# Properties

 Degree 2 Conductor $37 \cdot 59$ Sign $-1$ Motivic weight 0 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + 2-s + 1.41i·5-s − 7-s − 8-s − 9-s + 1.41i·10-s − 1.41i·11-s − 13-s − 14-s − 16-s + 1.41i·17-s − 18-s − 1.41i·19-s − 1.41i·22-s − 1.00·25-s − 26-s + ⋯
 L(s)  = 1 + 2-s + 1.41i·5-s − 7-s − 8-s − 9-s + 1.41i·10-s − 1.41i·11-s − 13-s − 14-s − 16-s + 1.41i·17-s − 18-s − 1.41i·19-s − 1.41i·22-s − 1.00·25-s − 26-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 2183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 2183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$2183$$    =    $$37 \cdot 59$$ $$\varepsilon$$ = $-1$ motivic weight = $$0$$ character : $\chi_{2183} (2182, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 2183,\ (\ :0),\ -1)$ $L(\frac{1}{2})$ $\approx$ $0.3284938577$ $L(\frac12)$ $\approx$ $0.3284938577$ $L(1)$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{37,\;59\}$, $$F_p$$ is a polynomial of degree 2. If $p \in \{37,\;59\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad37 $$1 - T$$
59 $$1 + T$$
good2 $$1 - T + T^{2}$$
3 $$1 + T^{2}$$
5 $$1 - 1.41iT - T^{2}$$
7 $$1 + T + T^{2}$$
11 $$1 + 1.41iT - T^{2}$$
13 $$1 + T + T^{2}$$
17 $$1 - 1.41iT - T^{2}$$
19 $$1 + 1.41iT - T^{2}$$
23 $$1 + T^{2}$$
29 $$1 - 1.41iT - T^{2}$$
31 $$1 + T + T^{2}$$
41 $$1 + T + T^{2}$$
43 $$1 + T^{2}$$
47 $$1 - 1.41iT - T^{2}$$
53 $$1 + T + T^{2}$$
61 $$1 - T + T^{2}$$
67 $$1 + 1.41iT - T^{2}$$
71 $$1 + T + T^{2}$$
73 $$1 - 1.41iT - T^{2}$$
79 $$1 - T^{2}$$
83 $$1 - T^{2}$$
89 $$1 - T + T^{2}$$
97 $$1 - T + 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

−9.532465787475984365750170095985, −8.973596509034156032405431219345, −8.067692065930665380064693294736, −6.92197854746022076756463447281, −6.32617190487885466280737279220, −5.82966402251187479712063556801, −4.91271949515141280751528950564, −3.61310997320949362430893589971, −3.14290753438165019520082492940, −2.61822302852796594794184042386, 0.14930695123441591838264459465, 2.15574307593812869283521030604, 3.16925160918246236881168404631, 4.17968671018951219770062979042, 4.87764311277989735638761271227, 5.43238234926388620078296698820, 6.19714624328991293007348906898, 7.23854201772672234103692300408, 8.122234452757815685402598422198, 9.037722429598036170613219066183