# Properties

 Degree 2 Conductor $2 \cdot 3 \cdot 59$ Sign $0.883 + 0.468i$ Motivic weight 1 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 − 2-s + (1.64 + 0.548i)3-s + 4-s − 2.54i·5-s + (−1.64 − 0.548i)6-s − 0.0900·7-s − 8-s + (2.39 + 1.80i)9-s + 2.54i·10-s + 2.40·11-s + (1.64 + 0.548i)12-s − 2.81i·13-s + 0.0900·14-s + (1.39 − 4.18i)15-s + 16-s − 3.17i·17-s + ⋯
 L(s)  = 1 − 0.707·2-s + (0.948 + 0.316i)3-s + 0.5·4-s − 1.13i·5-s + (−0.670 − 0.223i)6-s − 0.0340·7-s − 0.353·8-s + (0.799 + 0.600i)9-s + 0.805i·10-s + 0.725·11-s + (0.474 + 0.158i)12-s − 0.781i·13-s + 0.0240·14-s + (0.360 − 1.08i)15-s + 0.250·16-s − 0.769i·17-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.883 + 0.468i)\, \overline{\Lambda}(2-s) \end{aligned}\n
\begin{aligned} \Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.883 + 0.468i)\, \overline{\Lambda}(1-s) \end{aligned}\n

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$354$$    =    $$2 \cdot 3 \cdot 59$$ $$\varepsilon$$ = $0.883 + 0.468i$ motivic weight = $$1$$ character : $\chi_{354} (353, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 354,\ (\ :1/2),\ 0.883 + 0.468i)$ $L(1)$ $\approx$ $1.33372 - 0.331505i$ $L(\frac12)$ $\approx$ $1.33372 - 0.331505i$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{2,\;3,\;59\}$, $$F_p(T)$$ is a polynomial of degree 2. If $p \in \{2,\;3,\;59\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1 + T$$
3 $$1 + (-1.64 - 0.548i)T$$
59 $$1 + (7.57 + 1.26i)T$$
good5 $$1 + 2.54iT - 5T^{2}$$
7 $$1 + 0.0900T + 7T^{2}$$
11 $$1 - 2.40T + 11T^{2}$$
13 $$1 + 2.81iT - 13T^{2}$$
17 $$1 + 3.17iT - 17T^{2}$$
19 $$1 + 2.49T + 19T^{2}$$
23 $$1 - 3.09T + 23T^{2}$$
29 $$1 - 6.21iT - 29T^{2}$$
31 $$1 - 2.16iT - 31T^{2}$$
37 $$1 + 11.5iT - 37T^{2}$$
41 $$1 - 0.762iT - 41T^{2}$$
43 $$1 - 7.62iT - 43T^{2}$$
47 $$1 - 7.90T + 47T^{2}$$
53 $$1 - 3.58iT - 53T^{2}$$
61 $$1 - 3.56iT - 61T^{2}$$
67 $$1 - 14.5iT - 67T^{2}$$
71 $$1 + 4.84iT - 71T^{2}$$
73 $$1 - 6.03iT - 73T^{2}$$
79 $$1 + 11.0T + 79T^{2}$$
83 $$1 + 1.68T + 83T^{2}$$
89 $$1 + 8.62T + 89T^{2}$$
97 $$1 - 0.394iT - 97T^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}