# Properties

 Degree 2 Conductor $3 \cdot 5 \cdot 7$ Sign $0.662 - 0.749i$ Motivic weight 1 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + 0.193i·2-s + i·3-s + 1.96·4-s + (−1.48 + 1.67i)5-s − 0.193·6-s − i·7-s + 0.768i·8-s − 9-s + (−0.324 − 0.287i)10-s + 2·11-s + 1.96i·12-s − 1.35i·13-s + 0.193·14-s + (−1.67 − 1.48i)15-s + 3.77·16-s − 3.35i·17-s + ⋯
 L(s)  = 1 + 0.137i·2-s + 0.577i·3-s + 0.981·4-s + (−0.662 + 0.749i)5-s − 0.0791·6-s − 0.377i·7-s + 0.271i·8-s − 0.333·9-s + (−0.102 − 0.0908i)10-s + 0.603·11-s + 0.566i·12-s − 0.374i·13-s + 0.0518·14-s + (−0.432 − 0.382i)15-s + 0.943·16-s − 0.812i·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.662 - 0.749i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.662 - 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$105$$    =    $$3 \cdot 5 \cdot 7$$ $$\varepsilon$$ = $0.662 - 0.749i$ motivic weight = $$1$$ character : $\chi_{105} (64, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 105,\ (\ :1/2),\ 0.662 - 0.749i)$$ $$L(1)$$ $$\approx$$ $$1.01753 + 0.458536i$$ $$L(\frac12)$$ $$\approx$$ $$1.01753 + 0.458536i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{3,\;5,\;7\}$,$$F_p(T)$$ is a polynomial of degree 2. If $p \in \{3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 $$1 - iT$$
5 $$1 + (1.48 - 1.67i)T$$
7 $$1 + iT$$
good2 $$1 - 0.193iT - 2T^{2}$$
11 $$1 - 2T + 11T^{2}$$
13 $$1 + 1.35iT - 13T^{2}$$
17 $$1 + 3.35iT - 17T^{2}$$
19 $$1 + 5.35T + 19T^{2}$$
23 $$1 + 4.96iT - 23T^{2}$$
29 $$1 + 7.92T + 29T^{2}$$
31 $$1 - 4.57T + 31T^{2}$$
37 $$1 + 0.775iT - 37T^{2}$$
41 $$1 - 3.73T + 41T^{2}$$
43 $$1 - 12.6iT - 43T^{2}$$
47 $$1 - 9.92iT - 47T^{2}$$
53 $$1 + 8.57iT - 53T^{2}$$
59 $$1 - 8.62T + 59T^{2}$$
61 $$1 + 8.70T + 61T^{2}$$
67 $$1 - 9.92iT - 67T^{2}$$
71 $$1 - 2T + 71T^{2}$$
73 $$1 - 9.35iT - 73T^{2}$$
79 $$1 + 10.7T + 79T^{2}$$
83 $$1 + 3.22iT - 83T^{2}$$
89 $$1 + 1.03T + 89T^{2}$$
97 $$1 + 18.4iT - 97T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}