# Properties

 Degree 2 Conductor $2^{4} \cdot 3 \cdot 11 \cdot 13^{2}$ Sign $-1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 1

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## Dirichlet series

 L(s)  = 1 − 3-s − 2·5-s − 2·7-s + 9-s + 11-s + 2·15-s + 4·17-s − 6·19-s + 2·21-s − 25-s − 27-s − 8·29-s − 8·31-s − 33-s + 4·35-s − 10·37-s − 8·41-s + 2·43-s − 2·45-s − 8·47-s − 3·49-s − 4·51-s − 2·53-s − 2·55-s + 6·57-s + 12·59-s + 10·61-s + ⋯
 L(s)  = 1 − 0.577·3-s − 0.894·5-s − 0.755·7-s + 1/3·9-s + 0.301·11-s + 0.516·15-s + 0.970·17-s − 1.37·19-s + 0.436·21-s − 1/5·25-s − 0.192·27-s − 1.48·29-s − 1.43·31-s − 0.174·33-s + 0.676·35-s − 1.64·37-s − 1.24·41-s + 0.304·43-s − 0.298·45-s − 1.16·47-s − 3/7·49-s − 0.560·51-s − 0.274·53-s − 0.269·55-s + 0.794·57-s + 1.56·59-s + 1.28·61-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 89232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 89232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$89232$$    =    $$2^{4} \cdot 3 \cdot 11 \cdot 13^{2}$$ $$\varepsilon$$ = $-1$ motivic weight = $$1$$ character : $\chi_{89232} (1, \cdot )$ Sato-Tate : $\mathrm{SU}(2)$ primitive : yes self-dual : yes analytic rank = 1 Selberg data = $(2,\ 89232,\ (\ :1/2),\ -1)$ $L(1)$ $=$ $0$ $L(\frac12)$ $=$ $0$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;3,\;11,\;13\}$,$F_p(T) = 1 - a_p T + p T^2 .$If $p \in \{2,\;3,\;11,\;13\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1$$
3 $$1 + T$$
11 $$1 - T$$
13 $$1$$
good5 $$1 + 2 T + p T^{2}$$
7 $$1 + 2 T + p T^{2}$$
17 $$1 - 4 T + p T^{2}$$
19 $$1 + 6 T + p T^{2}$$
23 $$1 + p T^{2}$$
29 $$1 + 8 T + p T^{2}$$
31 $$1 + 8 T + p T^{2}$$
37 $$1 + 10 T + p T^{2}$$
41 $$1 + 8 T + p T^{2}$$
43 $$1 - 2 T + p T^{2}$$
47 $$1 + 8 T + p T^{2}$$
53 $$1 + 2 T + p T^{2}$$
59 $$1 - 12 T + p T^{2}$$
61 $$1 - 10 T + p T^{2}$$
67 $$1 - 12 T + p T^{2}$$
71 $$1 - 8 T + p T^{2}$$
73 $$1 + 6 T + p T^{2}$$
79 $$1 - 2 T + p T^{2}$$
83 $$1 - 16 T + p T^{2}$$
89 $$1 - 14 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

−14.25496466858539, −13.43578243790896, −13.01170753875220, −12.57815227298574, −12.22844281975795, −11.56711952164505, −11.30382652191058, −10.70038381909452, −10.20694311894262, −9.684294167689569, −9.209746791396241, −8.511284175067852, −8.103102288694718, −7.479797735605734, −6.902407113849555, −6.612537544988117, −5.916232717756071, −5.314927475946371, −4.937035031155925, −3.926771317051120, −3.694685653246139, −3.331625935898923, −2.146196650675746, −1.706679487599166, −0.5805066102372999, 0, 0.5805066102372999, 1.706679487599166, 2.146196650675746, 3.331625935898923, 3.694685653246139, 3.926771317051120, 4.937035031155925, 5.314927475946371, 5.916232717756071, 6.612537544988117, 6.902407113849555, 7.479797735605734, 8.103102288694718, 8.511284175067852, 9.209746791396241, 9.684294167689569, 10.20694311894262, 10.70038381909452, 11.30382652191058, 11.56711952164505, 12.22844281975795, 12.57815227298574, 13.01170753875220, 13.43578243790896, 14.25496466858539