↑

Properties

Label 2-2e9-16.5-c3-0-6
Degree $2$
Conductor $512$
Sign $0.382 - 0.923i$
Analytic cond. $30.2089$
Root an. cond. $5.49626$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−4.47 + 4.47i)3-s + (−7 − 7i)5-s − 8.94i·7-s − 13.0i·9-s + (−40.2 − 40.2i)11-s + (−45 + 45i)13-s + 62.6·15-s − 16·17-s + (76.0 − 76.0i)19-s + (40.0 + 40.0i)21-s + 8.94i·23-s − 27i·25-s + (−62.6 − 62.6i)27-s + (−67 + 67i)29-s + 304.·31-s + ⋯
L(s)  = 1  + (−0.860 + 0.860i)3-s + (−0.626 − 0.626i)5-s − 0.482i·7-s − 0.481i·9-s + (−1.10 − 1.10i)11-s + (−0.960 + 0.960i)13-s + 1.07·15-s − 0.228·17-s + (0.917 − 0.917i)19-s + (0.415 + 0.415i)21-s + 0.0810i·23-s − 0.215i·25-s + (−0.446 − 0.446i)27-s + (−0.429 + 0.429i)29-s + 1.76·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 - 0.923i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(512\)    =    \(2^{9}\)
Sign: $0.382 - 0.923i$
Analytic conductor: \(30.2089\)
Root analytic conductor: \(5.49626\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{512} (385, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 512,\ (\ :3/2),\ 0.382 - 0.923i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.6379991701\)
\(L(\frac12)\) \(\approx\) \(0.6379991701\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
good3 \( 1 + (4.47 - 4.47i)T - 27iT^{2} \)
5 \( 1 + (7 + 7i)T + 125iT^{2} \)
7 \( 1 + 8.94iT - 343T^{2} \)
11 \( 1 + (40.2 + 40.2i)T + 1.33e3iT^{2} \)
13 \( 1 + (45 - 45i)T - 2.19e3iT^{2} \)
17 \( 1 + 16T + 4.91e3T^{2} \)
19 \( 1 + (-76.0 + 76.0i)T - 6.85e3iT^{2} \)
23 \( 1 - 8.94iT - 1.21e4T^{2} \)
29 \( 1 + (67 - 67i)T - 2.43e4iT^{2} \)
31 \( 1 - 304.T + 2.97e4T^{2} \)
37 \( 1 + (9 + 9i)T + 5.06e4iT^{2} \)
41 \( 1 - 328iT - 6.89e4T^{2} \)
43 \( 1 + (174. + 174. i)T + 7.95e4iT^{2} \)
47 \( 1 + 411.T + 1.03e5T^{2} \)
53 \( 1 + (-407 - 407i)T + 1.48e5iT^{2} \)
59 \( 1 + (-523. - 523. i)T + 2.05e5iT^{2} \)
61 \( 1 + (589 - 589i)T - 2.26e5iT^{2} \)
67 \( 1 + (210. - 210. i)T - 3.00e5iT^{2} \)
71 \( 1 - 688. iT - 3.57e5T^{2} \)
73 \( 1 + 782iT - 3.89e5T^{2} \)
79 \( 1 + 429.T + 4.93e5T^{2} \)
83 \( 1 + (22.3 - 22.3i)T - 5.71e5iT^{2} \)
89 \( 1 - 510iT - 7.04e5T^{2} \)
97 \( 1 - 1.21e3T + 9.12e5T^{2} \)
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

−10.62956993728188526933754817532, −9.990587350493314470122382894444, −8.942088723491057688332755058341, −7.996540539316849224712407606687, −7.02336110499907030934847945585, −5.72658803153033367460250323147, −4.83287408880591062696330494354, −4.30019901143732261886021228806, −2.84129864921127716709083890819, −0.69683083385847460203116310939, 0.35549380936840880650823049417, 2.07405449802428428582102697677, 3.24963419522186573071383279743, 4.90308200853927131452359115122, 5.66061489569108722202138577985, 6.78537895226381582252294155515, 7.52446327955591466527962752594, 8.078597998422646231060493811666, 9.734526316899985074926367763256, 10.35245051787633491742727808924

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