# Properties

 Degree 2 Conductor $2^{4} \cdot 3 \cdot 7^{2}$ Sign $-0.386 - 0.922i$ Motivic weight 1 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (0.5 + 0.866i)3-s + (0.5 − 0.866i)5-s + (−0.499 + 0.866i)9-s + (2.5 + 4.33i)11-s + 0.999·15-s + (−2 − 3.46i)17-s + (−4 + 6.92i)19-s + (−2 + 3.46i)23-s + (2 + 3.46i)25-s − 0.999·27-s − 5·29-s + (−1.5 − 2.59i)31-s + (−2.5 + 4.33i)33-s + (2 − 3.46i)37-s − 2·43-s + ⋯
 L(s)  = 1 + (0.288 + 0.499i)3-s + (0.223 − 0.387i)5-s + (−0.166 + 0.288i)9-s + (0.753 + 1.30i)11-s + 0.258·15-s + (−0.485 − 0.840i)17-s + (−0.917 + 1.58i)19-s + (−0.417 + 0.722i)23-s + (0.400 + 0.692i)25-s − 0.192·27-s − 0.928·29-s + (−0.269 − 0.466i)31-s + (−0.435 + 0.753i)33-s + (0.328 − 0.569i)37-s − 0.304·43-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.386 - 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$2352$$    =    $$2^{4} \cdot 3 \cdot 7^{2}$$ $$\varepsilon$$ = $-0.386 - 0.922i$ motivic weight = $$1$$ character : $\chi_{2352} (961, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 2352,\ (\ :1/2),\ -0.386 - 0.922i)$$ $$L(1)$$ $$\approx$$ $$1.583094808$$ $$L(\frac12)$$ $$\approx$$ $$1.583094808$$ $$L(\frac{3}{2})$$ 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 + (-0.5 - 0.866i)T$$
7 $$1$$
good5 $$1 + (-0.5 + 0.866i)T + (-2.5 - 4.33i)T^{2}$$
11 $$1 + (-2.5 - 4.33i)T + (-5.5 + 9.52i)T^{2}$$
13 $$1 + 13T^{2}$$
17 $$1 + (2 + 3.46i)T + (-8.5 + 14.7i)T^{2}$$
19 $$1 + (4 - 6.92i)T + (-9.5 - 16.4i)T^{2}$$
23 $$1 + (2 - 3.46i)T + (-11.5 - 19.9i)T^{2}$$
29 $$1 + 5T + 29T^{2}$$
31 $$1 + (1.5 + 2.59i)T + (-15.5 + 26.8i)T^{2}$$
37 $$1 + (-2 + 3.46i)T + (-18.5 - 32.0i)T^{2}$$
41 $$1 + 41T^{2}$$
43 $$1 + 2T + 43T^{2}$$
47 $$1 + (-3 + 5.19i)T + (-23.5 - 40.7i)T^{2}$$
53 $$1 + (-4.5 - 7.79i)T + (-26.5 + 45.8i)T^{2}$$
59 $$1 + (-5.5 - 9.52i)T + (-29.5 + 51.0i)T^{2}$$
61 $$1 + (3 - 5.19i)T + (-30.5 - 52.8i)T^{2}$$
67 $$1 + (1 + 1.73i)T + (-33.5 + 58.0i)T^{2}$$
71 $$1 + 2T + 71T^{2}$$
73 $$1 + (-5 - 8.66i)T + (-36.5 + 63.2i)T^{2}$$
79 $$1 + (-1.5 + 2.59i)T + (-39.5 - 68.4i)T^{2}$$
83 $$1 + 7T + 83T^{2}$$
89 $$1 + (3 - 5.19i)T + (-44.5 - 77.0i)T^{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}