# Properties

 Degree 2 Conductor $2 \cdot 3 \cdot 5 \cdot 7$ Sign $-0.850 + 0.526i$ Motivic weight 2 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (−1.36 − 0.366i)2-s + (−2.99 − 0.157i)3-s + (1.73 + i)4-s + (2.05 + 4.55i)5-s + (4.03 + 1.31i)6-s + (−6.80 − 1.63i)7-s + (−1.99 − 2i)8-s + (8.95 + 0.945i)9-s + (−1.14 − 6.97i)10-s + (7.66 + 4.42i)11-s + (−5.03 − 3.26i)12-s + (−5.26 − 5.26i)13-s + (8.69 + 4.72i)14-s + (−5.44 − 13.9i)15-s + (1.99 + 3.46i)16-s + (−9.09 + 2.43i)17-s + ⋯
 L(s)  = 1 + (−0.683 − 0.183i)2-s + (−0.998 − 0.0525i)3-s + (0.433 + 0.250i)4-s + (0.411 + 0.911i)5-s + (0.672 + 0.218i)6-s + (−0.972 − 0.233i)7-s + (−0.249 − 0.250i)8-s + (0.994 + 0.105i)9-s + (−0.114 − 0.697i)10-s + (0.696 + 0.402i)11-s + (−0.419 − 0.272i)12-s + (−0.405 − 0.405i)13-s + (0.621 + 0.337i)14-s + (−0.363 − 0.931i)15-s + (0.124 + 0.216i)16-s + (−0.534 + 0.143i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.850 + 0.526i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.850 + 0.526i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$210$$    =    $$2 \cdot 3 \cdot 5 \cdot 7$$ $$\varepsilon$$ = $-0.850 + 0.526i$ motivic weight = $$2$$ character : $\chi_{210} (17, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 210,\ (\ :1),\ -0.850 + 0.526i)$$ $$L(\frac{3}{2})$$ $$\approx$$ $$0.0429669 - 0.151003i$$ $$L(\frac12)$$ $$\approx$$ $$0.0429669 - 0.151003i$$ $$L(2)$$ 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 + (1.36 + 0.366i)T$$
3 $$1 + (2.99 + 0.157i)T$$
5 $$1 + (-2.05 - 4.55i)T$$
7 $$1 + (6.80 + 1.63i)T$$
good11 $$1 + (-7.66 - 4.42i)T + (60.5 + 104. i)T^{2}$$
13 $$1 + (5.26 + 5.26i)T + 169iT^{2}$$
17 $$1 + (9.09 - 2.43i)T + (250. - 144.5i)T^{2}$$
19 $$1 + (16.3 + 28.3i)T + (-180.5 + 312. i)T^{2}$$
23 $$1 + (-3.23 + 12.0i)T + (-458. - 264.5i)T^{2}$$
29 $$1 + 48.1T + 841T^{2}$$
31 $$1 + (42.0 + 24.2i)T + (480.5 + 832. i)T^{2}$$
37 $$1 + (-4.02 + 15.0i)T + (-1.18e3 - 684.5i)T^{2}$$
41 $$1 - 45.6T + 1.68e3T^{2}$$
43 $$1 + (17.0 - 17.0i)T - 1.84e3iT^{2}$$
47 $$1 + (-18.2 + 67.9i)T + (-1.91e3 - 1.10e3i)T^{2}$$
53 $$1 + (-23.5 + 6.32i)T + (2.43e3 - 1.40e3i)T^{2}$$
59 $$1 + (31.3 + 18.1i)T + (1.74e3 + 3.01e3i)T^{2}$$
61 $$1 + (-41.5 + 23.9i)T + (1.86e3 - 3.22e3i)T^{2}$$
67 $$1 + (-20.6 + 5.53i)T + (3.88e3 - 2.24e3i)T^{2}$$
71 $$1 - 42.3iT - 5.04e3T^{2}$$
73 $$1 + (118. - 31.8i)T + (4.61e3 - 2.66e3i)T^{2}$$
79 $$1 + (-53.6 + 31.0i)T + (3.12e3 - 5.40e3i)T^{2}$$
83 $$1 + (113. - 113. i)T - 6.88e3iT^{2}$$
89 $$1 + (104. - 60.5i)T + (3.96e3 - 6.85e3i)T^{2}$$
97 $$1 + (-72.8 + 72.8i)T - 9.40e3iT^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}