# Properties

 Label 2-336-21.17-c3-0-13 Degree $2$ Conductor $336$ Sign $-0.425 - 0.905i$ Analytic cond. $19.8246$ Root an. cond. $4.45248$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−5.19 + 0.00519i)3-s + (8.05 + 13.9i)5-s + (5.67 + 17.6i)7-s + (26.9 − 0.0539i)9-s + (30.8 + 17.7i)11-s − 7.40i·13-s + (−41.9 − 72.4i)15-s + (14.4 − 25.0i)17-s + (−30.4 + 17.5i)19-s + (−29.6 − 91.5i)21-s + (48.0 − 27.7i)23-s + (−67.3 + 116. i)25-s + (−140. + 0.420i)27-s + 68.1i·29-s + (154. + 89.3i)31-s + ⋯
 L(s)  = 1 + (−0.999 + 0.000999i)3-s + (0.720 + 1.24i)5-s + (0.306 + 0.951i)7-s + (0.999 − 0.00199i)9-s + (0.845 + 0.487i)11-s − 0.158i·13-s + (−0.722 − 1.24i)15-s + (0.206 − 0.357i)17-s + (−0.367 + 0.212i)19-s + (−0.307 − 0.951i)21-s + (0.435 − 0.251i)23-s + (−0.539 + 0.933i)25-s + (−0.999 + 0.00299i)27-s + 0.436i·29-s + (0.896 + 0.517i)31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.425 - 0.905i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.425 - 0.905i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$336$$    =    $$2^{4} \cdot 3 \cdot 7$$ Sign: $-0.425 - 0.905i$ Analytic conductor: $$19.8246$$ Root analytic conductor: $$4.45248$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: $\chi_{336} (17, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 336,\ (\ :3/2),\ -0.425 - 0.905i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$1.517433305$$ $$L(\frac12)$$ $$\approx$$ $$1.517433305$$ $$L(\frac{5}{2})$$ 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 + (5.19 - 0.00519i)T$$
7 $$1 + (-5.67 - 17.6i)T$$
good5 $$1 + (-8.05 - 13.9i)T + (-62.5 + 108. i)T^{2}$$
11 $$1 + (-30.8 - 17.7i)T + (665.5 + 1.15e3i)T^{2}$$
13 $$1 + 7.40iT - 2.19e3T^{2}$$
17 $$1 + (-14.4 + 25.0i)T + (-2.45e3 - 4.25e3i)T^{2}$$
19 $$1 + (30.4 - 17.5i)T + (3.42e3 - 5.94e3i)T^{2}$$
23 $$1 + (-48.0 + 27.7i)T + (6.08e3 - 1.05e4i)T^{2}$$
29 $$1 - 68.1iT - 2.43e4T^{2}$$
31 $$1 + (-154. - 89.3i)T + (1.48e4 + 2.57e4i)T^{2}$$
37 $$1 + (-116. - 202. i)T + (-2.53e4 + 4.38e4i)T^{2}$$
41 $$1 + 370.T + 6.89e4T^{2}$$
43 $$1 - 187.T + 7.95e4T^{2}$$
47 $$1 + (87.3 + 151. i)T + (-5.19e4 + 8.99e4i)T^{2}$$
53 $$1 + (-235. - 136. i)T + (7.44e4 + 1.28e5i)T^{2}$$
59 $$1 + (48.4 - 83.8i)T + (-1.02e5 - 1.77e5i)T^{2}$$
61 $$1 + (333. - 192. i)T + (1.13e5 - 1.96e5i)T^{2}$$
67 $$1 + (509. - 881. i)T + (-1.50e5 - 2.60e5i)T^{2}$$
71 $$1 + 125. iT - 3.57e5T^{2}$$
73 $$1 + (-195. - 112. i)T + (1.94e5 + 3.36e5i)T^{2}$$
79 $$1 + (532. + 921. i)T + (-2.46e5 + 4.26e5i)T^{2}$$
83 $$1 + 601.T + 5.71e5T^{2}$$
89 $$1 + (752. + 1.30e3i)T + (-3.52e5 + 6.10e5i)T^{2}$$
97 $$1 - 327. iT - 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$