# Properties

 Degree 2 Conductor $3 \cdot 5 \cdot 7$ Sign $0.957 - 0.288i$ Motivic weight 1 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (−0.167 + 0.167i)2-s + (0.707 − 0.707i)3-s + 1.94i·4-s + (2.23 − 0.0836i)5-s + 0.236i·6-s + (−2.64 + 0.0627i)7-s + (−0.658 − 0.658i)8-s − 1.00i·9-s + (−0.359 + 0.387i)10-s + 3.98·11-s + (1.37 + 1.37i)12-s + (0.500 − 0.500i)13-s + (0.431 − 0.452i)14-s + (1.52 − 1.63i)15-s − 3.66·16-s + (−1.67 − 1.67i)17-s + ⋯
 L(s)  = 1 + (−0.118 + 0.118i)2-s + (0.408 − 0.408i)3-s + 0.972i·4-s + (0.999 − 0.0373i)5-s + 0.0964i·6-s + (−0.999 + 0.0237i)7-s + (−0.232 − 0.232i)8-s − 0.333i·9-s + (−0.113 + 0.122i)10-s + 1.20·11-s + (0.396 + 0.396i)12-s + (0.138 − 0.138i)13-s + (0.115 − 0.120i)14-s + (0.392 − 0.423i)15-s − 0.917·16-s + (−0.407 − 0.407i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 - 0.288i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.957 - 0.288i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$105$$    =    $$3 \cdot 5 \cdot 7$$ $$\varepsilon$$ = $0.957 - 0.288i$ motivic weight = $$1$$ character : $\chi_{105} (13, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 105,\ (\ :1/2),\ 0.957 - 0.288i)$$ $$L(1)$$ $$\approx$$ $$1.12400 + 0.165813i$$ $$L(\frac12)$$ $$\approx$$ $$1.12400 + 0.165813i$$ $$L(\frac{3}{2})$$ 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 + (-0.707 + 0.707i)T$$
5 $$1 + (-2.23 + 0.0836i)T$$
7 $$1 + (2.64 - 0.0627i)T$$
good2 $$1 + (0.167 - 0.167i)T - 2iT^{2}$$
11 $$1 - 3.98T + 11T^{2}$$
13 $$1 + (-0.500 + 0.500i)T - 13iT^{2}$$
17 $$1 + (1.67 + 1.67i)T + 17iT^{2}$$
19 $$1 + 7.21T + 19T^{2}$$
23 $$1 + (5.16 + 5.16i)T + 23iT^{2}$$
29 $$1 - 3.65iT - 29T^{2}$$
31 $$1 - 4.93iT - 31T^{2}$$
37 $$1 + (-0.292 + 0.292i)T - 37iT^{2}$$
41 $$1 + 7.63iT - 41T^{2}$$
43 $$1 + (-3.65 - 3.65i)T + 43iT^{2}$$
47 $$1 + (0.305 + 0.305i)T + 47iT^{2}$$
53 $$1 + (-5.39 - 5.39i)T + 53iT^{2}$$
59 $$1 - 6.10T + 59T^{2}$$
61 $$1 - 7.11iT - 61T^{2}$$
67 $$1 + (-0.944 + 0.944i)T - 67iT^{2}$$
71 $$1 - 1.19T + 71T^{2}$$
73 $$1 + (-1.38 + 1.38i)T - 73iT^{2}$$
79 $$1 + 8.64iT - 79T^{2}$$
83 $$1 + (11.9 - 11.9i)T - 83iT^{2}$$
89 $$1 - 7.82T + 89T^{2}$$
97 $$1 + (-7.43 - 7.43i)T + 97iT^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}