# Properties

 Degree 2 Conductor $2 \cdot 7^{2}$ Sign $-0.266 - 0.963i$ Motivic weight 1 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (0.5 + 0.866i)2-s + (−1 + 1.73i)3-s + (−0.499 + 0.866i)4-s − 1.99·6-s − 0.999·8-s + (−0.499 − 0.866i)9-s + (−0.999 − 1.73i)12-s + 4·13-s + (−0.5 − 0.866i)16-s + (3 − 5.19i)17-s + (0.499 − 0.866i)18-s + (1 + 1.73i)19-s + (0.999 − 1.73i)24-s + (2.5 − 4.33i)25-s + (2 + 3.46i)26-s − 4.00·27-s + ⋯
 L(s)  = 1 + (0.353 + 0.612i)2-s + (−0.577 + 0.999i)3-s + (−0.249 + 0.433i)4-s − 0.816·6-s − 0.353·8-s + (−0.166 − 0.288i)9-s + (−0.288 − 0.499i)12-s + 1.10·13-s + (−0.125 − 0.216i)16-s + (0.727 − 1.26i)17-s + (0.117 − 0.204i)18-s + (0.229 + 0.397i)19-s + (0.204 − 0.353i)24-s + (0.5 − 0.866i)25-s + (0.392 + 0.679i)26-s − 0.769·27-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.266 - 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.266 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$98$$    =    $$2 \cdot 7^{2}$$ $$\varepsilon$$ = $-0.266 - 0.963i$ motivic weight = $$1$$ character : $\chi_{98} (67, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 98,\ (\ :1/2),\ -0.266 - 0.963i)$$ $$L(1)$$ $$\approx$$ $$0.606661 + 0.797446i$$ $$L(\frac12)$$ $$\approx$$ $$0.606661 + 0.797446i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;7\}$,$$F_p(T)$$ is a polynomial of degree 2. If $p \in \{2,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1 + (-0.5 - 0.866i)T$$
7 $$1$$
good3 $$1 + (1 - 1.73i)T + (-1.5 - 2.59i)T^{2}$$
5 $$1 + (-2.5 + 4.33i)T^{2}$$
11 $$1 + (-5.5 - 9.52i)T^{2}$$
13 $$1 - 4T + 13T^{2}$$
17 $$1 + (-3 + 5.19i)T + (-8.5 - 14.7i)T^{2}$$
19 $$1 + (-1 - 1.73i)T + (-9.5 + 16.4i)T^{2}$$
23 $$1 + (-11.5 + 19.9i)T^{2}$$
29 $$1 + 6T + 29T^{2}$$
31 $$1 + (2 - 3.46i)T + (-15.5 - 26.8i)T^{2}$$
37 $$1 + (1 + 1.73i)T + (-18.5 + 32.0i)T^{2}$$
41 $$1 + 6T + 41T^{2}$$
43 $$1 - 8T + 43T^{2}$$
47 $$1 + (6 + 10.3i)T + (-23.5 + 40.7i)T^{2}$$
53 $$1 + (3 - 5.19i)T + (-26.5 - 45.8i)T^{2}$$
59 $$1 + (3 - 5.19i)T + (-29.5 - 51.0i)T^{2}$$
61 $$1 + (-4 - 6.92i)T + (-30.5 + 52.8i)T^{2}$$
67 $$1 + (-2 + 3.46i)T + (-33.5 - 58.0i)T^{2}$$
71 $$1 + 71T^{2}$$
73 $$1 + (-1 + 1.73i)T + (-36.5 - 63.2i)T^{2}$$
79 $$1 + (4 + 6.92i)T + (-39.5 + 68.4i)T^{2}$$
83 $$1 - 6T + 83T^{2}$$
89 $$1 + (3 + 5.19i)T + (-44.5 + 77.0i)T^{2}$$
97 $$1 - 10T + 97T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}