# Properties

 Degree 2 Conductor $2 \cdot 5 \cdot 17$ Sign $-0.976 - 0.216i$ Motivic weight 1 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 − i·2-s − 3-s − 4-s + (−2 − i)5-s + i·6-s − 2·7-s + i·8-s − 2·9-s + (−1 + 2i)10-s + 12-s − i·13-s + 2i·14-s + (2 + i)15-s + 16-s + (−1 − 4i)17-s + 2i·18-s + ⋯
 L(s)  = 1 − 0.707i·2-s − 0.577·3-s − 0.5·4-s + (−0.894 − 0.447i)5-s + 0.408i·6-s − 0.755·7-s + 0.353i·8-s − 0.666·9-s + (−0.316 + 0.632i)10-s + 0.288·12-s − 0.277i·13-s + 0.534i·14-s + (0.516 + 0.258i)15-s + 0.250·16-s + (−0.242 − 0.970i)17-s + 0.471i·18-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.976 - 0.216i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.976 - 0.216i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$170$$    =    $$2 \cdot 5 \cdot 17$$ $$\varepsilon$$ = $-0.976 - 0.216i$ motivic weight = $$1$$ character : $\chi_{170} (169, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 170,\ (\ :1/2),\ -0.976 - 0.216i)$ $L(1)$ $\approx$ $0.0315332 + 0.287260i$ $L(\frac12)$ $\approx$ $0.0315332 + 0.287260i$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{2,\;5,\;17\}$, $$F_p$$ is a polynomial of degree 2. If $p \in \{2,\;5,\;17\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 $$1 + iT$$
5 $$1 + (2 + i)T$$
17 $$1 + (1 + 4i)T$$
good3 $$1 + T + 3T^{2}$$
7 $$1 + 2T + 7T^{2}$$
11 $$1 - 11T^{2}$$
13 $$1 + iT - 13T^{2}$$
19 $$1 + 5T + 19T^{2}$$
23 $$1 - 4T + 23T^{2}$$
29 $$1 + 9iT - 29T^{2}$$
31 $$1 - 5iT - 31T^{2}$$
37 $$1 + 2T + 37T^{2}$$
41 $$1 + 10iT - 41T^{2}$$
43 $$1 + 6iT - 43T^{2}$$
47 $$1 - 7iT - 47T^{2}$$
53 $$1 + iT - 53T^{2}$$
59 $$1 + 5T + 59T^{2}$$
61 $$1 - 5iT - 61T^{2}$$
67 $$1 - 2iT - 67T^{2}$$
71 $$1 - 5iT - 71T^{2}$$
73 $$1 + 11T + 73T^{2}$$
79 $$1 - 16iT - 79T^{2}$$
83 $$1 + 6iT - 83T^{2}$$
89 $$1 - 5T + 89T^{2}$$
97 $$1 + 7T + 97T^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}