Properties

Label 2-135-1.1-c3-0-3
Degree $2$
Conductor $135$
Sign $1$
Analytic cond. $7.96525$
Root an. cond. $2.82227$
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  − 5.20·2-s + 19.0·4-s + 5·5-s + 24.4·7-s − 57.4·8-s − 26.0·10-s + 28.9·11-s − 65.3·13-s − 126.·14-s + 146.·16-s − 68.1·17-s + 104.·19-s + 95.2·20-s − 150.·22-s + 154.·23-s + 25·25-s + 340.·26-s + 464.·28-s + 205.·29-s − 18.2·31-s − 301.·32-s + 354.·34-s + 122.·35-s − 337.·37-s − 543.·38-s − 287.·40-s + 195.·41-s + ⋯
L(s)  = 1  − 1.83·2-s + 2.38·4-s + 0.447·5-s + 1.31·7-s − 2.53·8-s − 0.822·10-s + 0.794·11-s − 1.39·13-s − 2.42·14-s + 2.28·16-s − 0.972·17-s + 1.26·19-s + 1.06·20-s − 1.46·22-s + 1.40·23-s + 0.200·25-s + 2.56·26-s + 3.13·28-s + 1.31·29-s − 0.105·31-s − 1.66·32-s + 1.78·34-s + 0.589·35-s − 1.50·37-s − 2.31·38-s − 1.13·40-s + 0.746·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(135\)    =    \(3^{3} \cdot 5\)
Sign: $1$
Analytic conductor: \(7.96525\)
Root analytic conductor: \(2.82227\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{135} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 135,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(0.8901940760\)
\(L(\frac12)\) \(\approx\) \(0.8901940760\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 - 5T \)
good2 \( 1 + 5.20T + 8T^{2} \)
7 \( 1 - 24.4T + 343T^{2} \)
11 \( 1 - 28.9T + 1.33e3T^{2} \)
13 \( 1 + 65.3T + 2.19e3T^{2} \)
17 \( 1 + 68.1T + 4.91e3T^{2} \)
19 \( 1 - 104.T + 6.85e3T^{2} \)
23 \( 1 - 154.T + 1.21e4T^{2} \)
29 \( 1 - 205.T + 2.43e4T^{2} \)
31 \( 1 + 18.2T + 2.97e4T^{2} \)
37 \( 1 + 337.T + 5.06e4T^{2} \)
41 \( 1 - 195.T + 6.89e4T^{2} \)
43 \( 1 - 334.T + 7.95e4T^{2} \)
47 \( 1 - 5.00T + 1.03e5T^{2} \)
53 \( 1 - 319.T + 1.48e5T^{2} \)
59 \( 1 + 430.T + 2.05e5T^{2} \)
61 \( 1 - 594.T + 2.26e5T^{2} \)
67 \( 1 - 195.T + 3.00e5T^{2} \)
71 \( 1 + 425.T + 3.57e5T^{2} \)
73 \( 1 - 929.T + 3.89e5T^{2} \)
79 \( 1 - 24.4T + 4.93e5T^{2} \)
83 \( 1 - 545.T + 5.71e5T^{2} \)
89 \( 1 - 84.1T + 7.04e5T^{2} \)
97 \( 1 - 827.T + 9.12e5T^{2} \)
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

−12.20313426021897394366406670674, −11.38871870626451435951230400215, −10.52834706060518849615971317137, −9.440195046436596098741513155775, −8.748874984565395795904699133598, −7.58604885732771397060705404899, −6.77982788961548332106954320230, −5.03793412252658427284035642149, −2.41680980255820951534523683057, −1.11031820612410364928005082638, 1.11031820612410364928005082638, 2.41680980255820951534523683057, 5.03793412252658427284035642149, 6.77982788961548332106954320230, 7.58604885732771397060705404899, 8.748874984565395795904699133598, 9.440195046436596098741513155775, 10.52834706060518849615971317137, 11.38871870626451435951230400215, 12.20313426021897394366406670674

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