# Properties

 Label 2-35-5.3-c4-0-7 Degree $2$ Conductor $35$ Sign $-0.245 + 0.969i$ Analytic cond. $3.61794$ Root an. cond. $1.90209$ Motivic weight $4$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−2.97 + 2.97i)2-s + (−1.22 − 1.22i)3-s − 1.69i·4-s + (−22.1 − 11.5i)5-s + 7.31·6-s + (13.0 − 13.0i)7-s + (−42.5 − 42.5i)8-s − 77.9i·9-s + (100. − 31.6i)10-s + 17.6·11-s + (−2.07 + 2.07i)12-s + (−160. − 160. i)13-s + 77.8i·14-s + (13.0 + 41.4i)15-s + 280.·16-s + (−324. + 324. i)17-s + ⋯
 L(s)  = 1 + (−0.743 + 0.743i)2-s + (−0.136 − 0.136i)3-s − 0.105i·4-s + (−0.887 − 0.461i)5-s + 0.203·6-s + (0.267 − 0.267i)7-s + (−0.664 − 0.664i)8-s − 0.962i·9-s + (1.00 − 0.316i)10-s + 0.145·11-s + (−0.0144 + 0.0144i)12-s + (−0.949 − 0.949i)13-s + 0.397i·14-s + (0.0581 + 0.184i)15-s + 1.09·16-s + (−1.12 + 1.12i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.245 + 0.969i)\, \overline{\Lambda}(5-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.245 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$35$$    =    $$5 \cdot 7$$ Sign: $-0.245 + 0.969i$ Analytic conductor: $$3.61794$$ Root analytic conductor: $$1.90209$$ Motivic weight: $$4$$ Rational: no Arithmetic: yes Character: $\chi_{35} (8, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 35,\ (\ :2),\ -0.245 + 0.969i)$$

## Particular Values

 $$L(\frac{5}{2})$$ $$\approx$$ $$0.171381 - 0.220213i$$ $$L(\frac12)$$ $$\approx$$ $$0.171381 - 0.220213i$$ $$L(3)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad5 $$1 + (22.1 + 11.5i)T$$
7 $$1 + (-13.0 + 13.0i)T$$
good2 $$1 + (2.97 - 2.97i)T - 16iT^{2}$$
3 $$1 + (1.22 + 1.22i)T + 81iT^{2}$$
11 $$1 - 17.6T + 1.46e4T^{2}$$
13 $$1 + (160. + 160. i)T + 2.85e4iT^{2}$$
17 $$1 + (324. - 324. i)T - 8.35e4iT^{2}$$
19 $$1 - 468. iT - 1.30e5T^{2}$$
23 $$1 + (625. + 625. i)T + 2.79e5iT^{2}$$
29 $$1 + 755. iT - 7.07e5T^{2}$$
31 $$1 - 553.T + 9.23e5T^{2}$$
37 $$1 + (-555. + 555. i)T - 1.87e6iT^{2}$$
41 $$1 + 1.68e3T + 2.82e6T^{2}$$
43 $$1 + (416. + 416. i)T + 3.41e6iT^{2}$$
47 $$1 + (319. - 319. i)T - 4.87e6iT^{2}$$
53 $$1 + (-3.35e3 - 3.35e3i)T + 7.89e6iT^{2}$$
59 $$1 + 4.67e3iT - 1.21e7T^{2}$$
61 $$1 - 848.T + 1.38e7T^{2}$$
67 $$1 + (-2.47e3 + 2.47e3i)T - 2.01e7iT^{2}$$
71 $$1 + 2.45e3T + 2.54e7T^{2}$$
73 $$1 + (2.34e3 + 2.34e3i)T + 2.83e7iT^{2}$$
79 $$1 + 1.76e3iT - 3.89e7T^{2}$$
83 $$1 + (885. + 885. i)T + 4.74e7iT^{2}$$
89 $$1 - 3.51e3iT - 6.27e7T^{2}$$
97 $$1 + (-421. + 421. i)T - 8.85e7iT^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$