# Properties

 Label 2-6-1.1-c15-0-2 Degree $2$ Conductor $6$ Sign $-1$ Analytic cond. $8.56161$ Root an. cond. $2.92602$ Motivic weight $15$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $1$

# Origins

## Dirichlet series

 L(s)  = 1 + 128·2-s − 2.18e3·3-s + 1.63e4·4-s − 1.14e5·5-s − 2.79e5·6-s − 3.03e6·7-s + 2.09e6·8-s + 4.78e6·9-s − 1.46e7·10-s − 1.03e8·11-s − 3.58e7·12-s − 1.04e8·13-s − 3.88e8·14-s + 2.51e8·15-s + 2.68e8·16-s + 9.97e8·17-s + 6.12e8·18-s + 4.93e9·19-s − 1.88e9·20-s + 6.63e9·21-s − 1.32e10·22-s + 8.32e9·23-s − 4.58e9·24-s − 1.73e10·25-s − 1.33e10·26-s − 1.04e10·27-s − 4.97e10·28-s + ⋯
 L(s)  = 1 + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.657·5-s − 0.408·6-s − 1.39·7-s + 0.353·8-s + 1/3·9-s − 0.464·10-s − 1.60·11-s − 0.288·12-s − 0.461·13-s − 0.984·14-s + 0.379·15-s + 1/4·16-s + 0.589·17-s + 0.235·18-s + 1.26·19-s − 0.328·20-s + 0.804·21-s − 1.13·22-s + 0.509·23-s − 0.204·24-s − 0.568·25-s − 0.326·26-s − 0.192·27-s − 0.696·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(16-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$6$$    =    $$2 \cdot 3$$ Sign: $-1$ Analytic conductor: $$8.56161$$ Root analytic conductor: $$2.92602$$ Motivic weight: $$15$$ Rational: yes Arithmetic: yes Character: $\chi_{6} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 6,\ (\ :15/2),\ -1)$$

## Particular Values

 $$L(8)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{17}{2})$$ 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^{7} T$$
3 $$1 + p^{7} T$$
good5 $$1 + 22962 p T + p^{15} T^{2}$$
7 $$1 + 433504 p T + p^{15} T^{2}$$
11 $$1 + 9404700 p T + p^{15} T^{2}$$
13 $$1 + 104365834 T + p^{15} T^{2}$$
17 $$1 - 997689762 T + p^{15} T^{2}$$
19 $$1 - 4934015444 T + p^{15} T^{2}$$
23 $$1 - 8324920200 T + p^{15} T^{2}$$
29 $$1 - 104128242846 T + p^{15} T^{2}$$
31 $$1 + 296696681512 T + p^{15} T^{2}$$
37 $$1 + 178337455666 T + p^{15} T^{2}$$
41 $$1 + 1790882416086 T + p^{15} T^{2}$$
43 $$1 + 2863459422772 T + p^{15} T^{2}$$
47 $$1 - 4332907521600 T + p^{15} T^{2}$$
53 $$1 - 9732317104422 T + p^{15} T^{2}$$
59 $$1 + 13514837176500 T + p^{15} T^{2}$$
61 $$1 - 5352663511190 T + p^{15} T^{2}$$
67 $$1 + 53233909720108 T + p^{15} T^{2}$$
71 $$1 + 20229661643400 T + p^{15} T^{2}$$
73 $$1 - 26264166466106 T + p^{15} T^{2}$$
79 $$1 + 339031361615128 T + p^{15} T^{2}$$
83 $$1 - 131684771045076 T + p^{15} T^{2}$$
89 $$1 + 39352148322678 T + p^{15} T^{2}$$
97 $$1 - 1128750908801474 T + p^{15} T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$