# Properties

 Degree 2 Conductor $2 \cdot 13691$ Sign $-1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 3

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## Dirichlet series

 L(s)  = 1 − 2-s − 2·3-s + 4-s − 3·5-s + 2·6-s − 2·7-s − 8-s + 9-s + 3·10-s − 6·11-s − 2·12-s − 6·13-s + 2·14-s + 6·15-s + 16-s − 8·17-s − 18-s − 5·19-s − 3·20-s + 4·21-s + 6·22-s + 2·24-s + 4·25-s + 6·26-s + 4·27-s − 2·28-s + 2·29-s + ⋯
 L(s)  = 1 − 0.707·2-s − 1.15·3-s + 1/2·4-s − 1.34·5-s + 0.816·6-s − 0.755·7-s − 0.353·8-s + 1/3·9-s + 0.948·10-s − 1.80·11-s − 0.577·12-s − 1.66·13-s + 0.534·14-s + 1.54·15-s + 1/4·16-s − 1.94·17-s − 0.235·18-s − 1.14·19-s − 0.670·20-s + 0.872·21-s + 1.27·22-s + 0.408·24-s + 4/5·25-s + 1.17·26-s + 0.769·27-s − 0.377·28-s + 0.371·29-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 27382 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 27382 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$27382$$    =    $$2 \cdot 13691$$ $$\varepsilon$$ = $-1$ motivic weight = $$1$$ character : $\chi_{27382} (1, \cdot )$ Sato-Tate : $\mathrm{SU}(2)$ primitive : yes self-dual : yes analytic rank = 3 Selberg data = $(2,\ 27382,\ (\ :1/2),\ -1)$ $L(1)$ $=$ $0$ $L(\frac12)$ $=$ $0$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{2,\;13691\}$, $F_p(T) = 1 - a_p T + p T^2 .$If $p \in \{2,\;13691\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 $$1 + T$$
13691 $$1 + T$$
good3 $$1 + 2 T + p T^{2}$$
5 $$1 + 3 T + p T^{2}$$
7 $$1 + 2 T + p T^{2}$$
11 $$1 + 6 T + p T^{2}$$
13 $$1 + 6 T + p T^{2}$$
17 $$1 + 8 T + p T^{2}$$
19 $$1 + 5 T + p T^{2}$$
23 $$1 + p T^{2}$$
29 $$1 - 2 T + p T^{2}$$
31 $$1 - 7 T + p T^{2}$$
37 $$1 + 7 T + p T^{2}$$
41 $$1 + 10 T + p T^{2}$$
43 $$1 + 5 T + p T^{2}$$
47 $$1 + 7 T + p T^{2}$$
53 $$1 + 14 T + p T^{2}$$
59 $$1 + 9 T + p T^{2}$$
61 $$1 - 2 T + p T^{2}$$
67 $$1 + 12 T + p T^{2}$$
71 $$1 - 6 T + p T^{2}$$
73 $$1 - 2 T + p T^{2}$$
79 $$1 + 10 T + p T^{2}$$
83 $$1 + 14 T + p T^{2}$$
89 $$1 + 12 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

−15.87582347637714, −15.70780124686833, −15.35614079179596, −14.81892398534507, −13.82355012214809, −13.11162112284388, −12.58867974955580, −12.28488832062087, −11.62418411877250, −11.20634735858314, −10.68914615363681, −10.19453953236317, −9.802370203255513, −8.815475997565422, −8.275011310751548, −7.923815118233866, −7.047577191738038, −6.755521562785137, −6.240244888362818, −5.247559203346068, −4.745270607684209, −4.380379024877120, −3.076726845990189, −2.765774972069143, −1.815865570564512, 0, 0, 0, 1.815865570564512, 2.765774972069143, 3.076726845990189, 4.380379024877120, 4.745270607684209, 5.247559203346068, 6.240244888362818, 6.755521562785137, 7.047577191738038, 7.923815118233866, 8.275011310751548, 8.815475997565422, 9.802370203255513, 10.19453953236317, 10.68914615363681, 11.20634735858314, 11.62418411877250, 12.28488832062087, 12.58867974955580, 13.11162112284388, 13.82355012214809, 14.81892398534507, 15.35614079179596, 15.70780124686833, 15.87582347637714