# Properties

 Label 2-15-3.2-c8-0-7 Degree $2$ Conductor $15$ Sign $-0.929 + 0.368i$ Analytic cond. $6.11067$ Root an. cond. $2.47197$ Motivic weight $8$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 7.97i·2-s + (−29.8 − 75.3i)3-s + 192.·4-s − 279. i·5-s + (−600. + 238. i)6-s − 3.21e3·7-s − 3.57e3i·8-s + (−4.77e3 + 4.49e3i)9-s − 2.22e3·10-s − 6.48e3i·11-s + (−5.74e3 − 1.44e4i)12-s − 1.05e4·13-s + 2.56e4i·14-s + (−2.10e4 + 8.34e3i)15-s + 2.07e4·16-s − 5.77e4i·17-s + ⋯
 L(s)  = 1 − 0.498i·2-s + (−0.368 − 0.929i)3-s + 0.751·4-s − 0.447i·5-s + (−0.463 + 0.183i)6-s − 1.33·7-s − 0.873i·8-s + (−0.728 + 0.685i)9-s − 0.222·10-s − 0.442i·11-s + (−0.276 − 0.698i)12-s − 0.367·13-s + 0.666i·14-s + (−0.415 + 0.164i)15-s + 0.316·16-s − 0.691i·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.929 + 0.368i)\, \overline{\Lambda}(9-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.929 + 0.368i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$15$$    =    $$3 \cdot 5$$ Sign: $-0.929 + 0.368i$ Analytic conductor: $$6.11067$$ Root analytic conductor: $$2.47197$$ Motivic weight: $$8$$ Rational: no Arithmetic: yes Character: $\chi_{15} (11, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 15,\ (\ :4),\ -0.929 + 0.368i)$$

## Particular Values

 $$L(\frac{9}{2})$$ $$\approx$$ $$0.232765 - 1.21913i$$ $$L(\frac12)$$ $$\approx$$ $$0.232765 - 1.21913i$$ $$L(5)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 + (29.8 + 75.3i)T$$
5 $$1 + 279. iT$$
good2 $$1 + 7.97iT - 256T^{2}$$
7 $$1 + 3.21e3T + 5.76e6T^{2}$$
11 $$1 + 6.48e3iT - 2.14e8T^{2}$$
13 $$1 + 1.05e4T + 8.15e8T^{2}$$
17 $$1 + 5.77e4iT - 6.97e9T^{2}$$
19 $$1 - 2.28e5T + 1.69e10T^{2}$$
23 $$1 + 1.13e5iT - 7.83e10T^{2}$$
29 $$1 + 1.11e6iT - 5.00e11T^{2}$$
31 $$1 + 3.04e5T + 8.52e11T^{2}$$
37 $$1 - 6.30e5T + 3.51e12T^{2}$$
41 $$1 - 4.62e6iT - 7.98e12T^{2}$$
43 $$1 - 5.44e6T + 1.16e13T^{2}$$
47 $$1 + 6.69e6iT - 2.38e13T^{2}$$
53 $$1 - 1.25e7iT - 6.22e13T^{2}$$
59 $$1 + 5.55e6iT - 1.46e14T^{2}$$
61 $$1 + 1.31e7T + 1.91e14T^{2}$$
67 $$1 - 1.70e7T + 4.06e14T^{2}$$
71 $$1 - 1.08e7iT - 6.45e14T^{2}$$
73 $$1 + 2.67e7T + 8.06e14T^{2}$$
79 $$1 + 3.07e6T + 1.51e15T^{2}$$
83 $$1 - 2.32e7iT - 2.25e15T^{2}$$
89 $$1 + 3.21e7iT - 3.93e15T^{2}$$
97 $$1 + 9.03e7T + 7.83e15T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$