Properties

 Label 2-7e2-7.4-c5-0-4 Degree $2$ Conductor $49$ Sign $-0.947 + 0.318i$ Analytic cond. $7.85880$ Root an. cond. $2.80335$ Motivic weight $5$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

Related objects

Dirichlet series

 L(s)  = 1 + (3.90 + 6.76i)2-s + (−11.7 + 20.3i)3-s + (−14.5 + 25.1i)4-s + (37.1 + 64.3i)5-s − 183.·6-s + 22.8·8-s + (−155. − 268. i)9-s + (−290. + 502. i)10-s + (212. − 367. i)11-s + (−341. − 592. i)12-s − 252.·13-s − 1.74e3·15-s + (554. + 960. i)16-s + (552. − 956. i)17-s + (1.21e3 − 2.09e3i)18-s + (3.23 + 5.60i)19-s + ⋯
 L(s)  = 1 + (0.690 + 1.19i)2-s + (−0.754 + 1.30i)3-s + (−0.454 + 0.786i)4-s + (0.664 + 1.15i)5-s − 2.08·6-s + 0.126·8-s + (−0.638 − 1.10i)9-s + (−0.917 + 1.58i)10-s + (0.528 − 0.915i)11-s + (−0.685 − 1.18i)12-s − 0.413·13-s − 2.00·15-s + (0.541 + 0.937i)16-s + (0.463 − 0.802i)17-s + (0.881 − 1.52i)18-s + (0.00205 + 0.00356i)19-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.947 + 0.318i)\, \overline{\Lambda}(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.947 + 0.318i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 Degree: $$2$$ Conductor: $$49$$    =    $$7^{2}$$ Sign: $-0.947 + 0.318i$ Analytic conductor: $$7.85880$$ Root analytic conductor: $$2.80335$$ Motivic weight: $$5$$ Rational: no Arithmetic: yes Character: $\chi_{49} (18, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 49,\ (\ :5/2),\ -0.947 + 0.318i)$$

Particular Values

 $$L(3)$$ $$\approx$$ $$0.331300 - 2.02573i$$ $$L(\frac12)$$ $$\approx$$ $$0.331300 - 2.02573i$$ $$L(\frac{7}{2})$$ not available $$L(1)$$ not available

Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad7 $$1$$
good2 $$1 + (-3.90 - 6.76i)T + (-16 + 27.7i)T^{2}$$
3 $$1 + (11.7 - 20.3i)T + (-121.5 - 210. i)T^{2}$$
5 $$1 + (-37.1 - 64.3i)T + (-1.56e3 + 2.70e3i)T^{2}$$
11 $$1 + (-212. + 367. i)T + (-8.05e4 - 1.39e5i)T^{2}$$
13 $$1 + 252.T + 3.71e5T^{2}$$
17 $$1 + (-552. + 956. i)T + (-7.09e5 - 1.22e6i)T^{2}$$
19 $$1 + (-3.23 - 5.60i)T + (-1.23e6 + 2.14e6i)T^{2}$$
23 $$1 + (-1.80e3 - 3.12e3i)T + (-3.21e6 + 5.57e6i)T^{2}$$
29 $$1 + 5.00e3T + 2.05e7T^{2}$$
31 $$1 + (1.41e3 - 2.44e3i)T + (-1.43e7 - 2.47e7i)T^{2}$$
37 $$1 + (-1.02e3 - 1.77e3i)T + (-3.46e7 + 6.00e7i)T^{2}$$
41 $$1 - 9.39e3T + 1.15e8T^{2}$$
43 $$1 - 1.03e4T + 1.47e8T^{2}$$
47 $$1 + (8.51e3 + 1.47e4i)T + (-1.14e8 + 1.98e8i)T^{2}$$
53 $$1 + (-1.97e4 + 3.42e4i)T + (-2.09e8 - 3.62e8i)T^{2}$$
59 $$1 + (1.69e4 - 2.94e4i)T + (-3.57e8 - 6.19e8i)T^{2}$$
61 $$1 + (1.41e4 + 2.45e4i)T + (-4.22e8 + 7.31e8i)T^{2}$$
67 $$1 + (2.80e4 - 4.85e4i)T + (-6.75e8 - 1.16e9i)T^{2}$$
71 $$1 + 1.55e4T + 1.80e9T^{2}$$
73 $$1 + (-3.91e4 + 6.77e4i)T + (-1.03e9 - 1.79e9i)T^{2}$$
79 $$1 + (-2.26e4 - 3.92e4i)T + (-1.53e9 + 2.66e9i)T^{2}$$
83 $$1 - 1.38e3T + 3.93e9T^{2}$$
89 $$1 + (-3.44e4 - 5.96e4i)T + (-2.79e9 + 4.83e9i)T^{2}$$
97 $$1 - 1.08e5T + 8.58e9T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$