# Properties

 Label 2-350-5.4-c5-0-34 Degree $2$ Conductor $350$ Sign $0.447 + 0.894i$ Analytic cond. $56.1343$ Root an. cond. $7.49228$ Motivic weight $5$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 4i·2-s + 10i·3-s − 16·4-s − 40·6-s − 49i·7-s − 64i·8-s + 143·9-s − 336·11-s − 160i·12-s + 584i·13-s + 196·14-s + 256·16-s + 1.45e3i·17-s + 572i·18-s − 470·19-s + ⋯
 L(s)  = 1 + 0.707i·2-s + 0.641i·3-s − 0.5·4-s − 0.453·6-s − 0.377i·7-s − 0.353i·8-s + 0.588·9-s − 0.837·11-s − 0.320i·12-s + 0.958i·13-s + 0.267·14-s + 0.250·16-s + 1.22i·17-s + 0.416i·18-s − 0.298·19-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$350$$    =    $$2 \cdot 5^{2} \cdot 7$$ Sign: $0.447 + 0.894i$ Analytic conductor: $$56.1343$$ Root analytic conductor: $$7.49228$$ Motivic weight: $$5$$ Rational: no Arithmetic: yes Character: $\chi_{350} (99, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 350,\ (\ :5/2),\ 0.447 + 0.894i)$$

## Particular Values

 $$L(3)$$ $$\approx$$ $$0.3936288571$$ $$L(\frac12)$$ $$\approx$$ $$0.3936288571$$ $$L(\frac{7}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 - 4iT$$
5 $$1$$
7 $$1 + 49iT$$
good3 $$1 - 10iT - 243T^{2}$$
11 $$1 + 336T + 1.61e5T^{2}$$
13 $$1 - 584iT - 3.71e5T^{2}$$
17 $$1 - 1.45e3iT - 1.41e6T^{2}$$
19 $$1 + 470T + 2.47e6T^{2}$$
23 $$1 + 4.20e3iT - 6.43e6T^{2}$$
29 $$1 + 4.86e3T + 2.05e7T^{2}$$
31 $$1 + 7.37e3T + 2.86e7T^{2}$$
37 $$1 + 1.43e4iT - 6.93e7T^{2}$$
41 $$1 - 6.22e3T + 1.15e8T^{2}$$
43 $$1 - 3.70e3iT - 1.47e8T^{2}$$
47 $$1 - 1.81e3iT - 2.29e8T^{2}$$
53 $$1 + 3.72e4iT - 4.18e8T^{2}$$
59 $$1 + 3.43e4T + 7.14e8T^{2}$$
61 $$1 - 2.44e4T + 8.44e8T^{2}$$
67 $$1 - 1.74e4iT - 1.35e9T^{2}$$
71 $$1 - 2.82e4T + 1.80e9T^{2}$$
73 $$1 - 3.60e3iT - 2.07e9T^{2}$$
79 $$1 + 4.28e4T + 3.07e9T^{2}$$
83 $$1 + 3.52e4iT - 3.93e9T^{2}$$
89 $$1 + 2.67e4T + 5.58e9T^{2}$$
97 $$1 - 1.69e4iT - 8.58e9T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$