# Properties

 Label 2-336-7.4-c5-0-28 Degree $2$ Conductor $336$ Sign $-0.276 + 0.960i$ Analytic cond. $53.8889$ Root an. cond. $7.34091$ Motivic weight $5$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−4.5 + 7.79i)3-s + (−19.3 − 33.5i)5-s + (87.5 − 95.6i)7-s + (−40.5 − 70.1i)9-s + (−288. + 499. i)11-s + 391.·13-s + 348.·15-s + (664. − 1.15e3i)17-s + (471. + 816. i)19-s + (351. + 1.11e3i)21-s + (−816. − 1.41e3i)23-s + (812. − 1.40e3i)25-s + 729·27-s − 1.46e3·29-s + (−1.95e3 + 3.38e3i)31-s + ⋯
 L(s)  = 1 + (−0.288 + 0.499i)3-s + (−0.346 − 0.599i)5-s + (0.674 − 0.737i)7-s + (−0.166 − 0.288i)9-s + (−0.718 + 1.24i)11-s + 0.642·13-s + 0.399·15-s + (0.557 − 0.966i)17-s + (0.299 + 0.518i)19-s + (0.174 + 0.550i)21-s + (−0.321 − 0.557i)23-s + (0.260 − 0.450i)25-s + 0.192·27-s − 0.323·29-s + (−0.365 + 0.633i)31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.276 + 0.960i)\, \overline{\Lambda}(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.276 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$336$$    =    $$2^{4} \cdot 3 \cdot 7$$ Sign: $-0.276 + 0.960i$ Analytic conductor: $$53.8889$$ Root analytic conductor: $$7.34091$$ Motivic weight: $$5$$ Rational: no Arithmetic: yes Character: $\chi_{336} (193, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 336,\ (\ :5/2),\ -0.276 + 0.960i)$$

## Particular Values

 $$L(3)$$ $$\approx$$ $$1.087171410$$ $$L(\frac12)$$ $$\approx$$ $$1.087171410$$ $$L(\frac{7}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
3 $$1 + (4.5 - 7.79i)T$$
7 $$1 + (-87.5 + 95.6i)T$$
good5 $$1 + (19.3 + 33.5i)T + (-1.56e3 + 2.70e3i)T^{2}$$
11 $$1 + (288. - 499. i)T + (-8.05e4 - 1.39e5i)T^{2}$$
13 $$1 - 391.T + 3.71e5T^{2}$$
17 $$1 + (-664. + 1.15e3i)T + (-7.09e5 - 1.22e6i)T^{2}$$
19 $$1 + (-471. - 816. i)T + (-1.23e6 + 2.14e6i)T^{2}$$
23 $$1 + (816. + 1.41e3i)T + (-3.21e6 + 5.57e6i)T^{2}$$
29 $$1 + 1.46e3T + 2.05e7T^{2}$$
31 $$1 + (1.95e3 - 3.38e3i)T + (-1.43e7 - 2.47e7i)T^{2}$$
37 $$1 + (-8.15e3 - 1.41e4i)T + (-3.46e7 + 6.00e7i)T^{2}$$
41 $$1 + 1.31e4T + 1.15e8T^{2}$$
43 $$1 + 1.47e4T + 1.47e8T^{2}$$
47 $$1 + (3.40e3 + 5.90e3i)T + (-1.14e8 + 1.98e8i)T^{2}$$
53 $$1 + (-1.00e3 + 1.74e3i)T + (-2.09e8 - 3.62e8i)T^{2}$$
59 $$1 + (-2.57e4 + 4.45e4i)T + (-3.57e8 - 6.19e8i)T^{2}$$
61 $$1 + (2.05e4 + 3.55e4i)T + (-4.22e8 + 7.31e8i)T^{2}$$
67 $$1 + (-2.52e4 + 4.38e4i)T + (-6.75e8 - 1.16e9i)T^{2}$$
71 $$1 + 3.99e4T + 1.80e9T^{2}$$
73 $$1 + (-2.78e4 + 4.82e4i)T + (-1.03e9 - 1.79e9i)T^{2}$$
79 $$1 + (3.15e4 + 5.46e4i)T + (-1.53e9 + 2.66e9i)T^{2}$$
83 $$1 + 4.55e4T + 3.93e9T^{2}$$
89 $$1 + (7.84e3 + 1.35e4i)T + (-2.79e9 + 4.83e9i)T^{2}$$
97 $$1 - 3.12e3T + 8.58e9T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$