# Properties

 Degree 2 Conductor $3 \cdot 5 \cdot 7 \cdot 11$ Sign $-1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 1

# Origins

## Dirichlet series

 L(s)  = 1 − 2-s − 3-s − 4-s − 5-s + 6-s − 7-s + 3·8-s + 9-s + 10-s − 11-s + 12-s − 2·13-s + 14-s + 15-s − 16-s + 6·17-s − 18-s + 4·19-s + 20-s + 21-s + 22-s − 3·24-s + 25-s + 2·26-s − 27-s + 28-s − 6·29-s + ⋯
 L(s)  = 1 − 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.447·5-s + 0.408·6-s − 0.377·7-s + 1.06·8-s + 1/3·9-s + 0.316·10-s − 0.301·11-s + 0.288·12-s − 0.554·13-s + 0.267·14-s + 0.258·15-s − 1/4·16-s + 1.45·17-s − 0.235·18-s + 0.917·19-s + 0.223·20-s + 0.218·21-s + 0.213·22-s − 0.612·24-s + 1/5·25-s + 0.392·26-s − 0.192·27-s + 0.188·28-s − 1.11·29-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$1155$$    =    $$3 \cdot 5 \cdot 7 \cdot 11$$ $$\varepsilon$$ = $-1$ motivic weight = $$1$$ character : $\chi_{1155} (1, \cdot )$ primitive : yes self-dual : yes analytic rank = 1 Selberg data = $(2,\ 1155,\ (\ :1/2),\ -1)$ $L(1)$ $=$ $0$ $L(\frac12)$ $=$ $0$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{3,\;5,\;7,\;11\}$, $F_p(T) = 1 - a_p T + p T^2 .$If $p \in \{3,\;5,\;7,\;11\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 $$1 + T$$
5 $$1 + T$$
7 $$1 + T$$
11 $$1 + T$$
good2 $$1 + T + p T^{2}$$
13 $$1 + 2 T + p T^{2}$$
17 $$1 - 6 T + p T^{2}$$
19 $$1 - 4 T + p T^{2}$$
23 $$1 + p T^{2}$$
29 $$1 + 6 T + p T^{2}$$
31 $$1 + p T^{2}$$
37 $$1 - 6 T + p T^{2}$$
41 $$1 + 10 T + p T^{2}$$
43 $$1 - 4 T + p T^{2}$$
47 $$1 - 8 T + p T^{2}$$
53 $$1 + 6 T + p T^{2}$$
59 $$1 + 8 T + p T^{2}$$
61 $$1 - 2 T + p T^{2}$$
67 $$1 + 8 T + p T^{2}$$
71 $$1 + 8 T + p T^{2}$$
73 $$1 + 14 T + p T^{2}$$
79 $$1 - 4 T + p T^{2}$$
83 $$1 + 16 T + p T^{2}$$
89 $$1 + 10 T + p T^{2}$$
97 $$1 + 2 T + p 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

−9.571266661364712875338592102175, −8.623778385242342894395938674674, −7.55148413955994068836952145934, −7.37358077810077022104911371712, −5.89163547152298247464596699510, −5.15689425543262389753761609955, −4.19129942022627334449750341566, −3.10974320419115283648760707984, −1.30580464467610317835068015061, 0, 1.30580464467610317835068015061, 3.10974320419115283648760707984, 4.19129942022627334449750341566, 5.15689425543262389753761609955, 5.89163547152298247464596699510, 7.37358077810077022104911371712, 7.55148413955994068836952145934, 8.623778385242342894395938674674, 9.571266661364712875338592102175