# Properties

 Label 2-26-13.3-c7-0-3 Degree $2$ Conductor $26$ Sign $-0.236 - 0.971i$ 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 + (35.5 + 61.6i)3-s + (−31.9 + 55.4i)4-s + 523.·5-s + (−284. + 492. i)6-s + (208. − 360. i)7-s − 511.·8-s + (−1.43e3 + 2.49e3i)9-s + (2.09e3 + 3.62e3i)10-s + (−1.71e3 − 2.96e3i)11-s − 4.55e3·12-s + (−6.74e3 − 4.15e3i)13-s + 3.33e3·14-s + (1.86e4 + 3.22e4i)15-s + (−2.04e3 − 3.54e3i)16-s + (3.26e3 − 5.66e3i)17-s + ⋯
 L(s)  = 1 + (0.353 + 0.612i)2-s + (0.760 + 1.31i)3-s + (−0.249 + 0.433i)4-s + 1.87·5-s + (−0.537 + 0.931i)6-s + (0.229 − 0.397i)7-s − 0.353·8-s + (−0.657 + 1.13i)9-s + (0.662 + 1.14i)10-s + (−0.387 − 0.671i)11-s − 0.760·12-s + (−0.851 − 0.525i)13-s + 0.324·14-s + (1.42 + 2.46i)15-s + (−0.125 − 0.216i)16-s + (0.161 − 0.279i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.236 - 0.971i)\, \overline{\Lambda}(8-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.236 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$26$$    =    $$2 \cdot 13$$ Sign: $-0.236 - 0.971i$ 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: $$(2,\ 26,\ (\ :7/2),\ -0.236 - 0.971i)$$

## Particular Values

 $$L(4)$$ $$\approx$$ $$1.86555 + 2.37452i$$ $$L(\frac12)$$ $$\approx$$ $$1.86555 + 2.37452i$$ $$L(\frac{9}{2})$$ 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 + (6.74e3 + 4.15e3i)T$$
good3 $$1 + (-35.5 - 61.6i)T + (-1.09e3 + 1.89e3i)T^{2}$$
5 $$1 - 523.T + 7.81e4T^{2}$$
7 $$1 + (-208. + 360. i)T + (-4.11e5 - 7.13e5i)T^{2}$$
11 $$1 + (1.71e3 + 2.96e3i)T + (-9.74e6 + 1.68e7i)T^{2}$$
17 $$1 + (-3.26e3 + 5.66e3i)T + (-2.05e8 - 3.55e8i)T^{2}$$
19 $$1 + (1.39e4 - 2.42e4i)T + (-4.46e8 - 7.74e8i)T^{2}$$
23 $$1 + (5.30e4 + 9.19e4i)T + (-1.70e9 + 2.94e9i)T^{2}$$
29 $$1 + (3.42e4 + 5.93e4i)T + (-8.62e9 + 1.49e10i)T^{2}$$
31 $$1 - 5.15e4T + 2.75e10T^{2}$$
37 $$1 + (2.40e4 + 4.16e4i)T + (-4.74e10 + 8.22e10i)T^{2}$$
41 $$1 + (-3.01e5 - 5.21e5i)T + (-9.73e10 + 1.68e11i)T^{2}$$
43 $$1 + (4.58e5 - 7.93e5i)T + (-1.35e11 - 2.35e11i)T^{2}$$
47 $$1 + 3.26e5T + 5.06e11T^{2}$$
53 $$1 - 9.34e5T + 1.17e12T^{2}$$
59 $$1 + (5.87e5 - 1.01e6i)T + (-1.24e12 - 2.15e12i)T^{2}$$
61 $$1 + (-1.44e6 + 2.50e6i)T + (-1.57e12 - 2.72e12i)T^{2}$$
67 $$1 + (1.59e5 + 2.75e5i)T + (-3.03e12 + 5.24e12i)T^{2}$$
71 $$1 + (6.41e5 - 1.11e6i)T + (-4.54e12 - 7.87e12i)T^{2}$$
73 $$1 + 1.67e6T + 1.10e13T^{2}$$
79 $$1 + 8.09e6T + 1.92e13T^{2}$$
83 $$1 - 5.57e6T + 2.71e13T^{2}$$
89 $$1 + (3.93e6 + 6.82e6i)T + (-2.21e13 + 3.83e13i)T^{2}$$
97 $$1 + (-3.40e6 + 5.89e6i)T + (-4.03e13 - 6.99e13i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$