Properties

 Label 2-70-35.2-c2-0-6 Degree $2$ Conductor $70$ Sign $0.967 + 0.254i$ Analytic cond. $1.90736$ Root an. cond. $1.38107$ Motivic weight $2$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

Related objects

Dirichlet series

 L(s)  = 1 + (1.36 − 0.366i)2-s + (2.37 + 0.636i)3-s + (1.73 − i)4-s + (−1.96 − 4.59i)5-s + 3.47·6-s + (0.219 + 6.99i)7-s + (1.99 − 2i)8-s + (−2.55 − 1.47i)9-s + (−4.36 − 5.56i)10-s + (3.73 + 6.47i)11-s + (4.74 − 1.27i)12-s + (−10.9 + 10.9i)13-s + (2.86 + 9.47i)14-s + (−1.73 − 12.1i)15-s + (1.99 − 3.46i)16-s + (5.47 − 20.4i)17-s + ⋯
 L(s)  = 1 + (0.683 − 0.183i)2-s + (0.791 + 0.212i)3-s + (0.433 − 0.250i)4-s + (−0.392 − 0.919i)5-s + 0.579·6-s + (0.0313 + 0.999i)7-s + (0.249 − 0.250i)8-s + (−0.284 − 0.164i)9-s + (−0.436 − 0.556i)10-s + (0.339 + 0.588i)11-s + (0.395 − 0.106i)12-s + (−0.842 + 0.842i)13-s + (0.204 + 0.676i)14-s + (−0.115 − 0.811i)15-s + (0.124 − 0.216i)16-s + (0.321 − 1.20i)17-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 + 0.254i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.967 + 0.254i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 Degree: $$2$$ Conductor: $$70$$    =    $$2 \cdot 5 \cdot 7$$ Sign: $0.967 + 0.254i$ Analytic conductor: $$1.90736$$ Root analytic conductor: $$1.38107$$ Motivic weight: $$2$$ Rational: no Arithmetic: yes Character: $\chi_{70} (37, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 70,\ (\ :1),\ 0.967 + 0.254i)$$

Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$1.95952 - 0.253215i$$ $$L(\frac12)$$ $$\approx$$ $$1.95952 - 0.253215i$$ $$L(2)$$ not available $$L(1)$$ not available

Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (-1.36 + 0.366i)T$$
5 $$1 + (1.96 + 4.59i)T$$
7 $$1 + (-0.219 - 6.99i)T$$
good3 $$1 + (-2.37 - 0.636i)T + (7.79 + 4.5i)T^{2}$$
11 $$1 + (-3.73 - 6.47i)T + (-60.5 + 104. i)T^{2}$$
13 $$1 + (10.9 - 10.9i)T - 169iT^{2}$$
17 $$1 + (-5.47 + 20.4i)T + (-250. - 144.5i)T^{2}$$
19 $$1 + (12.4 + 7.21i)T + (180.5 + 312. i)T^{2}$$
23 $$1 + (-5.37 - 20.0i)T + (-458. + 264.5i)T^{2}$$
29 $$1 - 15.8iT - 841T^{2}$$
31 $$1 + (8 + 13.8i)T + (-480.5 + 832. i)T^{2}$$
37 $$1 + (-64.1 + 17.1i)T + (1.18e3 - 684.5i)T^{2}$$
41 $$1 + 27.9T + 1.68e3T^{2}$$
43 $$1 + (-39.6 + 39.6i)T - 1.84e3iT^{2}$$
47 $$1 + (-69.4 + 18.6i)T + (1.91e3 - 1.10e3i)T^{2}$$
53 $$1 + (-40.9 - 10.9i)T + (2.43e3 + 1.40e3i)T^{2}$$
59 $$1 + (-55.8 + 32.2i)T + (1.74e3 - 3.01e3i)T^{2}$$
61 $$1 + (29.5 - 51.1i)T + (-1.86e3 - 3.22e3i)T^{2}$$
67 $$1 + (14.3 - 53.5i)T + (-3.88e3 - 2.24e3i)T^{2}$$
71 $$1 + 36.3T + 5.04e3T^{2}$$
73 $$1 + (63.8 + 17.1i)T + (4.61e3 + 2.66e3i)T^{2}$$
79 $$1 + (35.6 + 20.6i)T + (3.12e3 + 5.40e3i)T^{2}$$
83 $$1 + (9.12 - 9.12i)T - 6.88e3iT^{2}$$
89 $$1 + (-42.1 - 24.3i)T + (3.96e3 + 6.85e3i)T^{2}$$
97 $$1 + (63.7 + 63.7i)T + 9.40e3iT^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$