# Properties

 Label 2-26-1.1-c7-0-4 Degree $2$ Conductor $26$ Sign $-1$ Analytic cond. $8.12201$ Root an. cond. $2.84991$ Motivic weight $7$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $1$

# Origins

## Dirichlet series

 L(s)  = 1 − 8·2-s − 39·3-s + 64·4-s + 385·5-s + 312·6-s − 293·7-s − 512·8-s − 666·9-s − 3.08e3·10-s − 5.40e3·11-s − 2.49e3·12-s + 2.19e3·13-s + 2.34e3·14-s − 1.50e4·15-s + 4.09e3·16-s − 2.10e4·17-s + 5.32e3·18-s − 2.73e4·19-s + 2.46e4·20-s + 1.14e4·21-s + 4.32e4·22-s − 6.30e4·23-s + 1.99e4·24-s + 7.01e4·25-s − 1.75e4·26-s + 1.11e5·27-s − 1.87e4·28-s + ⋯
 L(s)  = 1 − 0.707·2-s − 0.833·3-s + 1/2·4-s + 1.37·5-s + 0.589·6-s − 0.322·7-s − 0.353·8-s − 0.304·9-s − 0.973·10-s − 1.22·11-s − 0.416·12-s + 0.277·13-s + 0.228·14-s − 1.14·15-s + 1/4·16-s − 1.03·17-s + 0.215·18-s − 0.913·19-s + 0.688·20-s + 0.269·21-s + 0.865·22-s − 1.08·23-s + 0.294·24-s + 0.897·25-s − 0.196·26-s + 1.08·27-s − 0.161·28-s + ⋯

## Functional equation

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

## Invariants

 Degree: $$2$$ Conductor: $$26$$    =    $$2 \cdot 13$$ Sign: $-1$ Analytic conductor: $$8.12201$$ Root analytic conductor: $$2.84991$$ Motivic weight: $$7$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 26,\ (\ :7/2),\ -1)$$

## Particular Values

 $$L(4)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{9}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + p^{3} T$$
13 $$1 - p^{3} T$$
good3 $$1 + 13 p T + p^{7} T^{2}$$
5 $$1 - 77 p T + p^{7} T^{2}$$
7 $$1 + 293 T + p^{7} T^{2}$$
11 $$1 + 5402 T + p^{7} T^{2}$$
17 $$1 + 21011 T + p^{7} T^{2}$$
19 $$1 + 27326 T + p^{7} T^{2}$$
23 $$1 + 63072 T + p^{7} T^{2}$$
29 $$1 - 122238 T + p^{7} T^{2}$$
31 $$1 + 208396 T + p^{7} T^{2}$$
37 $$1 + 442379 T + p^{7} T^{2}$$
41 $$1 - 58000 T + p^{7} T^{2}$$
43 $$1 + 202025 T + p^{7} T^{2}$$
47 $$1 - 588511 T + p^{7} T^{2}$$
53 $$1 - 1684336 T + p^{7} T^{2}$$
59 $$1 + 442630 T + p^{7} T^{2}$$
61 $$1 + 1083608 T + p^{7} T^{2}$$
67 $$1 - 3443486 T + p^{7} T^{2}$$
71 $$1 - 2084705 T + p^{7} T^{2}$$
73 $$1 - 5937890 T + p^{7} T^{2}$$
79 $$1 + 6609256 T + p^{7} T^{2}$$
83 $$1 + 142740 T + p^{7} T^{2}$$
89 $$1 + 6985286 T + p^{7} T^{2}$$
97 $$1 + 200762 T + p^{7} T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$