# Properties

 Degree 2 Conductor $2 \cdot 3^{2} \cdot 5^{2} \cdot 7$ Sign $-0.503 + 0.863i$ Motivic weight 1 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + 2-s + 4-s + (−2.64 − 0.0951i)7-s + 8-s + 5.28i·11-s − 2.19·13-s + (−2.64 − 0.0951i)14-s + 16-s − 1.04i·17-s − 6.43i·19-s + 5.28i·22-s − 7.47·23-s − 2.19·26-s + (−2.64 − 0.0951i)28-s − 7.47i·29-s + ⋯
 L(s)  = 1 + 0.707·2-s + 0.5·4-s + (−0.999 − 0.0359i)7-s + 0.353·8-s + 1.59i·11-s − 0.607·13-s + (−0.706 − 0.0254i)14-s + 0.250·16-s − 0.253i·17-s − 1.47i·19-s + 1.12i·22-s − 1.55·23-s − 0.429·26-s + (−0.499 − 0.0179i)28-s − 1.38i·29-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.503 + 0.863i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.503 + 0.863i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$3150$$    =    $$2 \cdot 3^{2} \cdot 5^{2} \cdot 7$$ $$\varepsilon$$ = $-0.503 + 0.863i$ motivic weight = $$1$$ character : $\chi_{3150} (3149, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 3150,\ (\ :1/2),\ -0.503 + 0.863i)$ $L(1)$ $\approx$ $1.086077824$ $L(\frac12)$ $\approx$ $1.086077824$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;3,\;5,\;7\}$,$$F_p(T)$$ is a polynomial of degree 2. If $p \in \{2,\;3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1 - T$$
3 $$1$$
5 $$1$$
7 $$1 + (2.64 + 0.0951i)T$$
good11 $$1 - 5.28iT - 11T^{2}$$
13 $$1 + 2.19T + 13T^{2}$$
17 $$1 + 1.04iT - 17T^{2}$$
19 $$1 + 6.43iT - 19T^{2}$$
23 $$1 + 7.47T + 23T^{2}$$
29 $$1 + 7.47iT - 29T^{2}$$
31 $$1 + 9.09iT - 31T^{2}$$
37 $$1 - 0.855iT - 37T^{2}$$
41 $$1 + 2.19T + 41T^{2}$$
43 $$1 + 0.954iT - 43T^{2}$$
47 $$1 + 11.0iT - 47T^{2}$$
53 $$1 - 3.09T + 53T^{2}$$
59 $$1 - 13.7T + 59T^{2}$$
61 $$1 - 8.05iT - 61T^{2}$$
67 $$1 + 5.33iT - 67T^{2}$$
71 $$1 + 6.43iT - 71T^{2}$$
73 $$1 + 4.57T + 73T^{2}$$
79 $$1 + 15.6T + 79T^{2}$$
83 $$1 - 4.38iT - 83T^{2}$$
89 $$1 - 4.28T + 89T^{2}$$
97 $$1 + 11.8T + 97T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}