# Properties

 Degree 2 Conductor $2 \cdot 7 \cdot 13$ Sign $1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 0

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## Dirichlet series

 L(s)  = 1 + 2-s + 3·3-s + 4-s − 4·5-s + 3·6-s − 7-s + 8-s + 6·9-s − 4·10-s + 11-s + 3·12-s − 13-s − 14-s − 12·15-s + 16-s + 6·18-s − 6·19-s − 4·20-s − 3·21-s + 22-s − 7·23-s + 3·24-s + 11·25-s − 26-s + 9·27-s − 28-s − 4·29-s + ⋯
 L(s)  = 1 + 0.707·2-s + 1.73·3-s + 1/2·4-s − 1.78·5-s + 1.22·6-s − 0.377·7-s + 0.353·8-s + 2·9-s − 1.26·10-s + 0.301·11-s + 0.866·12-s − 0.277·13-s − 0.267·14-s − 3.09·15-s + 1/4·16-s + 1.41·18-s − 1.37·19-s − 0.894·20-s − 0.654·21-s + 0.213·22-s − 1.45·23-s + 0.612·24-s + 11/5·25-s − 0.196·26-s + 1.73·27-s − 0.188·28-s − 0.742·29-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 182 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 182 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$182$$    =    $$2 \cdot 7 \cdot 13$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : $\chi_{182} (1, \cdot )$ Sato-Tate : $\mathrm{SU}(2)$ primitive : yes self-dual : yes analytic rank = 0 Selberg data = $(2,\ 182,\ (\ :1/2),\ 1)$ $L(1)$ $\approx$ $2.135366401$ $L(\frac12)$ $\approx$ $2.135366401$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

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

−19.88247752011130, −19.25377868977211, −18.62711522199385, −16.75088312096775, −15.74934755474166, −15.24756131521109, −14.61287591414215, −13.72057454122583, −12.68476149313497, −12.02186514366721, −10.77192769214820, −9.447300131639462, −8.238523811690696, −7.780552892986458, −6.627661893273811, −4.291243091362351, −3.806297220695687, −2.563810376497470, 2.563810376497470, 3.806297220695687, 4.291243091362351, 6.627661893273811, 7.780552892986458, 8.238523811690696, 9.447300131639462, 10.77192769214820, 12.02186514366721, 12.68476149313497, 13.72057454122583, 14.61287591414215, 15.24756131521109, 15.74934755474166, 16.75088312096775, 18.62711522199385, 19.25377868977211, 19.88247752011130