# Properties

 Degree 2 Conductor $2^{4} \cdot 3 \cdot 7$ Sign $-0.761 + 0.648i$ Motivic weight 1 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (−0.291 − 1.70i)3-s + (−0.0726 − 0.125i)5-s + (−1.05 − 2.42i)7-s + (−2.83 + 0.993i)9-s + (−2.13 − 1.23i)11-s − 2.04i·13-s + (−0.193 + 0.160i)15-s + (−0.878 + 1.52i)17-s + (3.68 − 2.12i)19-s + (−3.83 + 2.50i)21-s + (−7.46 + 4.30i)23-s + (2.48 − 4.31i)25-s + (2.52 + 4.54i)27-s − 7.08i·29-s + (−3.11 − 1.80i)31-s + ⋯
 L(s)  = 1 + (−0.168 − 0.985i)3-s + (−0.0324 − 0.0562i)5-s + (−0.398 − 0.917i)7-s + (−0.943 + 0.331i)9-s + (−0.644 − 0.372i)11-s − 0.566i·13-s + (−0.0500 + 0.0414i)15-s + (−0.213 + 0.369i)17-s + (0.844 − 0.487i)19-s + (−0.837 + 0.547i)21-s + (−1.55 + 0.898i)23-s + (0.497 − 0.862i)25-s + (0.485 + 0.874i)27-s − 1.31i·29-s + (−0.560 − 0.323i)31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.761 + 0.648i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.761 + 0.648i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$336$$    =    $$2^{4} \cdot 3 \cdot 7$$ $$\varepsilon$$ = $-0.761 + 0.648i$ motivic weight = $$1$$ character : $\chi_{336} (17, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 336,\ (\ :1/2),\ -0.761 + 0.648i)$ $L(1)$ $\approx$ $0.307460 - 0.835089i$ $L(\frac12)$ $\approx$ $0.307460 - 0.835089i$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;3,\;7\}$,$$F_p(T)$$ is a polynomial of degree 2. If $p \in \{2,\;3,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1$$
3 $$1 + (0.291 + 1.70i)T$$
7 $$1 + (1.05 + 2.42i)T$$
good5 $$1 + (0.0726 + 0.125i)T + (-2.5 + 4.33i)T^{2}$$
11 $$1 + (2.13 + 1.23i)T + (5.5 + 9.52i)T^{2}$$
13 $$1 + 2.04iT - 13T^{2}$$
17 $$1 + (0.878 - 1.52i)T + (-8.5 - 14.7i)T^{2}$$
19 $$1 + (-3.68 + 2.12i)T + (9.5 - 16.4i)T^{2}$$
23 $$1 + (7.46 - 4.30i)T + (11.5 - 19.9i)T^{2}$$
29 $$1 + 7.08iT - 29T^{2}$$
31 $$1 + (3.11 + 1.80i)T + (15.5 + 26.8i)T^{2}$$
37 $$1 + (2.93 + 5.08i)T + (-18.5 + 32.0i)T^{2}$$
41 $$1 - 5.33T + 41T^{2}$$
43 $$1 - 9.19T + 43T^{2}$$
47 $$1 + (-4.65 - 8.05i)T + (-23.5 + 40.7i)T^{2}$$
53 $$1 + (-4.49 - 2.59i)T + (26.5 + 45.8i)T^{2}$$
59 $$1 + (-5.60 + 9.70i)T + (-29.5 - 51.0i)T^{2}$$
61 $$1 + (-4.66 + 2.69i)T + (30.5 - 52.8i)T^{2}$$
67 $$1 + (2.57 - 4.45i)T + (-33.5 - 58.0i)T^{2}$$
71 $$1 + 7.79iT - 71T^{2}$$
73 $$1 + (-11.3 - 6.52i)T + (36.5 + 63.2i)T^{2}$$
79 $$1 + (2.86 + 4.95i)T + (-39.5 + 68.4i)T^{2}$$
83 $$1 - 15.9T + 83T^{2}$$
89 $$1 + (4.34 + 7.52i)T + (-44.5 + 77.0i)T^{2}$$
97 $$1 - 6.65iT - 97T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}