Properties

 Degree 2 Conductor $2 \cdot 3 \cdot 5 \cdot 7$ Sign $-0.935 + 0.354i$ Motivic weight 2 Primitive yes Self-dual no Analytic rank 0

Related objects

Dirichlet series

 L(s)  = 1 + (−0.366 − 1.36i)2-s + (−0.448 + 1.67i)3-s + (−1.73 + i)4-s + (−3.58 + 3.48i)5-s + 2.44·6-s + (3.00 − 6.32i)7-s + (2 + 1.99i)8-s + (−2.59 − 1.50i)9-s + (6.07 + 3.61i)10-s + (−1.95 − 3.38i)11-s + (−0.896 − 3.34i)12-s + (−8.93 − 8.93i)13-s + (−9.73 − 1.78i)14-s + (−4.22 − 7.55i)15-s + (1.99 − 3.46i)16-s + (−14.3 − 3.83i)17-s + ⋯
 L(s)  = 1 + (−0.183 − 0.683i)2-s + (−0.149 + 0.557i)3-s + (−0.433 + 0.250i)4-s + (−0.716 + 0.697i)5-s + 0.408·6-s + (0.428 − 0.903i)7-s + (0.250 + 0.249i)8-s + (−0.288 − 0.166i)9-s + (0.607 + 0.361i)10-s + (−0.177 − 0.308i)11-s + (−0.0747 − 0.278i)12-s + (−0.687 − 0.687i)13-s + (−0.695 − 0.127i)14-s + (−0.281 − 0.503i)15-s + (0.124 − 0.216i)16-s + (−0.841 − 0.225i)17-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.935 + 0.354i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.935 + 0.354i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 $$d$$ = $$2$$ $$N$$ = $$210$$    =    $$2 \cdot 3 \cdot 5 \cdot 7$$ $$\varepsilon$$ = $-0.935 + 0.354i$ motivic weight = $$2$$ character : $\chi_{210} (163, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 210,\ (\ :1),\ -0.935 + 0.354i)$$ $$L(\frac{3}{2})$$ $$\approx$$ $$0.0687204 - 0.375132i$$ $$L(\frac12)$$ $$\approx$$ $$0.0687204 - 0.375132i$$ $$L(2)$$ not available $$L(1)$$ not available

Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;3,\;5,\;7\}$,$$F_p(T)$$ is a polynomial of degree 2. If $p \in \{2,\;3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1 + (0.366 + 1.36i)T$$
3 $$1 + (0.448 - 1.67i)T$$
5 $$1 + (3.58 - 3.48i)T$$
7 $$1 + (-3.00 + 6.32i)T$$
good11 $$1 + (1.95 + 3.38i)T + (-60.5 + 104. i)T^{2}$$
13 $$1 + (8.93 + 8.93i)T + 169iT^{2}$$
17 $$1 + (14.3 + 3.83i)T + (250. + 144.5i)T^{2}$$
19 $$1 + (28.0 + 16.1i)T + (180.5 + 312. i)T^{2}$$
23 $$1 + (-12.3 + 3.30i)T + (458. - 264.5i)T^{2}$$
29 $$1 - 26.6iT - 841T^{2}$$
31 $$1 + (17.0 + 29.4i)T + (-480.5 + 832. i)T^{2}$$
37 $$1 + (-1.67 - 6.25i)T + (-1.18e3 + 684.5i)T^{2}$$
41 $$1 + 26.0T + 1.68e3T^{2}$$
43 $$1 + (-21.0 - 21.0i)T + 1.84e3iT^{2}$$
47 $$1 + (-10.7 - 40.1i)T + (-1.91e3 + 1.10e3i)T^{2}$$
53 $$1 + (-11.7 + 43.7i)T + (-2.43e3 - 1.40e3i)T^{2}$$
59 $$1 + (20.7 - 12.0i)T + (1.74e3 - 3.01e3i)T^{2}$$
61 $$1 + (-8.39 + 14.5i)T + (-1.86e3 - 3.22e3i)T^{2}$$
67 $$1 + (-57.0 - 15.2i)T + (3.88e3 + 2.24e3i)T^{2}$$
71 $$1 + 132.T + 5.04e3T^{2}$$
73 $$1 + (-2.29 + 8.56i)T + (-4.61e3 - 2.66e3i)T^{2}$$
79 $$1 + (-90.6 - 52.3i)T + (3.12e3 + 5.40e3i)T^{2}$$
83 $$1 + (-94.1 - 94.1i)T + 6.88e3iT^{2}$$
89 $$1 + (43.2 + 24.9i)T + (3.96e3 + 6.85e3i)T^{2}$$
97 $$1 + (-118. + 118. i)T - 9.40e3iT^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}