# Properties

 Degree 2 Conductor $2^{2} \cdot 5$ Sign $-0.429 + 0.902i$ Motivic weight 3 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (−0.510 − 2.78i)2-s + (−4.02 − 4.02i)3-s + (−7.47 + 2.83i)4-s + (10.9 − 2.32i)5-s + (−9.15 + 13.2i)6-s + (14.4 − 14.4i)7-s + (11.7 + 19.3i)8-s + 5.46i·9-s + (−12.0 − 29.2i)10-s + 47.0i·11-s + (41.5 + 18.6i)12-s + (−8.79 + 8.79i)13-s + (−47.5 − 32.8i)14-s + (−53.4 − 34.6i)15-s + (47.8 − 42.4i)16-s + (−26.4 − 26.4i)17-s + ⋯
 L(s)  = 1 + (−0.180 − 0.983i)2-s + (−0.775 − 0.775i)3-s + (−0.934 + 0.354i)4-s + (0.978 − 0.208i)5-s + (−0.622 + 0.902i)6-s + (0.779 − 0.779i)7-s + (0.517 + 0.855i)8-s + 0.202i·9-s + (−0.381 − 0.924i)10-s + 1.28i·11-s + (1.00 + 0.449i)12-s + (−0.187 + 0.187i)13-s + (−0.907 − 0.626i)14-s + (−0.919 − 0.596i)15-s + (0.747 − 0.663i)16-s + (−0.377 − 0.377i)17-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.429 + 0.902i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.429 + 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$20$$    =    $$2^{2} \cdot 5$$ $$\varepsilon$$ = $-0.429 + 0.902i$ motivic weight = $$3$$ character : $\chi_{20} (7, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 20,\ (\ :3/2),\ -0.429 + 0.902i)$ $L(2)$ $\approx$ $0.466631 - 0.739033i$ $L(\frac12)$ $\approx$ $0.466631 - 0.739033i$ $L(\frac{5}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{2,\;5\}$, $$F_p$$ is a polynomial of degree 2. If $p \in \{2,\;5\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 $$1 + (0.510 + 2.78i)T$$
5 $$1 + (-10.9 + 2.32i)T$$
good3 $$1 + (4.02 + 4.02i)T + 27iT^{2}$$
7 $$1 + (-14.4 + 14.4i)T - 343iT^{2}$$
11 $$1 - 47.0iT - 1.33e3T^{2}$$
13 $$1 + (8.79 - 8.79i)T - 2.19e3iT^{2}$$
17 $$1 + (26.4 + 26.4i)T + 4.91e3iT^{2}$$
19 $$1 - 49.8T + 6.85e3T^{2}$$
23 $$1 + (-41.2 - 41.2i)T + 1.21e4iT^{2}$$
29 $$1 - 247. iT - 2.43e4T^{2}$$
31 $$1 + 62.3iT - 2.97e4T^{2}$$
37 $$1 + (73.2 + 73.2i)T + 5.06e4iT^{2}$$
41 $$1 - 118.T + 6.89e4T^{2}$$
43 $$1 + (245. + 245. i)T + 7.95e4iT^{2}$$
47 $$1 + (125. - 125. i)T - 1.03e5iT^{2}$$
53 $$1 + (326. - 326. i)T - 1.48e5iT^{2}$$
59 $$1 - 365.T + 2.05e5T^{2}$$
61 $$1 + 268.T + 2.26e5T^{2}$$
67 $$1 + (-112. + 112. i)T - 3.00e5iT^{2}$$
71 $$1 + 559. iT - 3.57e5T^{2}$$
73 $$1 + (-215. + 215. i)T - 3.89e5iT^{2}$$
79 $$1 + 1.17e3T + 4.93e5T^{2}$$
83 $$1 + (-592. - 592. i)T + 5.71e5iT^{2}$$
89 $$1 + 552. iT - 7.04e5T^{2}$$
97 $$1 + (-460. - 460. i)T + 9.12e5iT^{2}$$
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\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

−17.69228910626314187536542704257, −17.01398480099287287596753840720, −14.34179980883879406150067092396, −13.14562394125720404168773497423, −12.10321468443508349104157916001, −10.78819775661994631196785931301, −9.399528730580484261776589120337, −7.23190358823218872578164335035, −4.98389133627466619250463836693, −1.51891437296429275105910588701, 5.09659175642776025805495108126, 6.08003447560574532295963258695, 8.439904173195141790459013438083, 9.915470233499977202168121370849, 11.24698935806875743926966083811, 13.43960017639732019534965628059, 14.67404770927266189144315739987, 15.89353724487323927610275982809, 16.97164713760731699950011859544, 17.82774945949674882394337286596