Properties

Degree 2
Conductor $ 2^{3} $
Sign $0.640 - 0.767i$
Motivic weight 13
Primitive yes
Self-dual no
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + (65.8 + 62.0i)2-s − 1.23e3i·3-s + (489. + 8.17e3i)4-s + 2.52e4i·5-s + (7.64e4 − 8.11e4i)6-s + 6.08e5·7-s + (−4.75e5 + 5.69e5i)8-s + 7.79e4·9-s + (−1.56e6 + 1.66e6i)10-s + 7.12e6i·11-s + (1.00e7 − 6.02e5i)12-s − 1.12e7i·13-s + (4.00e7 + 3.77e7i)14-s + 3.11e7·15-s + (−6.66e7 + 8.00e6i)16-s − 5.09e7·17-s + ⋯
L(s)  = 1  + (0.727 + 0.685i)2-s − 0.975i·3-s + (0.0597 + 0.998i)4-s + 0.723i·5-s + (0.668 − 0.709i)6-s + 1.95·7-s + (−0.640 + 0.767i)8-s + 0.0488·9-s + (−0.495 + 0.526i)10-s + 1.21i·11-s + (0.973 − 0.0582i)12-s − 0.648i·13-s + (1.42 + 1.33i)14-s + 0.705·15-s + (−0.992 + 0.119i)16-s − 0.511·17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.640 - 0.767i)\, \overline{\Lambda}(14-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (0.640 - 0.767i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(8\)    =    \(2^{3}\)
\( \varepsilon \)  =  $0.640 - 0.767i$
motivic weight  =  \(13\)
character  :  $\chi_{8} (5, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 8,\ (\ :13/2),\ 0.640 - 0.767i)$
$L(7)$  $\approx$  $2.55512 + 1.19527i$
$L(\frac12)$  $\approx$  $2.55512 + 1.19527i$
$L(\frac{15}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 2$, \(F_p\) is a polynomial of degree 2. If $p = 2$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 + (-65.8 - 62.0i)T \)
good3 \( 1 + 1.23e3iT - 1.59e6T^{2} \)
5 \( 1 - 2.52e4iT - 1.22e9T^{2} \)
7 \( 1 - 6.08e5T + 9.68e10T^{2} \)
11 \( 1 - 7.12e6iT - 3.45e13T^{2} \)
13 \( 1 + 1.12e7iT - 3.02e14T^{2} \)
17 \( 1 + 5.09e7T + 9.90e15T^{2} \)
19 \( 1 + 1.86e8iT - 4.20e16T^{2} \)
23 \( 1 + 3.07e8T + 5.04e17T^{2} \)
29 \( 1 - 4.74e8iT - 1.02e19T^{2} \)
31 \( 1 + 2.41e9T + 2.44e19T^{2} \)
37 \( 1 + 1.55e10iT - 2.43e20T^{2} \)
41 \( 1 + 2.32e10T + 9.25e20T^{2} \)
43 \( 1 - 6.55e10iT - 1.71e21T^{2} \)
47 \( 1 + 6.84e10T + 5.46e21T^{2} \)
53 \( 1 + 9.64e10iT - 2.60e22T^{2} \)
59 \( 1 + 3.22e11iT - 1.04e23T^{2} \)
61 \( 1 + 6.66e11iT - 1.61e23T^{2} \)
67 \( 1 - 5.91e11iT - 5.48e23T^{2} \)
71 \( 1 - 8.43e11T + 1.16e24T^{2} \)
73 \( 1 + 7.21e11T + 1.67e24T^{2} \)
79 \( 1 - 5.78e11T + 4.66e24T^{2} \)
83 \( 1 + 1.54e11iT - 8.87e24T^{2} \)
89 \( 1 - 2.38e12T + 2.19e25T^{2} \)
97 \( 1 + 1.26e13T + 6.73e25T^{2} \)
show more
show less
\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−18.01190991297890747425172424913, −17.71589917394948224495098286825, −15.22840407429498477697154768548, −14.29071813278238700296139301621, −12.76940851115830205490987300544, −11.28660286471362163711734934589, −7.977332520532549214505395441524, −6.95079489358404702701397500004, −4.79021458568723709447934504244, −2.05597772219470668917871605546, 1.47206363951694865860601046652, 4.13847733670970732735505770859, 5.23353068686638011330789676137, 8.727530566140184871235034888812, 10.63564263509288000676105637554, 11.78285288767715316408471987830, 13.81902721714609421196886197479, 15.02879127992900801465866422669, 16.55995466000176649178516657766, 18.52347794703156014354139906089

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