# Properties

 Degree 2 Conductor $2 \cdot 3 \cdot 59$ Sign $1$ Motivic weight 3 Primitive yes Self-dual yes Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 − 2·2-s + 3·3-s + 4·4-s + 5.85·5-s − 6·6-s + 28.5·7-s − 8·8-s + 9·9-s − 11.7·10-s − 2.56·11-s + 12·12-s + 82.9·13-s − 57.1·14-s + 17.5·15-s + 16·16-s − 74.8·17-s − 18·18-s + 34·19-s + 23.4·20-s + 85.6·21-s + 5.13·22-s + 15.1·23-s − 24·24-s − 90.6·25-s − 165.·26-s + 27·27-s + 114.·28-s + ⋯
 L(s)  = 1 − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.524·5-s − 0.408·6-s + 1.54·7-s − 0.353·8-s + 0.333·9-s − 0.370·10-s − 0.0703·11-s + 0.288·12-s + 1.77·13-s − 1.09·14-s + 0.302·15-s + 0.250·16-s − 1.06·17-s − 0.235·18-s + 0.410·19-s + 0.262·20-s + 0.890·21-s + 0.0497·22-s + 0.137·23-s − 0.204·24-s − 0.725·25-s − 1.25·26-s + 0.192·27-s + 0.771·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$354$$    =    $$2 \cdot 3 \cdot 59$$ $$\varepsilon$$ = $1$ motivic weight = $$3$$ character : $\chi_{354} (1, \cdot )$ primitive : yes self-dual : yes analytic rank = 0 Selberg data = $(2,\ 354,\ (\ :3/2),\ 1)$ $L(2)$ $\approx$ $2.322692999$ $L(\frac12)$ $\approx$ $2.322692999$ $L(\frac{5}{2})$ 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 + 2T$$
3 $$1 - 3T$$
59 $$1 - 59T$$
good5 $$1 - 5.85T + 125T^{2}$$
7 $$1 - 28.5T + 343T^{2}$$
11 $$1 + 2.56T + 1.33e3T^{2}$$
13 $$1 - 82.9T + 2.19e3T^{2}$$
17 $$1 + 74.8T + 4.91e3T^{2}$$
19 $$1 - 34T + 6.85e3T^{2}$$
23 $$1 - 15.1T + 1.21e4T^{2}$$
29 $$1 - 239.T + 2.43e4T^{2}$$
31 $$1 + 242.T + 2.97e4T^{2}$$
37 $$1 + 269.T + 5.06e4T^{2}$$
41 $$1 + 179.T + 6.89e4T^{2}$$
43 $$1 - 441.T + 7.95e4T^{2}$$
47 $$1 + 249.T + 1.03e5T^{2}$$
53 $$1 + 19.2T + 1.48e5T^{2}$$
61 $$1 - 789.T + 2.26e5T^{2}$$
67 $$1 - 829.T + 3.00e5T^{2}$$
71 $$1 + 825.T + 3.57e5T^{2}$$
73 $$1 - 913.T + 3.89e5T^{2}$$
79 $$1 + 1.22e3T + 4.93e5T^{2}$$
83 $$1 - 873.T + 5.71e5T^{2}$$
89 $$1 + 888.T + 7.04e5T^{2}$$
97 $$1 - 384.T + 9.12e5T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}