# Properties

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

# Related objects

## Dirichlet series

 L(s)  = 1 − 3·3-s − 3·5-s − 2·7-s + 6·9-s − 11-s + 9·15-s − 6·17-s + 4·19-s + 6·21-s + 23-s + 4·25-s − 9·27-s − 8·29-s − 7·31-s + 3·33-s + 6·35-s − 37-s + 4·41-s + 6·43-s − 18·45-s − 8·47-s − 3·49-s + 18·51-s + 2·53-s + 3·55-s − 12·57-s − 59-s + ⋯
 L(s)  = 1 − 1.73·3-s − 1.34·5-s − 0.755·7-s + 2·9-s − 0.301·11-s + 2.32·15-s − 1.45·17-s + 0.917·19-s + 1.30·21-s + 0.208·23-s + 4/5·25-s − 1.73·27-s − 1.48·29-s − 1.25·31-s + 0.522·33-s + 1.01·35-s − 0.164·37-s + 0.624·41-s + 0.914·43-s − 2.68·45-s − 1.16·47-s − 3/7·49-s + 2.52·51-s + 0.274·53-s + 0.404·55-s − 1.58·57-s − 0.130·59-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$88$$    =    $$2^{3} \cdot 11$$ $$\varepsilon$$ = $-1$ motivic weight = $$1$$ character : $\chi_{88} (1, \cdot )$ Sato-Tate : $\mathrm{SU}(2)$ primitive : yes self-dual : yes analytic rank = 1 Selberg data = $(2,\ 88,\ (\ :1/2),\ -1)$ $L(1)$ $=$ $0$ $L(\frac12)$ $=$ $0$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{2,\;11\}$, $F_p(T) = 1 - a_p T + p T^2 .$If $p \in \{2,\;11\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 $$1$$
11 $$1 + T$$
good3 $$1 + p T + p T^{2}$$
5 $$1 + 3 T + p T^{2}$$
7 $$1 + 2 T + p T^{2}$$
13 $$1 + p T^{2}$$
17 $$1 + 6 T + p T^{2}$$
19 $$1 - 4 T + p T^{2}$$
23 $$1 - T + p T^{2}$$
29 $$1 + 8 T + p T^{2}$$
31 $$1 + 7 T + p T^{2}$$
37 $$1 + T + p T^{2}$$
41 $$1 - 4 T + p T^{2}$$
43 $$1 - 6 T + p T^{2}$$
47 $$1 + 8 T + p T^{2}$$
53 $$1 - 2 T + p T^{2}$$
59 $$1 + T + p T^{2}$$
61 $$1 - 4 T + p T^{2}$$
67 $$1 + 5 T + p T^{2}$$
71 $$1 - 3 T + p T^{2}$$
73 $$1 - 16 T + p T^{2}$$
79 $$1 - 2 T + p T^{2}$$
83 $$1 + 2 T + p T^{2}$$
89 $$1 - 15 T + p T^{2}$$
97 $$1 + 7 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

−19.39814407971896, −18.44840376758474, −17.63042282918627, −16.41501109154892, −16.02085093624741, −15.13896432548456, −13.16442893918529, −12.38704213422327, −11.36466946676257, −10.89204680724752, −9.401670233685537, −7.573565195405382, −6.610175306374676, −5.267180620239121, −3.921312004127912, 0, 3.921312004127912, 5.267180620239121, 6.610175306374676, 7.573565195405382, 9.401670233685537, 10.89204680724752, 11.36466946676257, 12.38704213422327, 13.16442893918529, 15.13896432548456, 16.02085093624741, 16.41501109154892, 17.63042282918627, 18.44840376758474, 19.39814407971896