↑

Properties

Label 2-810-15.14-c2-0-38
Degree $2$
Conductor $810$
Sign $0.569 + 0.822i$
Analytic cond. $22.0709$
Root an. cond. $4.69796$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 1.41·2-s + 2.00·4-s + (4.11 − 2.84i)5-s − 9.58i·7-s + 2.82·8-s + (5.81 − 4.02i)10-s + 10.2i·11-s + 3.45i·13-s − 13.5i·14-s + 4.00·16-s + 30.5·17-s + 10.0·19-s + (8.22 − 5.69i)20-s + 14.4i·22-s − 30.6·23-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.500·4-s + (0.822 − 0.569i)5-s − 1.36i·7-s + 0.353·8-s + (0.581 − 0.402i)10-s + 0.927i·11-s + 0.265i·13-s − 0.968i·14-s + 0.250·16-s + 1.79·17-s + 0.530·19-s + (0.411 − 0.284i)20-s + 0.656i·22-s − 1.33·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.569 + 0.822i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.569 + 0.822i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(810\)    =    \(2 \cdot 3^{4} \cdot 5\)
Sign: $0.569 + 0.822i$
Analytic conductor: \(22.0709\)
Root analytic conductor: \(4.69796\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{810} (809, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 810,\ (\ :1),\ 0.569 + 0.822i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.511621094\)
\(L(\frac12)\) \(\approx\) \(3.511621094\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 1.41T \)
3 \( 1 \)
5 \( 1 + (-4.11 + 2.84i)T \)
good7 \( 1 + 9.58iT - 49T^{2} \)
11 \( 1 - 10.2iT - 121T^{2} \)
13 \( 1 - 3.45iT - 169T^{2} \)
17 \( 1 - 30.5T + 289T^{2} \)
19 \( 1 - 10.0T + 361T^{2} \)
23 \( 1 + 30.6T + 529T^{2} \)
29 \( 1 + 42.0iT - 841T^{2} \)
31 \( 1 + 10.0T + 961T^{2} \)
37 \( 1 + 21.9iT - 1.36e3T^{2} \)
41 \( 1 + 39.6iT - 1.68e3T^{2} \)
43 \( 1 - 21.6iT - 1.84e3T^{2} \)
47 \( 1 - 12.5T + 2.20e3T^{2} \)
53 \( 1 - 47.0T + 2.80e3T^{2} \)
59 \( 1 - 32.3iT - 3.48e3T^{2} \)
61 \( 1 + 114.T + 3.72e3T^{2} \)
67 \( 1 - 32.2iT - 4.48e3T^{2} \)
71 \( 1 + 65.9iT - 5.04e3T^{2} \)
73 \( 1 - 54.1iT - 5.32e3T^{2} \)
79 \( 1 - 88.1T + 6.24e3T^{2} \)
83 \( 1 - 41.1T + 6.88e3T^{2} \)
89 \( 1 + 38.5iT - 7.92e3T^{2} \)
97 \( 1 - 128. iT - 9.40e3T^{2} \)
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

−10.06076721964719217316592671862, −9.361315307011499895798785560831, −7.86355058552134603239408679582, −7.38666753570831236436049643770, −6.24329089167554200296580735443, −5.44098754147489657525913835718, −4.44272788177815112086404146065, −3.69251075420150344362817548270, −2.15945064326197649754890833920, −1.01192673484240071187170260292, 1.58248644561266537681539871066, 2.83559346383730179304193824621, 3.42947067432803213143992418333, 5.20298149271658088061387749002, 5.70646100602232042130426215540, 6.30902549245564044638739814054, 7.53406538456448643519759681772, 8.465732716929503270631896396420, 9.453508974983739569425998498995, 10.22285409432367551756707663778

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