Properties

Label 2-387-1.1-c7-0-82
Degree $2$
Conductor $387$
Sign $-1$
Analytic cond. $120.893$
Root an. cond. $10.9951$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3.37·2-s − 116.·4-s + 241.·5-s − 173.·7-s + 826.·8-s − 814.·10-s − 682.·11-s + 168.·13-s + 584.·14-s + 1.21e4·16-s − 5.56e3·17-s − 1.38e4·19-s − 2.81e4·20-s + 2.30e3·22-s + 2.46e4·23-s − 1.99e4·25-s − 567.·26-s + 2.01e4·28-s + 3.36e4·29-s + 2.27e4·31-s − 1.46e5·32-s + 1.87e4·34-s − 4.17e4·35-s + 1.81e5·37-s + 4.67e4·38-s + 1.99e5·40-s − 3.35e5·41-s + ⋯
L(s)  = 1  − 0.298·2-s − 0.910·4-s + 0.862·5-s − 0.190·7-s + 0.570·8-s − 0.257·10-s − 0.154·11-s + 0.0212·13-s + 0.0569·14-s + 0.740·16-s − 0.274·17-s − 0.462·19-s − 0.785·20-s + 0.0461·22-s + 0.422·23-s − 0.255·25-s − 0.00633·26-s + 0.173·28-s + 0.256·29-s + 0.136·31-s − 0.791·32-s + 0.0820·34-s − 0.164·35-s + 0.589·37-s + 0.138·38-s + 0.492·40-s − 0.759·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(387\)    =    \(3^{2} \cdot 43\)
Sign: $-1$
Analytic conductor: \(120.893\)
Root analytic conductor: \(10.9951\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{387} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 387,\ (\ :7/2),\ -1)\)

Particular Values

\(L(4)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
43 \( 1 + 7.95e4T \)
good2 \( 1 + 3.37T + 128T^{2} \)
5 \( 1 - 241.T + 7.81e4T^{2} \)
7 \( 1 + 173.T + 8.23e5T^{2} \)
11 \( 1 + 682.T + 1.94e7T^{2} \)
13 \( 1 - 168.T + 6.27e7T^{2} \)
17 \( 1 + 5.56e3T + 4.10e8T^{2} \)
19 \( 1 + 1.38e4T + 8.93e8T^{2} \)
23 \( 1 - 2.46e4T + 3.40e9T^{2} \)
29 \( 1 - 3.36e4T + 1.72e10T^{2} \)
31 \( 1 - 2.27e4T + 2.75e10T^{2} \)
37 \( 1 - 1.81e5T + 9.49e10T^{2} \)
41 \( 1 + 3.35e5T + 1.94e11T^{2} \)
47 \( 1 - 7.43e5T + 5.06e11T^{2} \)
53 \( 1 - 1.26e4T + 1.17e12T^{2} \)
59 \( 1 - 8.30e5T + 2.48e12T^{2} \)
61 \( 1 - 2.22e6T + 3.14e12T^{2} \)
67 \( 1 - 3.88e6T + 6.06e12T^{2} \)
71 \( 1 + 3.88e6T + 9.09e12T^{2} \)
73 \( 1 - 5.22e6T + 1.10e13T^{2} \)
79 \( 1 + 2.64e6T + 1.92e13T^{2} \)
83 \( 1 + 6.76e6T + 2.71e13T^{2} \)
89 \( 1 + 3.27e5T + 4.42e13T^{2} \)
97 \( 1 - 5.05e6T + 8.07e13T^{2} \)
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

−9.709781877435118466605261760572, −8.866014653009412210739777927815, −8.067245303352862256690625261378, −6.83493546166278422643610090348, −5.77291577521282009311851016442, −4.88178812857600409731129405393, −3.79040180666967503508356582325, −2.40118614154969101758647349965, −1.18658828312449388506593558455, 0, 1.18658828312449388506593558455, 2.40118614154969101758647349965, 3.79040180666967503508356582325, 4.88178812857600409731129405393, 5.77291577521282009311851016442, 6.83493546166278422643610090348, 8.067245303352862256690625261378, 8.866014653009412210739777927815, 9.709781877435118466605261760572

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