↑

Properties

Label 2-195-1.1-c5-0-34
Degree 22
Conductor 195195
Sign −1-1
Analytic cond. 31.274831.2748
Root an. cond. 5.592395.59239
Motivic weight 55
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 10.1·2-s + 9·3-s + 71.2·4-s + 25·5-s − 91.4·6-s + 177.·7-s − 399.·8-s + 81·9-s − 254.·10-s − 242.·11-s + 641.·12-s − 169·13-s − 1.80e3·14-s + 225·15-s + 1.77e3·16-s − 2.15e3·17-s − 823.·18-s − 2.44e3·19-s + 1.78e3·20-s + 1.59e3·21-s + 2.46e3·22-s − 2.31e3·23-s − 3.59e3·24-s + 625·25-s + 1.71e3·26-s + 729·27-s + 1.26e4·28-s + ⋯
L(s)  = 1  − 1.79·2-s + 0.577·3-s + 2.22·4-s + 0.447·5-s − 1.03·6-s + 1.36·7-s − 2.20·8-s + 0.333·9-s − 0.803·10-s − 0.604·11-s + 1.28·12-s − 0.277·13-s − 2.45·14-s + 0.258·15-s + 1.73·16-s − 1.80·17-s − 0.598·18-s − 1.55·19-s + 0.996·20-s + 0.789·21-s + 1.08·22-s − 0.913·23-s − 1.27·24-s + 0.200·25-s + 0.498·26-s + 0.192·27-s + 3.04·28-s + ⋯

Functional equation

Λ(s)=(195s/2 ΓC(s) L(s)=(− Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}
Λ(s)=(195s/2 ΓC(s+5/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 195195    =    3⋅5⋅133 \cdot 5 \cdot 13
Sign: −1-1
Analytic conductor: 31.274831.2748
Root analytic conductor: 5.592395.59239
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 195, ( :5/2), −1)(2,\ 195,\ (\ :5/2),\ -1)

Particular Values

L(3)L(3) == 00
L(12)L(\frac12) == 00
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)
bad3 1−9T 1 - 9T
5 1−25T 1 - 25T
13 1+169T 1 + 169T
good2 1+10.1T+32T2 1 + 10.1T + 32T^{2}
7 1−177.T+1.68e4T2 1 - 177.T + 1.68e4T^{2}
11 1+242.T+1.61e5T2 1 + 242.T + 1.61e5T^{2}
17 1+2.15e3T+1.41e6T2 1 + 2.15e3T + 1.41e6T^{2}
19 1+2.44e3T+2.47e6T2 1 + 2.44e3T + 2.47e6T^{2}
23 1+2.31e3T+6.43e6T2 1 + 2.31e3T + 6.43e6T^{2}
29 1−550.T+2.05e7T2 1 - 550.T + 2.05e7T^{2}
31 1−1.68e3T+2.86e7T2 1 - 1.68e3T + 2.86e7T^{2}
37 1+9.96e3T+6.93e7T2 1 + 9.96e3T + 6.93e7T^{2}
41 1+3.86e3T+1.15e8T2 1 + 3.86e3T + 1.15e8T^{2}
43 1−1.15e4T+1.47e8T2 1 - 1.15e4T + 1.47e8T^{2}
47 1+1.98e4T+2.29e8T2 1 + 1.98e4T + 2.29e8T^{2}
53 1+7.57e3T+4.18e8T2 1 + 7.57e3T + 4.18e8T^{2}
59 1−2.03e3T+7.14e8T2 1 - 2.03e3T + 7.14e8T^{2}
61 1−3.08e4T+8.44e8T2 1 - 3.08e4T + 8.44e8T^{2}
67 1+4.64e4T+1.35e9T2 1 + 4.64e4T + 1.35e9T^{2}
71 1−2.42e3T+1.80e9T2 1 - 2.42e3T + 1.80e9T^{2}
73 1+7.42e4T+2.07e9T2 1 + 7.42e4T + 2.07e9T^{2}
79 1−7.18e4T+3.07e9T2 1 - 7.18e4T + 3.07e9T^{2}
83 1+6.94e4T+3.93e9T2 1 + 6.94e4T + 3.93e9T^{2}
89 1+6.32e4T+5.58e9T2 1 + 6.32e4T + 5.58e9T^{2}
97 1−1.37e5T+8.58e9T2 1 - 1.37e5T + 8.58e9T^{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

−10.76019327914351845726276339339, −10.09476970092954004458432378741, −8.808389236938870029138926531982, −8.442528139930943302911173233562, −7.47097419004719226147916187528, −6.37714676921522357119539044271, −4.61264535085130128486391072591, −2.33886415484967366231048390978, −1.75760940851970976289839247184, 0, 1.75760940851970976289839247184, 2.33886415484967366231048390978, 4.61264535085130128486391072591, 6.37714676921522357119539044271, 7.47097419004719226147916187528, 8.442528139930943302911173233562, 8.808389236938870029138926531982, 10.09476970092954004458432378741, 10.76019327914351845726276339339

Graph of the ZZ-function along the critical line