# Properties

 Label 2-175-175.103-c1-0-4 Degree $2$ Conductor $175$ Sign $0.321 - 0.946i$ Analytic cond. $1.39738$ Root an. cond. $1.18210$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (0.000942 + 4.94e−5i)2-s + (1.67 + 2.07i)3-s + (−1.98 − 0.209i)4-s + (2.16 + 0.564i)5-s + (0.00147 + 0.00203i)6-s + (−2.21 + 1.45i)7-s + (−0.00372 − 0.000590i)8-s + (−0.853 + 4.01i)9-s + (0.00201 + 0.000639i)10-s + (4.83 − 1.02i)11-s + (−2.90 − 4.47i)12-s + (−4.82 − 2.45i)13-s + (−0.00215 + 0.00126i)14-s + (2.45 + 5.43i)15-s + (3.91 + 0.831i)16-s + (0.276 + 0.720i)17-s + ⋯
 L(s)  = 1 + (0.000666 + 3.49e−5i)2-s + (0.968 + 1.19i)3-s + (−0.994 − 0.104i)4-s + (0.967 + 0.252i)5-s + (0.000603 + 0.000831i)6-s + (−0.835 + 0.549i)7-s + (−0.00131 − 0.000208i)8-s + (−0.284 + 1.33i)9-s + (0.000636 + 0.000202i)10-s + (1.45 − 0.309i)11-s + (−0.838 − 1.29i)12-s + (−1.33 − 0.681i)13-s + (−0.000576 + 0.000337i)14-s + (0.635 + 1.40i)15-s + (0.978 + 0.207i)16-s + (0.0671 + 0.174i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.321 - 0.946i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.321 - 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$175$$    =    $$5^{2} \cdot 7$$ Sign: $0.321 - 0.946i$ Analytic conductor: $$1.39738$$ Root analytic conductor: $$1.18210$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{175} (103, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 175,\ (\ :1/2),\ 0.321 - 0.946i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.11072 + 0.795517i$$ $$L(\frac12)$$ $$\approx$$ $$1.11072 + 0.795517i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad5 $$1 + (-2.16 - 0.564i)T$$
7 $$1 + (2.21 - 1.45i)T$$
good2 $$1 + (-0.000942 - 4.94e-5i)T + (1.98 + 0.209i)T^{2}$$
3 $$1 + (-1.67 - 2.07i)T + (-0.623 + 2.93i)T^{2}$$
11 $$1 + (-4.83 + 1.02i)T + (10.0 - 4.47i)T^{2}$$
13 $$1 + (4.82 + 2.45i)T + (7.64 + 10.5i)T^{2}$$
17 $$1 + (-0.276 - 0.720i)T + (-12.6 + 11.3i)T^{2}$$
19 $$1 + (-0.101 - 0.962i)T + (-18.5 + 3.95i)T^{2}$$
23 $$1 + (-0.143 + 2.73i)T + (-22.8 - 2.40i)T^{2}$$
29 $$1 + (-2.32 + 3.19i)T + (-8.96 - 27.5i)T^{2}$$
31 $$1 + (-3.67 + 8.24i)T + (-20.7 - 23.0i)T^{2}$$
37 $$1 + (1.77 - 1.15i)T + (15.0 - 33.8i)T^{2}$$
41 $$1 + (6.23 - 2.02i)T + (33.1 - 24.0i)T^{2}$$
43 $$1 + (2.74 + 2.74i)T + 43iT^{2}$$
47 $$1 + (4.18 + 1.60i)T + (34.9 + 31.4i)T^{2}$$
53 $$1 + (0.179 - 0.145i)T + (11.0 - 51.8i)T^{2}$$
59 $$1 + (-2.95 + 3.28i)T + (-6.16 - 58.6i)T^{2}$$
61 $$1 + (9.04 - 8.14i)T + (6.37 - 60.6i)T^{2}$$
67 $$1 + (-0.672 + 0.258i)T + (49.7 - 44.8i)T^{2}$$
71 $$1 + (-3.49 - 2.53i)T + (21.9 + 67.5i)T^{2}$$
73 $$1 + (-0.442 + 0.681i)T + (-29.6 - 66.6i)T^{2}$$
79 $$1 + (-2.01 - 4.51i)T + (-52.8 + 58.7i)T^{2}$$
83 $$1 + (2.09 - 13.2i)T + (-78.9 - 25.6i)T^{2}$$
89 $$1 + (7.76 + 8.62i)T + (-9.30 + 88.5i)T^{2}$$
97 $$1 + (-2.13 - 13.4i)T + (-92.2 + 29.9i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$