# Properties

 Degree 2 Conductor $2^{4}$ Sign $0.998 + 0.0551i$ Motivic weight 3 Primitive yes Self-dual no Analytic rank 0

# Learn more about

## Dirichlet series

 L(s)  = 1 + (2.81 + 0.316i)2-s + (−3.27 − 3.27i)3-s + (7.80 + 1.77i)4-s + (−12.6 + 12.6i)5-s + (−8.16 − 10.2i)6-s − 13.8i·7-s + (21.3 + 7.45i)8-s − 5.59i·9-s + (−39.5 + 31.5i)10-s + (1.54 − 1.54i)11-s + (−19.7 − 31.3i)12-s + (32.7 + 32.7i)13-s + (4.38 − 38.9i)14-s + 82.7·15-s + (57.6 + 27.7i)16-s + 18.6·17-s + ⋯
 L(s)  = 1 + (0.993 + 0.111i)2-s + (−0.629 − 0.629i)3-s + (0.975 + 0.222i)4-s + (−1.13 + 1.13i)5-s + (−0.555 − 0.695i)6-s − 0.749i·7-s + (0.944 + 0.329i)8-s − 0.207i·9-s + (−1.25 + 0.997i)10-s + (0.0424 − 0.0424i)11-s + (−0.474 − 0.753i)12-s + (0.699 + 0.699i)13-s + (0.0837 − 0.744i)14-s + 1.42·15-s + (0.901 + 0.433i)16-s + 0.266·17-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0551i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0551i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$16$$    =    $$2^{4}$$ $$\varepsilon$$ = $0.998 + 0.0551i$ motivic weight = $$3$$ character : $\chi_{16} (13, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 16,\ (\ :3/2),\ 0.998 + 0.0551i)$ $L(2)$ $\approx$ $1.29158 - 0.0356701i$ $L(\frac12)$ $\approx$ $1.29158 - 0.0356701i$ $L(\frac{5}{2})$ 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 + (-2.81 - 0.316i)T$$
good3 $$1 + (3.27 + 3.27i)T + 27iT^{2}$$
5 $$1 + (12.6 - 12.6i)T - 125iT^{2}$$
7 $$1 + 13.8iT - 343T^{2}$$
11 $$1 + (-1.54 + 1.54i)T - 1.33e3iT^{2}$$
13 $$1 + (-32.7 - 32.7i)T + 2.19e3iT^{2}$$
17 $$1 - 18.6T + 4.91e3T^{2}$$
19 $$1 + (86.4 + 86.4i)T + 6.85e3iT^{2}$$
23 $$1 - 134. iT - 1.21e4T^{2}$$
29 $$1 + (59.7 + 59.7i)T + 2.43e4iT^{2}$$
31 $$1 + 31.5T + 2.97e4T^{2}$$
37 $$1 + (-89.1 + 89.1i)T - 5.06e4iT^{2}$$
41 $$1 - 210. iT - 6.89e4T^{2}$$
43 $$1 + (-119. + 119. i)T - 7.95e4iT^{2}$$
47 $$1 + 182.T + 1.03e5T^{2}$$
53 $$1 + (26.1 - 26.1i)T - 1.48e5iT^{2}$$
59 $$1 + (-441. + 441. i)T - 2.05e5iT^{2}$$
61 $$1 + (174. + 174. i)T + 2.26e5iT^{2}$$
67 $$1 + (-91.7 - 91.7i)T + 3.00e5iT^{2}$$
71 $$1 + 348. iT - 3.57e5T^{2}$$
73 $$1 - 299. iT - 3.89e5T^{2}$$
79 $$1 + 943.T + 4.93e5T^{2}$$
83 $$1 + (-313. - 313. i)T + 5.71e5iT^{2}$$
89 $$1 + 1.41e3iT - 7.04e5T^{2}$$
97 $$1 - 1.51e3T + 9.12e5T^{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

−18.80428636821950363259418671615, −17.24409571198555303481118615532, −15.75319592822826627784935850727, −14.62429414166078430699001073400, −13.20205984725613540839392196906, −11.65697398613831378058459120435, −11.00004024031805071090457338786, −7.43192695854085614181681259807, −6.47822961820532445296571330633, −3.81746618896785242225556130638, 4.23731031807583349022840487675, 5.62036500732862361628891850858, 8.248769012487150467291391348964, 10.67481878802626459797891779872, 11.96202222694314605717937679653, 12.87078476741698750657766775968, 14.96433181711139586642192676086, 16.01113049375911288393669519023, 16.67359888057166418265649385568, 19.01838602946845692153318283135