# Properties

 Label 2-150-15.14-c10-0-38 Degree $2$ Conductor $150$ Sign $0.926 - 0.376i$ Analytic cond. $95.3035$ Root an. cond. $9.76235$ Motivic weight $10$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 22.6·2-s + (−18.8 + 242. i)3-s + 512.·4-s + (−426. + 5.48e3i)6-s − 670. i·7-s + 1.15e4·8-s + (−5.83e4 − 9.12e3i)9-s − 2.33e5i·11-s + (−9.64e3 + 1.24e5i)12-s + 3.07e5i·13-s − 1.51e4i·14-s + 2.62e5·16-s − 6.72e5·17-s + (−1.32e6 − 2.06e5i)18-s + 1.55e6·19-s + ⋯
 L(s)  = 1 + 0.707·2-s + (−0.0775 + 0.996i)3-s + 0.500·4-s + (−0.0548 + 0.704i)6-s − 0.0398i·7-s + 0.353·8-s + (−0.987 − 0.154i)9-s − 1.44i·11-s + (−0.0387 + 0.498i)12-s + 0.828i·13-s − 0.0282i·14-s + 0.250·16-s − 0.473·17-s + (−0.698 − 0.109i)18-s + 0.626·19-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.926 - 0.376i)\, \overline{\Lambda}(11-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.926 - 0.376i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$150$$    =    $$2 \cdot 3 \cdot 5^{2}$$ Sign: $0.926 - 0.376i$ Analytic conductor: $$95.3035$$ Root analytic conductor: $$9.76235$$ Motivic weight: $$10$$ Rational: no Arithmetic: yes Character: $\chi_{150} (149, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 150,\ (\ :5),\ 0.926 - 0.376i)$$

## Particular Values

 $$L(\frac{11}{2})$$ $$\approx$$ $$3.326104291$$ $$L(\frac12)$$ $$\approx$$ $$3.326104291$$ $$L(6)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 - 22.6T$$
3 $$1 + (18.8 - 242. i)T$$
5 $$1$$
good7 $$1 + 670. iT - 2.82e8T^{2}$$
11 $$1 + 2.33e5iT - 2.59e10T^{2}$$
13 $$1 - 3.07e5iT - 1.37e11T^{2}$$
17 $$1 + 6.72e5T + 2.01e12T^{2}$$
19 $$1 - 1.55e6T + 6.13e12T^{2}$$
23 $$1 - 5.57e6T + 4.14e13T^{2}$$
29 $$1 + 2.97e7iT - 4.20e14T^{2}$$
31 $$1 - 3.09e7T + 8.19e14T^{2}$$
37 $$1 - 8.56e7iT - 4.80e15T^{2}$$
41 $$1 + 3.59e7iT - 1.34e16T^{2}$$
43 $$1 + 3.66e7iT - 2.16e16T^{2}$$
47 $$1 + 3.28e7T + 5.25e16T^{2}$$
53 $$1 - 4.59e8T + 1.74e17T^{2}$$
59 $$1 + 4.88e8iT - 5.11e17T^{2}$$
61 $$1 + 6.12e7T + 7.13e17T^{2}$$
67 $$1 - 6.70e8iT - 1.82e18T^{2}$$
71 $$1 + 1.23e9iT - 3.25e18T^{2}$$
73 $$1 - 1.08e9iT - 4.29e18T^{2}$$
79 $$1 - 1.86e9T + 9.46e18T^{2}$$
83 $$1 + 1.09e9T + 1.55e19T^{2}$$
89 $$1 - 5.19e9iT - 3.11e19T^{2}$$
97 $$1 - 1.07e10iT - 7.37e19T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$