Properties

 Degree 2 Conductor $2^{6} \cdot 3^{2} \cdot 7$ Sign $0.755 - 0.654i$ Motivic weight 1 Primitive yes Self-dual no Analytic rank 0

Related objects

Dirichlet series

 L(s)  = 1 + (−2 + 1.73i)7-s + 3.46i·11-s − 6.92i·17-s + 4·19-s − 3.46i·23-s + 5·25-s − 6·29-s + 4·31-s + 2·37-s − 6.92i·41-s + 3.46i·43-s + (1.00 − 6.92i)49-s − 6·53-s + 12·59-s + 13.8i·61-s + ⋯
 L(s)  = 1 + (−0.755 + 0.654i)7-s + 1.04i·11-s − 1.68i·17-s + 0.917·19-s − 0.722i·23-s + 25-s − 1.11·29-s + 0.718·31-s + 0.328·37-s − 1.08i·41-s + 0.528i·43-s + (0.142 − 0.989i)49-s − 0.824·53-s + 1.56·59-s + 1.77i·61-s + ⋯

Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 - 0.654i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.755 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 $$d$$ = $$2$$ $$N$$ = $$4032$$    =    $$2^{6} \cdot 3^{2} \cdot 7$$ $$\varepsilon$$ = $0.755 - 0.654i$ motivic weight = $$1$$ character : $\chi_{4032} (3583, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 4032,\ (\ :1/2),\ 0.755 - 0.654i)$ $L(1)$ $\approx$ $1.608895491$ $L(\frac12)$ $\approx$ $1.608895491$ $L(\frac{3}{2})$ not available $L(1)$ not available

Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{2,\;3,\;7\}$, $$F_p$$ is a polynomial of degree 2. If $p \in \{2,\;3,\;7\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 $$1$$
3 $$1$$
7 $$1 + (2 - 1.73i)T$$
good5 $$1 - 5T^{2}$$
11 $$1 - 3.46iT - 11T^{2}$$
13 $$1 - 13T^{2}$$
17 $$1 + 6.92iT - 17T^{2}$$
19 $$1 - 4T + 19T^{2}$$
23 $$1 + 3.46iT - 23T^{2}$$
29 $$1 + 6T + 29T^{2}$$
31 $$1 - 4T + 31T^{2}$$
37 $$1 - 2T + 37T^{2}$$
41 $$1 + 6.92iT - 41T^{2}$$
43 $$1 - 3.46iT - 43T^{2}$$
47 $$1 + 47T^{2}$$
53 $$1 + 6T + 53T^{2}$$
59 $$1 - 12T + 59T^{2}$$
61 $$1 - 13.8iT - 61T^{2}$$
67 $$1 - 3.46iT - 67T^{2}$$
71 $$1 - 10.3iT - 71T^{2}$$
73 $$1 - 13.8iT - 73T^{2}$$
79 $$1 + 10.3iT - 79T^{2}$$
83 $$1 - 12T + 83T^{2}$$
89 $$1 - 6.92iT - 89T^{2}$$
97 $$1 - 97T^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}