# Properties

 Degree 2 Conductor $2^{5} \cdot 431$ 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 + 4·7-s + 6·9-s − 5·11-s + 4·13-s − 9·15-s − 6·17-s − 7·19-s − 12·21-s − 3·23-s + 4·25-s − 9·27-s + 3·29-s + 4·31-s + 15·33-s + 12·35-s + 8·37-s − 12·39-s − 6·41-s − 8·43-s + 18·45-s + 6·47-s + 9·49-s + 18·51-s + 53-s − 15·55-s + ⋯
 L(s)  = 1 − 1.73·3-s + 1.34·5-s + 1.51·7-s + 2·9-s − 1.50·11-s + 1.10·13-s − 2.32·15-s − 1.45·17-s − 1.60·19-s − 2.61·21-s − 0.625·23-s + 4/5·25-s − 1.73·27-s + 0.557·29-s + 0.718·31-s + 2.61·33-s + 2.02·35-s + 1.31·37-s − 1.92·39-s − 0.937·41-s − 1.21·43-s + 2.68·45-s + 0.875·47-s + 9/7·49-s + 2.52·51-s + 0.137·53-s − 2.02·55-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 13792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 13792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$13792$$    =    $$2^{5} \cdot 431$$ $$\varepsilon$$ = $-1$ motivic weight = $$1$$ character : $\chi_{13792} (1, \cdot )$ Sato-Tate : $\mathrm{SU}(2)$ primitive : yes self-dual : yes analytic rank = 1 Selberg data = $(2,\ 13792,\ (\ :1/2),\ -1)$ $L(1)$ $=$ $0$ $L(\frac12)$ $=$ $0$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{2,\;431\}$, $F_p(T) = 1 - a_p T + p T^2 .$If $p \in \{2,\;431\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1$$
431 $$1 + T$$
good3 $$1 + p T + p T^{2}$$
5 $$1 - 3 T + p T^{2}$$
7 $$1 - 4 T + p T^{2}$$
11 $$1 + 5 T + p T^{2}$$
13 $$1 - 4 T + p T^{2}$$
17 $$1 + 6 T + p T^{2}$$
19 $$1 + 7 T + p T^{2}$$
23 $$1 + 3 T + p T^{2}$$
29 $$1 - 3 T + p T^{2}$$
31 $$1 - 4 T + p T^{2}$$
37 $$1 - 8 T + p T^{2}$$
41 $$1 + 6 T + p T^{2}$$
43 $$1 + 8 T + p T^{2}$$
47 $$1 - 6 T + p T^{2}$$
53 $$1 - T + p T^{2}$$
59 $$1 + 9 T + p T^{2}$$
61 $$1 - 2 T + p T^{2}$$
67 $$1 - 10 T + p T^{2}$$
71 $$1 - 8 T + p T^{2}$$
73 $$1 + 6 T + p T^{2}$$
79 $$1 - 16 T + p T^{2}$$
83 $$1 + 16 T + p T^{2}$$
89 $$1 + 6 T + p T^{2}$$
97 $$1 + 5 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

−16.77938728403716, −15.87994617451942, −15.45410817472850, −14.99902208740289, −13.88013150201051, −13.69498352485712, −12.89850428938662, −12.70867820817584, −11.66552617875833, −11.23911597732312, −10.83475405631389, −10.32945017493787, −9.995176784409672, −8.849647103501781, −8.354879090174665, −7.723493883288323, −6.617356149871961, −6.351902031982537, −5.759898314910281, −5.113911739541168, −4.715302840201092, −4.105312731832032, −2.441740694491275, −1.959536553916767, −1.151807071572739, 0, 1.151807071572739, 1.959536553916767, 2.441740694491275, 4.105312731832032, 4.715302840201092, 5.113911739541168, 5.759898314910281, 6.351902031982537, 6.617356149871961, 7.723493883288323, 8.354879090174665, 8.849647103501781, 9.995176784409672, 10.32945017493787, 10.83475405631389, 11.23911597732312, 11.66552617875833, 12.70867820817584, 12.89850428938662, 13.69498352485712, 13.88013150201051, 14.99902208740289, 15.45410817472850, 15.87994617451942, 16.77938728403716