# Properties

 Label 2-26-13.12-c5-0-2 Degree $2$ Conductor $26$ Sign $-0.192 + 0.981i$ Analytic cond. $4.16997$ Root an. cond. $2.04205$ Motivic weight $5$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 4i·2-s − 13·3-s − 16·4-s − 51i·5-s − 52i·6-s − 105i·7-s − 64i·8-s − 74·9-s + 204·10-s − 120i·11-s + 208·12-s + (−598 − 117i)13-s + 420·14-s + 663i·15-s + 256·16-s − 1.10e3·17-s + ⋯
 L(s)  = 1 + 0.707i·2-s − 0.833·3-s − 0.5·4-s − 0.912i·5-s − 0.589i·6-s − 0.809i·7-s − 0.353i·8-s − 0.304·9-s + 0.645·10-s − 0.299i·11-s + 0.416·12-s + (−0.981 − 0.192i)13-s + 0.572·14-s + 0.760i·15-s + 0.250·16-s − 0.923·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.192 + 0.981i)\, \overline{\Lambda}(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.192 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$26$$    =    $$2 \cdot 13$$ Sign: $-0.192 + 0.981i$ Analytic conductor: $$4.16997$$ Root analytic conductor: $$2.04205$$ Motivic weight: $$5$$ Rational: no Arithmetic: yes Character: $\chi_{26} (25, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 26,\ (\ :5/2),\ -0.192 + 0.981i)$$

## Particular Values

 $$L(3)$$ $$\approx$$ $$0.326068 - 0.396047i$$ $$L(\frac12)$$ $$\approx$$ $$0.326068 - 0.396047i$$ $$L(\frac{7}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 - 4iT$$
13 $$1 + (598 + 117i)T$$
good3 $$1 + 13T + 243T^{2}$$
5 $$1 + 51iT - 3.12e3T^{2}$$
7 $$1 + 105iT - 1.68e4T^{2}$$
11 $$1 + 120iT - 1.61e5T^{2}$$
17 $$1 + 1.10e3T + 1.41e6T^{2}$$
19 $$1 - 1.17e3iT - 2.47e6T^{2}$$
23 $$1 - 1.05e3T + 6.43e6T^{2}$$
29 $$1 + 4.10e3T + 2.05e7T^{2}$$
31 $$1 + 9.62e3iT - 2.86e7T^{2}$$
37 $$1 + 8.70e3iT - 6.93e7T^{2}$$
41 $$1 - 9.48e3iT - 1.15e8T^{2}$$
43 $$1 - 9.99e3T + 1.47e8T^{2}$$
47 $$1 - 2.94e3iT - 2.29e8T^{2}$$
53 $$1 + 750T + 4.18e8T^{2}$$
59 $$1 - 4.09e4iT - 7.14e8T^{2}$$
61 $$1 + 5.79e4T + 8.44e8T^{2}$$
67 $$1 + 2.28e4iT - 1.35e9T^{2}$$
71 $$1 + 6.37e4iT - 1.80e9T^{2}$$
73 $$1 + 5.88e4iT - 2.07e9T^{2}$$
79 $$1 - 6.32e4T + 3.07e9T^{2}$$
83 $$1 + 5.54e4iT - 3.93e9T^{2}$$
89 $$1 - 1.04e5iT - 5.58e9T^{2}$$
97 $$1 + 1.60e5iT - 8.58e9T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$