Properties

 Label 2-294-7.2-c5-0-1 Degree $2$ Conductor $294$ Sign $0.605 - 0.795i$ Analytic cond. $47.1528$ Root an. cond. $6.86679$ Motivic weight $5$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

Related objects

Dirichlet series

 L(s)  = 1 + (2 − 3.46i)2-s + (−4.5 − 7.79i)3-s + (−7.99 − 13.8i)4-s + (−13 + 22.5i)5-s − 36·6-s − 63.9·8-s + (−40.5 + 70.1i)9-s + (51.9 + 90.0i)10-s + (−332 − 575. i)11-s + (−72 + 124. i)12-s + 318·13-s + 234·15-s + (−128 + 221. i)16-s + (−791 − 1.37e3i)17-s + (162 + 280. i)18-s + (−118 + 204. i)19-s + ⋯
 L(s)  = 1 + (0.353 − 0.612i)2-s + (−0.288 − 0.499i)3-s + (−0.249 − 0.433i)4-s + (−0.232 + 0.402i)5-s − 0.408·6-s − 0.353·8-s + (−0.166 + 0.288i)9-s + (0.164 + 0.284i)10-s + (−0.827 − 1.43i)11-s + (−0.144 + 0.249i)12-s + 0.521·13-s + 0.268·15-s + (−0.125 + 0.216i)16-s + (−0.663 − 1.14i)17-s + (0.117 + 0.204i)18-s + (−0.0749 + 0.129i)19-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 Degree: $$2$$ Conductor: $$294$$    =    $$2 \cdot 3 \cdot 7^{2}$$ Sign: $0.605 - 0.795i$ Analytic conductor: $$47.1528$$ Root analytic conductor: $$6.86679$$ Motivic weight: $$5$$ Rational: no Arithmetic: yes Character: $\chi_{294} (79, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 294,\ (\ :5/2),\ 0.605 - 0.795i)$$

Particular Values

 $$L(3)$$ $$\approx$$ $$0.5691900279$$ $$L(\frac12)$$ $$\approx$$ $$0.5691900279$$ $$L(\frac{7}{2})$$ not available $$L(1)$$ not available

Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (-2 + 3.46i)T$$
3 $$1 + (4.5 + 7.79i)T$$
7 $$1$$
good5 $$1 + (13 - 22.5i)T + (-1.56e3 - 2.70e3i)T^{2}$$
11 $$1 + (332 + 575. i)T + (-8.05e4 + 1.39e5i)T^{2}$$
13 $$1 - 318T + 3.71e5T^{2}$$
17 $$1 + (791 + 1.37e3i)T + (-7.09e5 + 1.22e6i)T^{2}$$
19 $$1 + (118 - 204. i)T + (-1.23e6 - 2.14e6i)T^{2}$$
23 $$1 + (1.10e3 - 1.91e3i)T + (-3.21e6 - 5.57e6i)T^{2}$$
29 $$1 + 4.95e3T + 2.05e7T^{2}$$
31 $$1 + (-3.56e3 - 6.17e3i)T + (-1.43e7 + 2.47e7i)T^{2}$$
37 $$1 + (2.17e3 - 3.77e3i)T + (-3.46e7 - 6.00e7i)T^{2}$$
41 $$1 - 1.05e4T + 1.15e8T^{2}$$
43 $$1 + 8.45e3T + 1.47e8T^{2}$$
47 $$1 + (2.67e3 - 4.63e3i)T + (-1.14e8 - 1.98e8i)T^{2}$$
53 $$1 + (-1.66e4 - 2.88e4i)T + (-2.09e8 + 3.62e8i)T^{2}$$
59 $$1 + (-7.71e3 - 1.33e4i)T + (-3.57e8 + 6.19e8i)T^{2}$$
61 $$1 + (-1.83e4 + 3.18e4i)T + (-4.22e8 - 7.31e8i)T^{2}$$
67 $$1 + (2.04e4 + 3.54e4i)T + (-6.75e8 + 1.16e9i)T^{2}$$
71 $$1 + 9.09e3T + 1.80e9T^{2}$$
73 $$1 + (-3.67e4 - 6.36e4i)T + (-1.03e9 + 1.79e9i)T^{2}$$
79 $$1 + (4.47e4 - 7.74e4i)T + (-1.53e9 - 2.66e9i)T^{2}$$
83 $$1 + 6.42e3T + 3.93e9T^{2}$$
89 $$1 + (-6.13e4 + 1.06e5i)T + (-2.79e9 - 4.83e9i)T^{2}$$
97 $$1 - 2.13e4T + 8.58e9T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$