# Properties

 Degree 2 Conductor $2 \cdot 5 \cdot 67$ Sign $-0.978 - 0.205i$ Motivic weight 1 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (0.5 + 0.866i)2-s + 3-s + (−0.499 + 0.866i)4-s − 5-s + (0.5 + 0.866i)6-s + (−1 + 1.73i)7-s − 0.999·8-s − 2·9-s + (−0.5 − 0.866i)10-s + (−3 + 5.19i)11-s + (−0.499 + 0.866i)12-s + (−1 − 1.73i)13-s − 1.99·14-s − 15-s + (−0.5 − 0.866i)16-s + (−3 − 5.19i)17-s + ⋯
 L(s)  = 1 + (0.353 + 0.612i)2-s + 0.577·3-s + (−0.249 + 0.433i)4-s − 0.447·5-s + (0.204 + 0.353i)6-s + (−0.377 + 0.654i)7-s − 0.353·8-s − 0.666·9-s + (−0.158 − 0.273i)10-s + (−0.904 + 1.56i)11-s + (−0.144 + 0.249i)12-s + (−0.277 − 0.480i)13-s − 0.534·14-s − 0.258·15-s + (−0.125 − 0.216i)16-s + (−0.727 − 1.26i)17-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.978 - 0.205i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.978 - 0.205i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$670$$    =    $$2 \cdot 5 \cdot 67$$ $$\varepsilon$$ = $-0.978 - 0.205i$ motivic weight = $$1$$ character : $\chi_{670} (171, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 670,\ (\ :1/2),\ -0.978 - 0.205i)$ $L(1)$ $\approx$ $0.110885 + 1.06793i$ $L(\frac12)$ $\approx$ $0.110885 + 1.06793i$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{2,\;5,\;67\}$, $$F_p(T)$$ is a polynomial of degree 2. If $p \in \{2,\;5,\;67\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1 + (-0.5 - 0.866i)T$$
5 $$1 + T$$
67 $$1 + (-2.5 + 7.79i)T$$
good3 $$1 - T + 3T^{2}$$
7 $$1 + (1 - 1.73i)T + (-3.5 - 6.06i)T^{2}$$
11 $$1 + (3 - 5.19i)T + (-5.5 - 9.52i)T^{2}$$
13 $$1 + (1 + 1.73i)T + (-6.5 + 11.2i)T^{2}$$
17 $$1 + (3 + 5.19i)T + (-8.5 + 14.7i)T^{2}$$
19 $$1 + (-2 - 3.46i)T + (-9.5 + 16.4i)T^{2}$$
23 $$1 + (-3 - 5.19i)T + (-11.5 + 19.9i)T^{2}$$
29 $$1 + (-3 + 5.19i)T + (-14.5 - 25.1i)T^{2}$$
31 $$1 + (2.5 - 4.33i)T + (-15.5 - 26.8i)T^{2}$$
37 $$1 + (-3.5 - 6.06i)T + (-18.5 + 32.0i)T^{2}$$
41 $$1 + (1.5 - 2.59i)T + (-20.5 - 35.5i)T^{2}$$
43 $$1 - 8T + 43T^{2}$$
47 $$1 + (3 - 5.19i)T + (-23.5 - 40.7i)T^{2}$$
53 $$1 + 3T + 53T^{2}$$
59 $$1 + 6T + 59T^{2}$$
61 $$1 + (4 + 6.92i)T + (-30.5 + 52.8i)T^{2}$$
71 $$1 + (-7.5 + 12.9i)T + (-35.5 - 61.4i)T^{2}$$
73 $$1 + (-2 - 3.46i)T + (-36.5 + 63.2i)T^{2}$$
79 $$1 + (4 - 6.92i)T + (-39.5 - 68.4i)T^{2}$$
83 $$1 + (-7.5 - 12.9i)T + (-41.5 + 71.8i)T^{2}$$
89 $$1 - 9T + 89T^{2}$$
97 $$1 + (-5 - 8.66i)T + (-48.5 + 84.0i)T^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}