316.T + 5.71e5T^{2}
89 1+1.61e3T+7.04e5T2 1 + 1.61e3T + 7.04e5T^{2}
97 1+1.23e3T+9.12e5T2 1 + 1.23e3T + 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

−9.565300761379069725071075913408, −8.549128425232762401851332835534, −8.081822090582430467459573901012, −7.35547771466274808680167967036, −6.34292192595067785689024680593, −4.59020259373986539081993937900, −4.19846332866758180011269420706, −2.99989576836360028894085390628, −0.968869054385734853890764973025, 0, 0.968869054385734853890764973025, 2.99989576836360028894085390628, 4.19846332866758180011269420706, 4.59020259373986539081993937900, 6.34292192595067785689024680593, 7.35547771466274808680167967036, 8.081822090582430467459573901012, 8.549128425232762401851332835534, 9.565300761379069725071075913408

Graph of the ZZ-function along the critical line