# Properties

 Degree 2 Conductor $2 \cdot 3$ Sign $1$ Motivic weight 11 Primitive yes Self-dual yes Analytic rank 0

# Origins

## Dirichlet series

 L(s)  = 1 − 32·2-s − 243·3-s + 1.02e3·4-s + 5.76e3·5-s + 7.77e3·6-s + 7.24e4·7-s − 3.27e4·8-s + 5.90e4·9-s − 1.84e5·10-s − 4.08e5·11-s − 2.48e5·12-s + 1.36e6·13-s − 2.31e6·14-s − 1.40e6·15-s + 1.04e6·16-s + 5.42e6·17-s − 1.88e6·18-s + 1.51e7·19-s + 5.90e6·20-s − 1.76e7·21-s + 1.30e7·22-s − 5.21e7·23-s + 7.96e6·24-s − 1.55e7·25-s − 4.37e7·26-s − 1.43e7·27-s + 7.42e7·28-s + ⋯
 L(s)  = 1 − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.825·5-s + 0.408·6-s + 1.62·7-s − 0.353·8-s + 1/3·9-s − 0.583·10-s − 0.765·11-s − 0.288·12-s + 1.02·13-s − 1.15·14-s − 0.476·15-s + 1/4·16-s + 0.926·17-s − 0.235·18-s + 1.40·19-s + 0.412·20-s − 0.940·21-s + 0.541·22-s − 1.69·23-s + 0.204·24-s − 0.319·25-s − 0.722·26-s − 0.192·27-s + 0.814·28-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$6$$    =    $$2 \cdot 3$$ $$\varepsilon$$ = $1$ motivic weight = $$11$$ character : $\chi_{6} (1, \cdot )$ primitive : yes self-dual : yes analytic rank = 0 Selberg data = $(2,\ 6,\ (\ :11/2),\ 1)$ $L(6)$ $\approx$ $1.20558$ $L(\frac12)$ $\approx$ $1.20558$ $L(\frac{13}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{2,\;3\}$, $$F_p$$ is a polynomial of degree 2. If $p \in \{2,\;3\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 $$1 + p^{5} T$$
3 $$1 + p^{5} T$$
good5 $$1 - 5766 T + p^{11} T^{2}$$
7 $$1 - 10352 p T + p^{11} T^{2}$$
11 $$1 + 408948 T + p^{11} T^{2}$$
13 $$1 - 1367558 T + p^{11} T^{2}$$
17 $$1 - 5422914 T + p^{11} T^{2}$$
19 $$1 - 15166100 T + p^{11} T^{2}$$
23 $$1 + 52194072 T + p^{11} T^{2}$$
29 $$1 - 118581150 T + p^{11} T^{2}$$
31 $$1 + 57652408 T + p^{11} T^{2}$$
37 $$1 + 375985186 T + p^{11} T^{2}$$
41 $$1 - 856316202 T + p^{11} T^{2}$$
43 $$1 + 1245189172 T + p^{11} T^{2}$$
47 $$1 + 1306762656 T + p^{11} T^{2}$$
53 $$1 - 409556358 T + p^{11} T^{2}$$
59 $$1 + 48862140 p T + p^{11} T^{2}$$
61 $$1 - 5731767302 T + p^{11} T^{2}$$
67 $$1 - 3893272244 T + p^{11} T^{2}$$
71 $$1 + 9075890088 T + p^{11} T^{2}$$
73 $$1 + 15571822822 T + p^{11} T^{2}$$
79 $$1 + 30196762600 T + p^{11} T^{2}$$
83 $$1 - 23135252628 T + p^{11} T^{2}$$
89 $$1 + 25614819990 T + p^{11} T^{2}$$
97 $$1 + 61937553406 T + p^{11} T^{2}$$
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\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}

## Imaginary part of the first few zeros on the critical line

−20.61408278124063655145673829357, −18.22185823557610957306388607067, −17.74481053097977942632133146030, −16.04683807740066808067323140928, −14.01498326645545396945292831922, −11.66318179051930405686989550799, −10.19821875859845610581844279656, −8.005005940346513327952156957974, −5.55489639023984419351049688700, −1.48555012435943654433178944737, 1.48555012435943654433178944737, 5.55489639023984419351049688700, 8.005005940346513327952156957974, 10.19821875859845610581844279656, 11.66318179051930405686989550799, 14.01498326645545396945292831922, 16.04683807740066808067323140928, 17.74481053097977942632133146030, 18.22185823557610957306388607067, 20.61408278124063655145673829357