# Properties

 Degree 2 Conductor $2 \cdot 3^{2} \cdot 5^{2} \cdot 7$ Sign $0.876 - 0.481i$ Motivic weight 1 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (0.707 + 0.707i)2-s + 1.00i·4-s + (−0.707 + 0.707i)7-s + (−0.707 + 0.707i)8-s − 0.585i·11-s + (0.0249 + 0.0249i)13-s − 1.00·14-s − 1.00·16-s + (−1.92 − 1.92i)17-s − 4.87i·19-s + (0.414 − 0.414i)22-s + (5.15 − 5.15i)23-s + 0.0352i·26-s + (−0.707 − 0.707i)28-s + 4.58·29-s + ⋯
 L(s)  = 1 + (0.499 + 0.499i)2-s + 0.500i·4-s + (−0.267 + 0.267i)7-s + (−0.250 + 0.250i)8-s − 0.176i·11-s + (0.00691 + 0.00691i)13-s − 0.267·14-s − 0.250·16-s + (−0.466 − 0.466i)17-s − 1.11i·19-s + (0.0883 − 0.0883i)22-s + (1.07 − 1.07i)23-s + 0.00691i·26-s + (−0.133 − 0.133i)28-s + 0.850·29-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.876 - 0.481i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.876 - 0.481i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$3150$$    =    $$2 \cdot 3^{2} \cdot 5^{2} \cdot 7$$ $$\varepsilon$$ = $0.876 - 0.481i$ motivic weight = $$1$$ character : $\chi_{3150} (2843, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 3150,\ (\ :1/2),\ 0.876 - 0.481i)$ $L(1)$ $\approx$ $2.328250389$ $L(\frac12)$ $\approx$ $2.328250389$ $L(\frac{3}{2})$ 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 + (-0.707 - 0.707i)T$$
3 $$1$$
5 $$1$$
7 $$1 + (0.707 - 0.707i)T$$
good11 $$1 + 0.585iT - 11T^{2}$$
13 $$1 + (-0.0249 - 0.0249i)T + 13iT^{2}$$
17 $$1 + (1.92 + 1.92i)T + 17iT^{2}$$
19 $$1 + 4.87iT - 19T^{2}$$
23 $$1 + (-5.15 + 5.15i)T - 23iT^{2}$$
29 $$1 - 4.58T + 29T^{2}$$
31 $$1 - 9.83T + 31T^{2}$$
37 $$1 + (2.87 - 2.87i)T - 37iT^{2}$$
41 $$1 + 0.979iT - 41T^{2}$$
43 $$1 + (-4.27 - 4.27i)T + 43iT^{2}$$
47 $$1 + (-6.32 - 6.32i)T + 47iT^{2}$$
53 $$1 + (-1.56 + 1.56i)T - 53iT^{2}$$
59 $$1 - 0.670T + 59T^{2}$$
61 $$1 - 2.35T + 61T^{2}$$
67 $$1 + (5.08 - 5.08i)T - 67iT^{2}$$
71 $$1 - 2.92iT - 71T^{2}$$
73 $$1 + (2.51 + 2.51i)T + 73iT^{2}$$
79 $$1 - 8.71iT - 79T^{2}$$
83 $$1 + (-2.99 + 2.99i)T - 83iT^{2}$$
89 $$1 - 9.72T + 89T^{2}$$
97 $$1 + (4.74 - 4.74i)T - 97iT^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}