Properties

 Label 2-14-7.5-c12-0-3 Degree $2$ Conductor $14$ Sign $0.963 - 0.268i$ Analytic cond. $12.7959$ Root an. cond. $3.57713$ Motivic weight $12$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

Related objects

Dirichlet series

 L(s)  = 1 + (−22.6 + 39.1i)2-s + (−745. + 430. i)3-s + (−1.02e3 − 1.77e3i)4-s + (−7.53e3 − 4.34e3i)5-s − 3.89e4i·6-s + (−9.37e4 − 7.10e4i)7-s + 9.26e4·8-s + (1.05e5 − 1.82e5i)9-s + (3.40e5 − 1.96e5i)10-s + (2.57e5 + 4.46e5i)11-s + (1.52e6 + 8.82e5i)12-s + 2.74e6i·13-s + (4.90e6 − 2.06e6i)14-s + 7.49e6·15-s + (−2.09e6 + 3.63e6i)16-s + (5.50e6 − 3.17e6i)17-s + ⋯
 L(s)  = 1 + (−0.353 + 0.612i)2-s + (−1.02 + 0.590i)3-s + (−0.249 − 0.433i)4-s + (−0.482 − 0.278i)5-s − 0.835i·6-s + (−0.797 − 0.603i)7-s + 0.353·8-s + (0.198 − 0.343i)9-s + (0.340 − 0.196i)10-s + (0.145 + 0.252i)11-s + (0.511 + 0.295i)12-s + 0.569i·13-s + (0.651 − 0.274i)14-s + 0.657·15-s + (−0.125 + 0.216i)16-s + (0.228 − 0.131i)17-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.963 - 0.268i)\, \overline{\Lambda}(13-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (0.963 - 0.268i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 Degree: $$2$$ Conductor: $$14$$    =    $$2 \cdot 7$$ Sign: $0.963 - 0.268i$ Analytic conductor: $$12.7959$$ Root analytic conductor: $$3.57713$$ Motivic weight: $$12$$ Rational: no Arithmetic: yes Character: $\chi_{14} (5, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 14,\ (\ :6),\ 0.963 - 0.268i)$$

Particular Values

 $$L(\frac{13}{2})$$ $$\approx$$ $$0.628997 + 0.0860986i$$ $$L(\frac12)$$ $$\approx$$ $$0.628997 + 0.0860986i$$ $$L(7)$$ not available $$L(1)$$ not available

Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (22.6 - 39.1i)T$$
7 $$1 + (9.37e4 + 7.10e4i)T$$
good3 $$1 + (745. - 430. i)T + (2.65e5 - 4.60e5i)T^{2}$$
5 $$1 + (7.53e3 + 4.34e3i)T + (1.22e8 + 2.11e8i)T^{2}$$
11 $$1 + (-2.57e5 - 4.46e5i)T + (-1.56e12 + 2.71e12i)T^{2}$$
13 $$1 - 2.74e6iT - 2.32e13T^{2}$$
17 $$1 + (-5.50e6 + 3.17e6i)T + (2.91e14 - 5.04e14i)T^{2}$$
19 $$1 + (-6.71e7 - 3.87e7i)T + (1.10e15 + 1.91e15i)T^{2}$$
23 $$1 + (-5.20e7 + 9.01e7i)T + (-1.09e16 - 1.89e16i)T^{2}$$
29 $$1 + 1.41e8T + 3.53e17T^{2}$$
31 $$1 + (6.99e8 - 4.03e8i)T + (3.93e17 - 6.82e17i)T^{2}$$
37 $$1 + (-1.86e9 + 3.23e9i)T + (-3.29e18 - 5.70e18i)T^{2}$$
41 $$1 + 4.75e9iT - 2.25e19T^{2}$$
43 $$1 - 8.94e9T + 3.99e19T^{2}$$
47 $$1 + (-1.79e10 - 1.03e10i)T + (5.80e19 + 1.00e20i)T^{2}$$
53 $$1 + (1.17e10 + 2.03e10i)T + (-2.45e20 + 4.25e20i)T^{2}$$
59 $$1 + (1.83e9 - 1.05e9i)T + (8.89e20 - 1.54e21i)T^{2}$$
61 $$1 + (7.28e9 + 4.20e9i)T + (1.32e21 + 2.29e21i)T^{2}$$
67 $$1 + (-3.27e10 - 5.66e10i)T + (-4.09e21 + 7.08e21i)T^{2}$$
71 $$1 + 2.46e11T + 1.64e22T^{2}$$
73 $$1 + (-8.87e10 + 5.12e10i)T + (1.14e22 - 1.98e22i)T^{2}$$
79 $$1 + (-1.38e11 + 2.39e11i)T + (-2.95e22 - 5.11e22i)T^{2}$$
83 $$1 - 8.18e10iT - 1.06e23T^{2}$$
89 $$1 + (4.69e10 + 2.71e10i)T + (1.23e23 + 2.13e23i)T^{2}$$
97 $$1 - 3.42e11iT - 6.93e23T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$