Properties

 Degree 2 Conductor $17 \cdot 43$ Sign $-0.911 + 0.410i$ Motivic weight 1 Primitive yes Self-dual no Analytic rank 0

Related objects

Dirichlet series

 L(s)  = 1 + (−1.41 − 1.41i)4-s + (−1.14 − 2.77i)9-s + (5.50 − 3.68i)11-s + (−4.77 + 4.77i)13-s + 4.00i·16-s + (−3.56 − 2.07i)17-s + (−0.868 − 1.29i)23-s + (−4.61 + 1.91i)25-s + (−8.40 − 5.61i)31-s + (−2.29 + 5.54i)36-s + (1.07 − 5.39i)41-s + (−2.50 − 6.05i)43-s + (−12.9 − 2.58i)44-s + (7.48 − 7.48i)47-s + (−6.46 − 2.67i)49-s + ⋯
 L(s)  = 1 + (−0.707 − 0.707i)4-s + (−0.382 − 0.923i)9-s + (1.66 − 1.11i)11-s + (−1.32 + 1.32i)13-s + 1.00i·16-s + (−0.864 − 0.502i)17-s + (−0.181 − 0.271i)23-s + (−0.923 + 0.382i)25-s + (−1.50 − 1.00i)31-s + (−0.382 + 0.923i)36-s + (0.167 − 0.842i)41-s + (−0.382 − 0.923i)43-s + (−1.95 − 0.389i)44-s + (1.09 − 1.09i)47-s + (−0.923 − 0.382i)49-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.911 + 0.410i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.911 + 0.410i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 $$d$$ = $$2$$ $$N$$ = $$731$$    =    $$17 \cdot 43$$ $$\varepsilon$$ = $-0.911 + 0.410i$ motivic weight = $$1$$ character : $\chi_{731} (687, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 731,\ (\ :1/2),\ -0.911 + 0.410i)$ $L(1)$ $\approx$ $0.132444 - 0.616418i$ $L(\frac12)$ $\approx$ $0.132444 - 0.616418i$ $L(\frac{3}{2})$ not available $L(1)$ not available

Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{17,\;43\}$,$$F_p(T)$$ is a polynomial of degree 2. If $p \in \{17,\;43\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad17 $$1 + (3.56 + 2.07i)T$$
43 $$1 + (2.50 + 6.05i)T$$
good2 $$1 + (1.41 + 1.41i)T^{2}$$
3 $$1 + (1.14 + 2.77i)T^{2}$$
5 $$1 + (4.61 - 1.91i)T^{2}$$
7 $$1 + (6.46 + 2.67i)T^{2}$$
11 $$1 + (-5.50 + 3.68i)T + (4.20 - 10.1i)T^{2}$$
13 $$1 + (4.77 - 4.77i)T - 13iT^{2}$$
19 $$1 + (13.4 + 13.4i)T^{2}$$
23 $$1 + (0.868 + 1.29i)T + (-8.80 + 21.2i)T^{2}$$
29 $$1 + (-26.7 + 11.0i)T^{2}$$
31 $$1 + (8.40 + 5.61i)T + (11.8 + 28.6i)T^{2}$$
37 $$1 + (14.1 + 34.1i)T^{2}$$
41 $$1 + (-1.07 + 5.39i)T + (-37.8 - 15.6i)T^{2}$$
47 $$1 + (-7.48 + 7.48i)T - 47iT^{2}$$
53 $$1 + (3.63 - 8.77i)T + (-37.4 - 37.4i)T^{2}$$
59 $$1 + (11.4 - 4.74i)T + (41.7 - 41.7i)T^{2}$$
61 $$1 + (-56.3 - 23.3i)T^{2}$$
67 $$1 - 11.7iT - 67T^{2}$$
71 $$1 + (27.1 + 65.5i)T^{2}$$
73 $$1 + (67.4 - 27.9i)T^{2}$$
79 $$1 + (-14.3 + 9.59i)T + (30.2 - 72.9i)T^{2}$$
83 $$1 + (-6.05 - 2.50i)T + (58.6 + 58.6i)T^{2}$$
89 $$1 - 89iT^{2}$$
97 $$1 + (-11.5 + 2.29i)T + (89.6 - 37.1i)T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}