Properties

Degree 2
Conductor $ 2^{4} $
Sign $-0.866 + 0.499i$
Motivic weight 11
Primitive yes
Self-dual no
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + (−30.9 + 33.0i)2-s + (585. + 585. i)3-s + (−133. − 2.04e3i)4-s + (−6.85e3 + 6.85e3i)5-s + (−3.74e4 + 1.22e3i)6-s − 1.99e4i·7-s + (7.16e4 + 5.88e4i)8-s + 5.08e5i·9-s + (−1.43e4 − 4.38e5i)10-s + (−9.64e4 + 9.64e4i)11-s + (1.11e6 − 1.27e6i)12-s + (−6.74e5 − 6.74e5i)13-s + (6.57e5 + 6.16e5i)14-s − 8.03e6·15-s + (−4.15e6 + 5.46e5i)16-s + 3.70e6·17-s + ⋯
L(s)  = 1  + (−0.683 + 0.729i)2-s + (1.39 + 1.39i)3-s + (−0.0652 − 0.997i)4-s + (−0.981 + 0.981i)5-s + (−1.96 + 0.0642i)6-s − 0.447i·7-s + (0.772 + 0.634i)8-s + 2.87i·9-s + (−0.0453 − 1.38i)10-s + (−0.180 + 0.180i)11-s + (1.29 − 1.47i)12-s + (−0.503 − 0.503i)13-s + (0.326 + 0.306i)14-s − 2.73·15-s + (−0.991 + 0.130i)16-s + 0.633·17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.866 + 0.499i)\, \overline{\Lambda}(12-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.866 + 0.499i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(16\)    =    \(2^{4}\)
\( \varepsilon \)  =  $-0.866 + 0.499i$
motivic weight  =  \(11\)
character  :  $\chi_{16} (13, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 16,\ (\ :11/2),\ -0.866 + 0.499i)$
$L(6)$  $\approx$  $0.323161 - 1.20673i$
$L(\frac12)$  $\approx$  $0.323161 - 1.20673i$
$L(\frac{13}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 2$, \(F_p(T)\) is a polynomial of degree 2. If $p = 2$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + (30.9 - 33.0i)T \)
good3 \( 1 + (-585. - 585. i)T + 1.77e5iT^{2} \)
5 \( 1 + (6.85e3 - 6.85e3i)T - 4.88e7iT^{2} \)
7 \( 1 + 1.99e4iT - 1.97e9T^{2} \)
11 \( 1 + (9.64e4 - 9.64e4i)T - 2.85e11iT^{2} \)
13 \( 1 + (6.74e5 + 6.74e5i)T + 1.79e12iT^{2} \)
17 \( 1 - 3.70e6T + 3.42e13T^{2} \)
19 \( 1 + (2.61e6 + 2.61e6i)T + 1.16e14iT^{2} \)
23 \( 1 + 2.97e7iT - 9.52e14T^{2} \)
29 \( 1 + (-4.14e7 - 4.14e7i)T + 1.22e16iT^{2} \)
31 \( 1 - 6.07e7T + 2.54e16T^{2} \)
37 \( 1 + (4.37e8 - 4.37e8i)T - 1.77e17iT^{2} \)
41 \( 1 - 1.20e9iT - 5.50e17T^{2} \)
43 \( 1 + (1.45e8 - 1.45e8i)T - 9.29e17iT^{2} \)
47 \( 1 - 1.52e9T + 2.47e18T^{2} \)
53 \( 1 + (2.52e9 - 2.52e9i)T - 9.26e18iT^{2} \)
59 \( 1 + (4.54e9 - 4.54e9i)T - 3.01e19iT^{2} \)
61 \( 1 + (-9.96e7 - 9.96e7i)T + 4.35e19iT^{2} \)
67 \( 1 + (-7.32e9 - 7.32e9i)T + 1.22e20iT^{2} \)
71 \( 1 - 1.31e9iT - 2.31e20T^{2} \)
73 \( 1 + 2.59e9iT - 3.13e20T^{2} \)
79 \( 1 + 4.08e9T + 7.47e20T^{2} \)
83 \( 1 + (-3.03e10 - 3.03e10i)T + 1.28e21iT^{2} \)
89 \( 1 + 5.68e10iT - 2.77e21T^{2} \)
97 \( 1 + 5.73e10T + 7.15e21T^{2} \)
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

−16.73717181730721466730860877238, −15.54622489915309019908774449605, −14.93723681412329129132574511194, −14.01507916185653770540552906523, −10.76654396898039693282678046114, −9.989480941281818736622296361774, −8.400972992053731004542324916463, −7.40308974551975897691961843387, −4.56142206897114557916900140381, −2.93811567187988018539479471950, 0.57503939181848003014329362622, 2.00081415741020161156402304292, 3.61665506502984511789911828194, 7.38932696691650370609757519566, 8.325278434336474833027695254909, 9.241823303791776572141905188733, 12.00941006338072447720172137799, 12.50063373807227503374166771646, 13.87348316300619107536043130641, 15.64199057645156149395812799438

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