# Properties

 Label 2-338-13.12-c7-0-38 Degree $2$ Conductor $338$ Sign $0.554 - 0.832i$ Analytic cond. $105.586$ Root an. cond. $10.2755$ Motivic weight $7$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 8i·2-s − 39·3-s − 64·4-s + 385i·5-s + 312i·6-s + 293i·7-s + 512i·8-s − 666·9-s + 3.08e3·10-s + 5.40e3i·11-s + 2.49e3·12-s + 2.34e3·14-s − 1.50e4i·15-s + 4.09e3·16-s + 2.10e4·17-s + 5.32e3i·18-s + ⋯
 L(s)  = 1 − 0.707i·2-s − 0.833·3-s − 0.5·4-s + 1.37i·5-s + 0.589i·6-s + 0.322i·7-s + 0.353i·8-s − 0.304·9-s + 0.973·10-s + 1.22i·11-s + 0.416·12-s + 0.228·14-s − 1.14i·15-s + 0.250·16-s + 1.03·17-s + 0.215i·18-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.554 - 0.832i)\, \overline{\Lambda}(8-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.554 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$338$$    =    $$2 \cdot 13^{2}$$ Sign: $0.554 - 0.832i$ Analytic conductor: $$105.586$$ Root analytic conductor: $$10.2755$$ Motivic weight: $$7$$ Rational: no Arithmetic: yes Character: $\chi_{338} (337, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 338,\ (\ :7/2),\ 0.554 - 0.832i)$$

## Particular Values

 $$L(4)$$ $$\approx$$ $$1.437175195$$ $$L(\frac12)$$ $$\approx$$ $$1.437175195$$ $$L(\frac{9}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + 8iT$$
13 $$1$$
good3 $$1 + 39T + 2.18e3T^{2}$$
5 $$1 - 385iT - 7.81e4T^{2}$$
7 $$1 - 293iT - 8.23e5T^{2}$$
11 $$1 - 5.40e3iT - 1.94e7T^{2}$$
17 $$1 - 2.10e4T + 4.10e8T^{2}$$
19 $$1 + 2.73e4iT - 8.93e8T^{2}$$
23 $$1 - 6.30e4T + 3.40e9T^{2}$$
29 $$1 - 1.22e5T + 1.72e10T^{2}$$
31 $$1 + 2.08e5iT - 2.75e10T^{2}$$
37 $$1 - 4.42e5iT - 9.49e10T^{2}$$
41 $$1 - 5.80e4iT - 1.94e11T^{2}$$
43 $$1 - 2.02e5T + 2.71e11T^{2}$$
47 $$1 + 5.88e5iT - 5.06e11T^{2}$$
53 $$1 - 1.68e6T + 1.17e12T^{2}$$
59 $$1 - 4.42e5iT - 2.48e12T^{2}$$
61 $$1 + 1.08e6T + 3.14e12T^{2}$$
67 $$1 - 3.44e6iT - 6.06e12T^{2}$$
71 $$1 - 2.08e6iT - 9.09e12T^{2}$$
73 $$1 + 5.93e6iT - 1.10e13T^{2}$$
79 $$1 + 6.60e6T + 1.92e13T^{2}$$
83 $$1 + 1.42e5iT - 2.71e13T^{2}$$
89 $$1 - 6.98e6iT - 4.42e13T^{2}$$
97 $$1 + 2.00e5iT - 8.07e13T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$