# Properties

 Degree 2 Conductor $2 \cdot 3 \cdot 59$ Sign $-0.652 - 0.757i$ Motivic weight 1 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 − 2-s + (−1.65 + 0.509i)3-s + 4-s − 2.90i·5-s + (1.65 − 0.509i)6-s − 3.06·7-s − 8-s + (2.48 − 1.68i)9-s + 2.90i·10-s + 1.35·11-s + (−1.65 + 0.509i)12-s + 7.06i·13-s + 3.06·14-s + (1.48 + 4.80i)15-s + 16-s − 0.568i·17-s + ⋯
 L(s)  = 1 − 0.707·2-s + (−0.955 + 0.294i)3-s + 0.5·4-s − 1.29i·5-s + (0.675 − 0.208i)6-s − 1.16·7-s − 0.353·8-s + (0.826 − 0.562i)9-s + 0.917i·10-s + 0.408·11-s + (−0.477 + 0.147i)12-s + 1.95i·13-s + 0.820·14-s + (0.382 + 1.24i)15-s + 0.250·16-s − 0.137i·17-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.652 - 0.757i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.652 - 0.757i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$354$$    =    $$2 \cdot 3 \cdot 59$$ $$\varepsilon$$ = $-0.652 - 0.757i$ motivic weight = $$1$$ character : $\chi_{354} (353, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 354,\ (\ :1/2),\ -0.652 - 0.757i)$ $L(1)$ $\approx$ $0.0771695 + 0.168225i$ $L(\frac12)$ $\approx$ $0.0771695 + 0.168225i$ $L(\frac{3}{2})$ 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 + (1.65 - 0.509i)T$$
59 $$1 + (3.07 + 7.03i)T$$
good5 $$1 + 2.90iT - 5T^{2}$$
7 $$1 + 3.06T + 7T^{2}$$
11 $$1 - 1.35T + 11T^{2}$$
13 $$1 - 7.06iT - 13T^{2}$$
17 $$1 + 0.568iT - 17T^{2}$$
19 $$1 + 4.42T + 19T^{2}$$
23 $$1 + 7.20T + 23T^{2}$$
29 $$1 - 2.41iT - 29T^{2}$$
31 $$1 - 6.44iT - 31T^{2}$$
37 $$1 - 1.74iT - 37T^{2}$$
41 $$1 - 8.09iT - 41T^{2}$$
43 $$1 - 0.770iT - 43T^{2}$$
47 $$1 + 4.48T + 47T^{2}$$
53 $$1 - 1.03iT - 53T^{2}$$
61 $$1 + 10.6iT - 61T^{2}$$
67 $$1 - 1.09iT - 67T^{2}$$
71 $$1 - 11.0iT - 71T^{2}$$
73 $$1 + 5.96iT - 73T^{2}$$
79 $$1 - 8.62T + 79T^{2}$$
83 $$1 - 14.6T + 83T^{2}$$
89 $$1 + 11.5T + 89T^{2}$$
97 $$1 - 8.16iT - 97T^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}