# Properties

 Degree 2 Conductor $2^{3} \cdot 7^{2}$ Sign $-0.575 - 0.817i$ Motivic weight 2 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (1.51 + 1.30i)2-s + (0.824 − 1.42i)3-s + (0.597 + 3.95i)4-s + (−3.95 + 2.28i)5-s + (3.11 − 1.08i)6-s + (−4.25 + 6.77i)8-s + (3.14 + 5.43i)9-s + (−8.96 − 1.69i)10-s + (6.18 − 10.7i)11-s + (6.13 + 2.40i)12-s + 18.3i·13-s + 7.52i·15-s + (−15.2 + 4.72i)16-s + (−6.51 + 11.2i)17-s + (−2.33 + 12.3i)18-s + (−1.51 − 2.61i)19-s + ⋯
 L(s)  = 1 + (0.758 + 0.652i)2-s + (0.274 − 0.475i)3-s + (0.149 + 0.988i)4-s + (−0.790 + 0.456i)5-s + (0.518 − 0.181i)6-s + (−0.531 + 0.846i)8-s + (0.348 + 0.604i)9-s + (−0.896 − 0.169i)10-s + (0.562 − 0.974i)11-s + (0.511 + 0.200i)12-s + 1.41i·13-s + 0.501i·15-s + (−0.955 + 0.295i)16-s + (−0.383 + 0.663i)17-s + (−0.129 + 0.685i)18-s + (−0.0796 − 0.137i)19-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.575 - 0.817i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.575 - 0.817i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$392$$    =    $$2^{3} \cdot 7^{2}$$ $$\varepsilon$$ = $-0.575 - 0.817i$ motivic weight = $$2$$ character : $\chi_{392} (67, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 392,\ (\ :1),\ -0.575 - 0.817i)$$ $$L(\frac{3}{2})$$ $$\approx$$ $$0.996610 + 1.92044i$$ $$L(\frac12)$$ $$\approx$$ $$0.996610 + 1.92044i$$ $$L(2)$$ 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 + (-1.51 - 1.30i)T$$
7 $$1$$
good3 $$1 + (-0.824 + 1.42i)T + (-4.5 - 7.79i)T^{2}$$
5 $$1 + (3.95 - 2.28i)T + (12.5 - 21.6i)T^{2}$$
11 $$1 + (-6.18 + 10.7i)T + (-60.5 - 104. i)T^{2}$$
13 $$1 - 18.3iT - 169T^{2}$$
17 $$1 + (6.51 - 11.2i)T + (-144.5 - 250. i)T^{2}$$
19 $$1 + (1.51 + 2.61i)T + (-180.5 + 312. i)T^{2}$$
23 $$1 + (26.2 - 15.1i)T + (264.5 - 458. i)T^{2}$$
29 $$1 - 22.7iT - 841T^{2}$$
31 $$1 + (19.5 + 11.2i)T + (480.5 + 832. i)T^{2}$$
37 $$1 + (-11.9 + 6.88i)T + (684.5 - 1.18e3i)T^{2}$$
41 $$1 - 60.5T + 1.68e3T^{2}$$
43 $$1 - 39.0T + 1.84e3T^{2}$$
47 $$1 + (-17.6 + 10.1i)T + (1.10e3 - 1.91e3i)T^{2}$$
53 $$1 + (-4.12 - 2.38i)T + (1.40e3 + 2.43e3i)T^{2}$$
59 $$1 + (-5.86 + 10.1i)T + (-1.74e3 - 3.01e3i)T^{2}$$
61 $$1 + (-94.3 + 54.4i)T + (1.86e3 - 3.22e3i)T^{2}$$
67 $$1 + (-39.5 + 68.5i)T + (-2.24e3 - 3.88e3i)T^{2}$$
71 $$1 + 12.9iT - 5.04e3T^{2}$$
73 $$1 + (49.2 - 85.3i)T + (-2.66e3 - 4.61e3i)T^{2}$$
79 $$1 + (-113. + 65.6i)T + (3.12e3 - 5.40e3i)T^{2}$$
83 $$1 + 28.3T + 6.88e3T^{2}$$
89 $$1 + (78.7 + 136. i)T + (-3.96e3 + 6.85e3i)T^{2}$$
97 $$1 - 39.6T + 9.40e3T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}