# Properties

 Degree 2 Conductor $2^{2} \cdot 7 \cdot 11 \cdot 13$ Sign $0.979 + 0.203i$ Motivic weight 1 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 − 2.10·3-s + 3.39i·5-s − i·7-s + 1.44·9-s + i·11-s + (−3.53 − 0.733i)13-s − 7.15i·15-s + 6.05·17-s + 1.08i·19-s + 2.10i·21-s − 6.38·23-s − 6.49·25-s + 3.26·27-s − 1.88·29-s − 5.69i·31-s + ⋯
 L(s)  = 1 − 1.21·3-s + 1.51i·5-s − 0.377i·7-s + 0.483·9-s + 0.301i·11-s + (−0.979 − 0.203i)13-s − 1.84i·15-s + 1.46·17-s + 0.248i·19-s + 0.460i·21-s − 1.33·23-s − 1.29·25-s + 0.629·27-s − 0.349·29-s − 1.02i·31-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.979 + 0.203i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.979 + 0.203i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$4004$$    =    $$2^{2} \cdot 7 \cdot 11 \cdot 13$$ $$\varepsilon$$ = $0.979 + 0.203i$ motivic weight = $$1$$ character : $\chi_{4004} (2157, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 4004,\ (\ :1/2),\ 0.979 + 0.203i)$ $L(1)$ $\approx$ $0.7471388668$ $L(\frac12)$ $\approx$ $0.7471388668$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{2,\;7,\;11,\;13\}$, $$F_p$$ is a polynomial of degree 2. If $p \in \{2,\;7,\;11,\;13\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 $$1$$
7 $$1 + iT$$
11 $$1 - iT$$
13 $$1 + (3.53 + 0.733i)T$$
good3 $$1 + 2.10T + 3T^{2}$$
5 $$1 - 3.39iT - 5T^{2}$$
17 $$1 - 6.05T + 17T^{2}$$
19 $$1 - 1.08iT - 19T^{2}$$
23 $$1 + 6.38T + 23T^{2}$$
29 $$1 + 1.88T + 29T^{2}$$
31 $$1 + 5.69iT - 31T^{2}$$
37 $$1 + 5.35iT - 37T^{2}$$
41 $$1 - 1.69iT - 41T^{2}$$
43 $$1 - 8.15T + 43T^{2}$$
47 $$1 + 2.26iT - 47T^{2}$$
53 $$1 + 12.6T + 53T^{2}$$
59 $$1 + 9.70iT - 59T^{2}$$
61 $$1 - 1.04T + 61T^{2}$$
67 $$1 - 6.67iT - 67T^{2}$$
71 $$1 + 13.9iT - 71T^{2}$$
73 $$1 - 8.29iT - 73T^{2}$$
79 $$1 + 1.23T + 79T^{2}$$
83 $$1 + 10.9iT - 83T^{2}$$
89 $$1 + 2.96iT - 89T^{2}$$
97 $$1 - 0.535iT - 97T^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}