# Properties

 Label 2-26-13.3-c7-0-1 Degree $2$ Conductor $26$ Sign $-0.732 - 0.680i$ Analytic cond. $8.12201$ Root an. cond. $2.84991$ Motivic weight $7$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (4 + 6.92i)2-s + (−39.6 − 68.6i)3-s + (−31.9 + 55.4i)4-s + 132.·5-s + (317. − 549. i)6-s + (−754. + 1.30e3i)7-s − 511.·8-s + (−2.04e3 + 3.54e3i)9-s + (530. + 918. i)10-s + (1.46e3 + 2.53e3i)11-s + 5.07e3·12-s + (−7.91e3 + 240. i)13-s − 1.20e4·14-s + (−5.25e3 − 9.10e3i)15-s + (−2.04e3 − 3.54e3i)16-s + (1.08e4 − 1.88e4i)17-s + ⋯
 L(s)  = 1 + (0.353 + 0.612i)2-s + (−0.847 − 1.46i)3-s + (−0.249 + 0.433i)4-s + 0.474·5-s + (0.599 − 1.03i)6-s + (−0.831 + 1.43i)7-s − 0.353·8-s + (−0.936 + 1.62i)9-s + (0.167 + 0.290i)10-s + (0.331 + 0.574i)11-s + 0.847·12-s + (−0.999 + 0.0303i)13-s − 1.17·14-s + (−0.402 − 0.696i)15-s + (−0.125 − 0.216i)16-s + (0.536 − 0.929i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.732 - 0.680i)\, \overline{\Lambda}(8-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.732 - 0.680i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$26$$    =    $$2 \cdot 13$$ Sign: $-0.732 - 0.680i$ Analytic conductor: $$8.12201$$ Root analytic conductor: $$2.84991$$ Motivic weight: $$7$$ Rational: no Arithmetic: yes Character: $\chi_{26} (3, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 26,\ (\ :7/2),\ -0.732 - 0.680i)$$

## Particular Values

 $$L(4)$$ $$\approx$$ $$0.218097 + 0.555195i$$ $$L(\frac12)$$ $$\approx$$ $$0.218097 + 0.555195i$$ $$L(\frac{9}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (-4 - 6.92i)T$$
13 $$1 + (7.91e3 - 240. i)T$$
good3 $$1 + (39.6 + 68.6i)T + (-1.09e3 + 1.89e3i)T^{2}$$
5 $$1 - 132.T + 7.81e4T^{2}$$
7 $$1 + (754. - 1.30e3i)T + (-4.11e5 - 7.13e5i)T^{2}$$
11 $$1 + (-1.46e3 - 2.53e3i)T + (-9.74e6 + 1.68e7i)T^{2}$$
17 $$1 + (-1.08e4 + 1.88e4i)T + (-2.05e8 - 3.55e8i)T^{2}$$
19 $$1 + (2.67e4 - 4.63e4i)T + (-4.46e8 - 7.74e8i)T^{2}$$
23 $$1 + (-1.42e4 - 2.46e4i)T + (-1.70e9 + 2.94e9i)T^{2}$$
29 $$1 + (7.63e4 + 1.32e5i)T + (-8.62e9 + 1.49e10i)T^{2}$$
31 $$1 - 8.33e4T + 2.75e10T^{2}$$
37 $$1 + (5.62e4 + 9.73e4i)T + (-4.74e10 + 8.22e10i)T^{2}$$
41 $$1 + (5.50e4 + 9.53e4i)T + (-9.73e10 + 1.68e11i)T^{2}$$
43 $$1 + (5.29e4 - 9.17e4i)T + (-1.35e11 - 2.35e11i)T^{2}$$
47 $$1 - 4.13e5T + 5.06e11T^{2}$$
53 $$1 + 8.89e5T + 1.17e12T^{2}$$
59 $$1 + (1.20e6 - 2.08e6i)T + (-1.24e12 - 2.15e12i)T^{2}$$
61 $$1 + (1.19e6 - 2.06e6i)T + (-1.57e12 - 2.72e12i)T^{2}$$
67 $$1 + (-1.84e5 - 3.20e5i)T + (-3.03e12 + 5.24e12i)T^{2}$$
71 $$1 + (-1.59e6 + 2.76e6i)T + (-4.54e12 - 7.87e12i)T^{2}$$
73 $$1 - 5.84e6T + 1.10e13T^{2}$$
79 $$1 - 1.07e6T + 1.92e13T^{2}$$
83 $$1 - 6.47e5T + 2.71e13T^{2}$$
89 $$1 + (-5.81e6 - 1.00e7i)T + (-2.21e13 + 3.83e13i)T^{2}$$
97 $$1 + (3.11e4 - 5.40e4i)T + (-4.03e13 - 6.99e13i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$