# Properties

 Label 2-175-7.6-c4-0-1 Degree $2$ Conductor $175$ Sign $i$ Analytic cond. $18.0897$ Root an. cond. $4.25320$ Motivic weight $4$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 17i·3-s − 16·4-s + 49i·7-s − 208·9-s − 73·11-s − 272i·12-s − 23i·13-s + 256·16-s + 263i·17-s − 833·21-s − 2.15e3i·27-s − 784i·28-s + 1.15e3·29-s − 1.24e3i·33-s + 3.32e3·36-s + ⋯
 L(s)  = 1 + 1.88i·3-s − 4-s + 0.999i·7-s − 2.56·9-s − 0.603·11-s − 1.88i·12-s − 0.136i·13-s + 16-s + 0.910i·17-s − 1.88·21-s − 2.96i·27-s − 0.999i·28-s + 1.37·29-s − 1.13i·33-s + 2.56·36-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(5-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$175$$    =    $$5^{2} \cdot 7$$ Sign: $i$ Analytic conductor: $$18.0897$$ Root analytic conductor: $$4.25320$$ Motivic weight: $$4$$ Rational: no Arithmetic: yes Character: $\chi_{175} (76, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 175,\ (\ :2),\ i)$$

## Particular Values

 $$L(\frac{5}{2})$$ $$\approx$$ $$0.4256445039$$ $$L(\frac12)$$ $$\approx$$ $$0.4256445039$$ $$L(3)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad5 $$1$$
7 $$1 - 49iT$$
good2 $$1 + 16T^{2}$$
3 $$1 - 17iT - 81T^{2}$$
11 $$1 + 73T + 1.46e4T^{2}$$
13 $$1 + 23iT - 2.85e4T^{2}$$
17 $$1 - 263iT - 8.35e4T^{2}$$
19 $$1 - 1.30e5T^{2}$$
23 $$1 + 2.79e5T^{2}$$
29 $$1 - 1.15e3T + 7.07e5T^{2}$$
31 $$1 - 9.23e5T^{2}$$
37 $$1 + 1.87e6T^{2}$$
41 $$1 - 2.82e6T^{2}$$
43 $$1 + 3.41e6T^{2}$$
47 $$1 + 3.45e3iT - 4.87e6T^{2}$$
53 $$1 + 7.89e6T^{2}$$
59 $$1 - 1.21e7T^{2}$$
61 $$1 - 1.38e7T^{2}$$
67 $$1 + 2.01e7T^{2}$$
71 $$1 + 1.00e4T + 2.54e7T^{2}$$
73 $$1 - 9.50e3iT - 2.83e7T^{2}$$
79 $$1 + 1.21e4T + 3.89e7T^{2}$$
83 $$1 - 6.38e3iT - 4.74e7T^{2}$$
89 $$1 - 6.27e7T^{2}$$
97 $$1 - 3.38e3iT - 8.85e7T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$