# Properties

 Label 2-3e2-1.1-c21-0-6 Degree $2$ Conductor $9$ Sign $-1$ Analytic cond. $25.1529$ Root an. cond. $5.01527$ Motivic weight $21$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $1$

# Origins

## Dirichlet series

 L(s)  = 1 − 2.09e6·4-s + 1.12e9·7-s − 3.70e11·13-s + 4.39e12·16-s − 3.55e13·19-s − 4.76e14·25-s − 2.35e15·28-s − 9.04e15·31-s − 5.77e16·37-s + 2.65e17·43-s + 7.04e17·49-s + 7.76e17·52-s − 1.08e19·61-s − 9.22e18·64-s + 6.94e18·67-s − 3.90e19·73-s + 7.45e19·76-s + 1.68e20·79-s − 4.15e20·91-s − 1.13e21·97-s + 1.00e21·100-s − 2.44e21·103-s + 4.46e21·109-s + 4.94e21·112-s + ⋯
 L(s)  = 1 − 4-s + 1.50·7-s − 0.744·13-s + 16-s − 1.32·19-s − 25-s − 1.50·28-s − 1.98·31-s − 1.97·37-s + 1.87·43-s + 1.26·49-s + 0.744·52-s − 1.94·61-s − 64-s + 0.465·67-s − 1.06·73-s + 1.32·76-s + 1.99·79-s − 1.11·91-s − 1.55·97-s + 100-s − 1.79·103-s + 1.80·109-s + 1.50·112-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(22-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$9$$    =    $$3^{2}$$ Sign: $-1$ Analytic conductor: $$25.1529$$ Root analytic conductor: $$5.01527$$ Motivic weight: $$21$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 9,\ (\ :21/2),\ -1)$$

## Particular Values

 $$L(11)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{23}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1$$
good2 $$1 + p^{21} T^{2}$$
5 $$1 + p^{21} T^{2}$$
7 $$1 - 1123983020 T + p^{21} T^{2}$$
11 $$1 + p^{21} T^{2}$$
13 $$1 + 370076825230 T + p^{21} T^{2}$$
17 $$1 + p^{21} T^{2}$$
19 $$1 + 35540635313176 T + p^{21} T^{2}$$
23 $$1 + p^{21} T^{2}$$
29 $$1 + p^{21} T^{2}$$
31 $$1 + 9040072142371732 T + p^{21} T^{2}$$
37 $$1 + 57776323439003290 T + p^{21} T^{2}$$
41 $$1 + p^{21} T^{2}$$
43 $$1 - 265258444413820520 T + p^{21} T^{2}$$
47 $$1 + p^{21} T^{2}$$
53 $$1 + p^{21} T^{2}$$
59 $$1 + p^{21} T^{2}$$
61 $$1 + 10860691764464843938 T + p^{21} T^{2}$$
67 $$1 - 6944332370266921520 T + p^{21} T^{2}$$
71 $$1 + p^{21} T^{2}$$
73 $$1 + 39098623343480501290 T + p^{21} T^{2}$$
79 $$1 -$$$$16\!\cdots\!96$$$$T + p^{21} T^{2}$$
83 $$1 + p^{21} T^{2}$$
89 $$1 + p^{21} T^{2}$$
97 $$1 +$$$$11\!\cdots\!30$$$$T + p^{21} T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$