↑

Properties

Label 2-1395-1395.1084-c0-0-1
Degree $2$
Conductor $1395$
Sign $-0.939 - 0.342i$
Analytic cond. $0.696195$
Root an. cond. $0.834383$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 + 0.866i)3-s + (−0.5 + 0.866i)4-s + (−0.5 + 0.866i)5-s + (−0.499 + 0.866i)9-s − 0.999·12-s + (−0.5 + 0.866i)13-s − 0.999·15-s + (−0.499 − 0.866i)16-s + 17-s − 19-s + (−0.499 − 0.866i)20-s + (1 − 1.73i)23-s + (−0.499 − 0.866i)25-s − 0.999·27-s + (−0.5 + 0.866i)31-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)3-s + (−0.5 + 0.866i)4-s + (−0.5 + 0.866i)5-s + (−0.499 + 0.866i)9-s − 0.999·12-s + (−0.5 + 0.866i)13-s − 0.999·15-s + (−0.499 − 0.866i)16-s + 17-s − 19-s + (−0.499 − 0.866i)20-s + (1 − 1.73i)23-s + (−0.499 − 0.866i)25-s − 0.999·27-s + (−0.5 + 0.866i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1395 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1395 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1395\)    =    \(3^{2} \cdot 5 \cdot 31\)
Sign: $-0.939 - 0.342i$
Analytic conductor: \(0.696195\)
Root analytic conductor: \(0.834383\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1395} (1084, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1395,\ (\ :0),\ -0.939 - 0.342i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8915609823\)
\(L(\frac12)\) \(\approx\) \(0.8915609823\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
31 \( 1 + (0.5 - 0.866i)T \)
good2 \( 1 + (0.5 - 0.866i)T^{2} \)
7 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
17 \( 1 - T + T^{2} \)
19 \( 1 + T + T^{2} \)
23 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 - T + T^{2} \)
41 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 - T + T^{2} \)
59 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 + T + T^{2} \)
73 \( 1 + 2T + T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (0.5 - 0.866i)T^{2} \)
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

−10.14019993583164816577505796274, −9.124800182770165479545611257544, −8.674409878270399535896206146597, −7.72453531440098749610325526180, −7.17771910386827418967469651255, −6.02730282933414767396184494588, −4.52502855373657117210145491752, −4.31149244031521960609674480718, −3.12999554363717579938571915455, −2.56889495042420291347104030893, 0.71799103131131201212197690063, 1.81799305482841325230290697010, 3.26258655458403652136986948237, 4.30682256298252175009719403360, 5.42332287831964364423837499173, 5.87490682267746146812107435783, 7.20372351583186424132979352840, 7.82064788852867906673176832885, 8.603321247502837111954577210779, 9.319527718104746411737978207038

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