# Properties

 Degree 2 Conductor 349 Sign $0.997 - 0.0635i$ Motivic weight 1 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (−0.788 + 0.455i)2-s + (−1.46 + 2.53i)3-s + (−0.585 + 1.01i)4-s + (1.45 − 2.51i)5-s − 2.66i·6-s + (3.29 − 1.90i)7-s − 2.88i·8-s + (−2.77 − 4.81i)9-s + 2.64i·10-s − 3.72i·11-s + (−1.71 − 2.96i)12-s + (3.22 − 1.86i)13-s + (−1.73 + 3.00i)14-s + (4.24 + 7.35i)15-s + (0.145 + 0.251i)16-s − 5.36·17-s + ⋯
 L(s)  = 1 + (−0.557 + 0.322i)2-s + (−0.844 + 1.46i)3-s + (−0.292 + 0.506i)4-s + (0.649 − 1.12i)5-s − 1.08i·6-s + (1.24 − 0.718i)7-s − 1.02i·8-s + (−0.925 − 1.60i)9-s + 0.837i·10-s − 1.12i·11-s + (−0.493 − 0.855i)12-s + (0.895 − 0.516i)13-s + (−0.462 + 0.801i)14-s + (1.09 + 1.90i)15-s + (0.0363 + 0.0629i)16-s − 1.30·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0635i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0635i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$349$$ $$\varepsilon$$ = $0.997 - 0.0635i$ motivic weight = $$1$$ character : $\chi_{349} (123, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 349,\ (\ :1/2),\ 0.997 - 0.0635i)$ $L(1)$ $\approx$ $0.780421 + 0.0248384i$ $L(\frac12)$ $\approx$ $0.780421 + 0.0248384i$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \neq 349$,$$F_p(T)$$ is a polynomial of degree 2. If $p = 349$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad349 $$1 + (-6.70 + 17.4i)T$$
good2 $$1 + (0.788 - 0.455i)T + (1 - 1.73i)T^{2}$$
3 $$1 + (1.46 - 2.53i)T + (-1.5 - 2.59i)T^{2}$$
5 $$1 + (-1.45 + 2.51i)T + (-2.5 - 4.33i)T^{2}$$
7 $$1 + (-3.29 + 1.90i)T + (3.5 - 6.06i)T^{2}$$
11 $$1 + 3.72iT - 11T^{2}$$
13 $$1 + (-3.22 + 1.86i)T + (6.5 - 11.2i)T^{2}$$
17 $$1 + 5.36T + 17T^{2}$$
19 $$1 + (-1.62 + 2.80i)T + (-9.5 - 16.4i)T^{2}$$
23 $$1 + (2.39 + 4.14i)T + (-11.5 + 19.9i)T^{2}$$
29 $$1 + (2.19 - 3.79i)T + (-14.5 - 25.1i)T^{2}$$
31 $$1 - 6.79T + 31T^{2}$$
37 $$1 + 7.48T + 37T^{2}$$
41 $$1 + 2.38T + 41T^{2}$$
43 $$1 + (1.70 + 0.985i)T + (21.5 + 37.2i)T^{2}$$
47 $$1 + 7.30iT - 47T^{2}$$
53 $$1 - 12.2iT - 53T^{2}$$
59 $$1 + (7.06 + 4.08i)T + (29.5 + 51.0i)T^{2}$$
61 $$1 + 4.05iT - 61T^{2}$$
67 $$1 - 15.3T + 67T^{2}$$
71 $$1 + (-6.81 + 3.93i)T + (35.5 - 61.4i)T^{2}$$
73 $$1 + (3.57 - 6.19i)T + (-36.5 - 63.2i)T^{2}$$
79 $$1 - 11.6iT - 79T^{2}$$
83 $$1 + (-7.11 - 12.3i)T + (-41.5 + 71.8i)T^{2}$$
89 $$1 + (-12.8 - 7.42i)T + (44.5 + 77.0i)T^{2}$$
97 $$1 + (5.04 - 2.91i)T + (48.5 - 84.0i)T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}