Properties

 Degree 2 Conductor $3 \cdot 5 \cdot 7$ Sign $0.999 - 0.0338i$ Motivic weight 2 Primitive yes Self-dual no Analytic rank 0

Related objects

Dirichlet series

 L(s)  = 1 + (0.707 + 0.707i)2-s + (2.70 − 1.29i)3-s − 3i·4-s + (0.707 + 4.94i)5-s + (2.82 + i)6-s + 7i·7-s + (4.94 − 4.94i)8-s + (5.65 − 7i)9-s + (−3.00 + 4.00i)10-s − 9.89i·11-s + (−3.87 − 8.12i)12-s + (8 − 8i)13-s + (−4.94 + 4.94i)14-s + (8.31 + 12.4i)15-s − 4.99·16-s + (−18.3 + 18.3i)17-s + ⋯
 L(s)  = 1 + (0.353 + 0.353i)2-s + (0.902 − 0.430i)3-s − 0.750i·4-s + (0.141 + 0.989i)5-s + (0.471 + 0.166i)6-s + i·7-s + (0.618 − 0.618i)8-s + (0.628 − 0.777i)9-s + (−0.300 + 0.400i)10-s − 0.899i·11-s + (−0.323 − 0.676i)12-s + (0.615 − 0.615i)13-s + (−0.353 + 0.353i)14-s + (0.554 + 0.832i)15-s − 0.312·16-s + (−1.08 + 1.08i)17-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0338i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.999 - 0.0338i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 $$d$$ = $$2$$ $$N$$ = $$105$$    =    $$3 \cdot 5 \cdot 7$$ $$\varepsilon$$ = $0.999 - 0.0338i$ motivic weight = $$2$$ character : $\chi_{105} (83, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 105,\ (\ :1),\ 0.999 - 0.0338i)$$ $$L(\frac{3}{2})$$ $$\approx$$ $$2.09602 + 0.0355141i$$ $$L(\frac12)$$ $$\approx$$ $$2.09602 + 0.0355141i$$ $$L(2)$$ not available $$L(1)$$ not available

Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{3,\;5,\;7\}$,$$F_p(T)$$ is a polynomial of degree 2. If $p \in \{3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 $$1 + (-2.70 + 1.29i)T$$
5 $$1 + (-0.707 - 4.94i)T$$
7 $$1 - 7iT$$
good2 $$1 + (-0.707 - 0.707i)T + 4iT^{2}$$
11 $$1 + 9.89iT - 121T^{2}$$
13 $$1 + (-8 + 8i)T - 169iT^{2}$$
17 $$1 + (18.3 - 18.3i)T - 289iT^{2}$$
19 $$1 + 10T + 361T^{2}$$
23 $$1 + (24.0 - 24.0i)T - 529iT^{2}$$
29 $$1 + 4.24T + 841T^{2}$$
31 $$1 - 14iT - 961T^{2}$$
37 $$1 + (-30 + 30i)T - 1.36e3iT^{2}$$
41 $$1 + 33.9T + 1.68e3T^{2}$$
43 $$1 + (-36 - 36i)T + 1.84e3iT^{2}$$
47 $$1 + (4.24 - 4.24i)T - 2.20e3iT^{2}$$
53 $$1 + (-70.7 + 70.7i)T - 2.80e3iT^{2}$$
59 $$1 - 29.6iT - 3.48e3T^{2}$$
61 $$1 + 14iT - 3.72e3T^{2}$$
67 $$1 + (-32 + 32i)T - 4.48e3iT^{2}$$
71 $$1 + 59.3iT - 5.04e3T^{2}$$
73 $$1 + (-39 + 39i)T - 5.32e3iT^{2}$$
79 $$1 - 56iT - 6.24e3T^{2}$$
83 $$1 + (36.7 + 36.7i)T + 6.88e3iT^{2}$$
89 $$1 - 19.7iT - 7.92e3T^{2}$$
97 $$1 + (-113 - 113i)T + 9.40e3iT^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}