# Properties

 Degree 2 Conductor $2^{2} \cdot 3^{2} \cdot 11 \cdot 19$ Sign $i$ Motivic weight 1 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 − 3.31i·11-s − 2.79·13-s + 4.88i·17-s + 4.35·19-s − 5·25-s + 7·49-s − 8.72i·53-s − 2.72i·59-s − 14.7i·71-s + 17.1·79-s + 1.75i·83-s + 3.27i·89-s + 13.2i·101-s − 17.4·109-s − 20.7i·113-s + ⋯
 L(s)  = 1 − 1.00i·11-s − 0.775·13-s + 1.18i·17-s + 1.00·19-s − 25-s + 49-s − 1.19i·53-s − 0.355i·59-s − 1.74i·71-s + 1.92·79-s + 0.192i·83-s + 0.346i·89-s + 1.32i·101-s − 1.67·109-s − 1.94i·113-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 7524 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 7524 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$7524$$    =    $$2^{2} \cdot 3^{2} \cdot 11 \cdot 19$$ $$\varepsilon$$ = $i$ motivic weight = $$1$$ character : $\chi_{7524} (2089, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 7524,\ (\ :1/2),\ i)$$ $$L(1)$$ $$\approx$$ $$1.320342450$$ $$L(\frac12)$$ $$\approx$$ $$1.320342450$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;3,\;11,\;19\}$,$$F_p(T)$$ is a polynomial of degree 2. If $p \in \{2,\;3,\;11,\;19\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1$$
3 $$1$$
11 $$1 + 3.31iT$$
19 $$1 - 4.35T$$
good5 $$1 + 5T^{2}$$
7 $$1 - 7T^{2}$$
13 $$1 + 2.79T + 13T^{2}$$
17 $$1 - 4.88iT - 17T^{2}$$
23 $$1 + 23T^{2}$$
29 $$1 + 29T^{2}$$
31 $$1 - 31T^{2}$$
37 $$1 - 37T^{2}$$
41 $$1 + 41T^{2}$$
43 $$1 - 43T^{2}$$
47 $$1 + 47T^{2}$$
53 $$1 + 8.72iT - 53T^{2}$$
59 $$1 + 2.72iT - 59T^{2}$$
61 $$1 - 61T^{2}$$
67 $$1 - 67T^{2}$$
71 $$1 + 14.7iT - 71T^{2}$$
73 $$1 - 73T^{2}$$
79 $$1 - 17.1T + 79T^{2}$$
83 $$1 - 1.75iT - 83T^{2}$$
89 $$1 - 3.27iT - 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}