Properties

Label 2-117-1.1-c3-0-2
Degree $2$
Conductor $117$
Sign $1$
Analytic cond. $6.90322$
Root an. cond. $2.62739$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 4.20·2-s + 9.71·4-s + 11.4·5-s − 11.2·7-s − 7.22·8-s − 48.1·10-s − 25.8·11-s + 13·13-s + 47.3·14-s − 47.3·16-s + 20.3·17-s + 154.·19-s + 111.·20-s + 108.·22-s + 180.·23-s + 5.69·25-s − 54.7·26-s − 109.·28-s + 20.4·29-s + 266.·31-s + 256.·32-s − 85.5·34-s − 128.·35-s + 115.·37-s − 651.·38-s − 82.5·40-s − 391.·41-s + ⋯
L(s)  = 1  − 1.48·2-s + 1.21·4-s + 1.02·5-s − 0.607·7-s − 0.319·8-s − 1.52·10-s − 0.709·11-s + 0.277·13-s + 0.904·14-s − 0.739·16-s + 0.290·17-s + 1.86·19-s + 1.24·20-s + 1.05·22-s + 1.63·23-s + 0.0455·25-s − 0.412·26-s − 0.738·28-s + 0.130·29-s + 1.54·31-s + 1.41·32-s − 0.431·34-s − 0.621·35-s + 0.515·37-s − 2.77·38-s − 0.326·40-s − 1.49·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $1$
Analytic conductor: \(6.90322\)
Root analytic conductor: \(2.62739\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(0.8608059666\)
\(L(\frac12)\) \(\approx\) \(0.8608059666\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 - 13T \)
good2 \( 1 + 4.20T + 8T^{2} \)
5 \( 1 - 11.4T + 125T^{2} \)
7 \( 1 + 11.2T + 343T^{2} \)
11 \( 1 + 25.8T + 1.33e3T^{2} \)
17 \( 1 - 20.3T + 4.91e3T^{2} \)
19 \( 1 - 154.T + 6.85e3T^{2} \)
23 \( 1 - 180.T + 1.21e4T^{2} \)
29 \( 1 - 20.4T + 2.43e4T^{2} \)
31 \( 1 - 266.T + 2.97e4T^{2} \)
37 \( 1 - 115.T + 5.06e4T^{2} \)
41 \( 1 + 391.T + 6.89e4T^{2} \)
43 \( 1 - 151.T + 7.95e4T^{2} \)
47 \( 1 - 467.T + 1.03e5T^{2} \)
53 \( 1 + 79.9T + 1.48e5T^{2} \)
59 \( 1 - 873.T + 2.05e5T^{2} \)
61 \( 1 + 187.T + 2.26e5T^{2} \)
67 \( 1 + 609.T + 3.00e5T^{2} \)
71 \( 1 + 248.T + 3.57e5T^{2} \)
73 \( 1 - 852.T + 3.89e5T^{2} \)
79 \( 1 + 331.T + 4.93e5T^{2} \)
83 \( 1 - 435.T + 5.71e5T^{2} \)
89 \( 1 + 259.T + 7.04e5T^{2} \)
97 \( 1 - 1.22e3T + 9.12e5T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.19380178243041017083216315785, −11.68758606210170029788769623936, −10.45589008053896860145497189166, −9.789808136460844883364061294352, −9.029318745885648176266668400799, −7.78670753134300063859372936665, −6.67832317975337838369391159663, −5.30267526773500931168090993651, −2.79389114844559719876005700761, −1.05559572769636285190084641436, 1.05559572769636285190084641436, 2.79389114844559719876005700761, 5.30267526773500931168090993651, 6.67832317975337838369391159663, 7.78670753134300063859372936665, 9.029318745885648176266668400799, 9.789808136460844883364061294352, 10.45589008053896860145497189166, 11.68758606210170029788769623936, 13.19380178243041017083216315785

Graph of the $Z$-function along the critical line