# Properties

 Degree 2 Conductor $17 \cdot 43$ Sign $-0.711 - 0.703i$ Motivic weight 1 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (1.41 + 1.41i)4-s + (−2.77 + 1.14i)9-s + (−4.38 − 0.872i)11-s + (−4.77 + 4.77i)13-s + 4.00i·16-s + (−2.07 + 3.56i)17-s + (1.52 − 7.64i)23-s + (1.91 + 4.61i)25-s + (10.2 − 2.03i)31-s + (−5.54 − 2.29i)36-s + (−10.0 + 6.70i)41-s + (−6.05 + 2.50i)43-s + (−4.97 − 7.44i)44-s + (6.16 − 6.16i)47-s + (2.67 − 6.46i)49-s + ⋯
 L(s)  = 1 + (0.707 + 0.707i)4-s + (−0.923 + 0.382i)9-s + (−1.32 − 0.263i)11-s + (−1.32 + 1.32i)13-s + 1.00i·16-s + (−0.502 + 0.864i)17-s + (0.317 − 1.59i)23-s + (0.382 + 0.923i)25-s + (1.84 − 0.366i)31-s + (−0.923 − 0.382i)36-s + (−1.56 + 1.04i)41-s + (−0.923 + 0.382i)43-s + (−0.749 − 1.12i)44-s + (0.898 − 0.898i)47-s + (0.382 − 0.923i)49-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.711 - 0.703i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.711 - 0.703i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$731$$    =    $$17 \cdot 43$$ $$\varepsilon$$ = $-0.711 - 0.703i$ motivic weight = $$1$$ character : $\chi_{731} (300, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 731,\ (\ :1/2),\ -0.711 - 0.703i)$ $L(1)$ $\approx$ $0.367470 + 0.894426i$ $L(\frac12)$ $\approx$ $0.367470 + 0.894426i$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{17,\;43\}$,$$F_p(T)$$ is a polynomial of degree 2. If $p \in \{17,\;43\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad17 $$1 + (2.07 - 3.56i)T$$
43 $$1 + (6.05 - 2.50i)T$$
good2 $$1 + (-1.41 - 1.41i)T^{2}$$
3 $$1 + (2.77 - 1.14i)T^{2}$$
5 $$1 + (-1.91 - 4.61i)T^{2}$$
7 $$1 + (-2.67 + 6.46i)T^{2}$$
11 $$1 + (4.38 + 0.872i)T + (10.1 + 4.20i)T^{2}$$
13 $$1 + (4.77 - 4.77i)T - 13iT^{2}$$
19 $$1 + (-13.4 - 13.4i)T^{2}$$
23 $$1 + (-1.52 + 7.64i)T + (-21.2 - 8.80i)T^{2}$$
29 $$1 + (11.0 + 26.7i)T^{2}$$
31 $$1 + (-10.2 + 2.03i)T + (28.6 - 11.8i)T^{2}$$
37 $$1 + (34.1 - 14.1i)T^{2}$$
41 $$1 + (10.0 - 6.70i)T + (15.6 - 37.8i)T^{2}$$
47 $$1 + (-6.16 + 6.16i)T - 47iT^{2}$$
53 $$1 + (-10.1 - 4.22i)T + (37.4 + 37.4i)T^{2}$$
59 $$1 + (-3.46 - 8.36i)T + (-41.7 + 41.7i)T^{2}$$
61 $$1 + (23.3 - 56.3i)T^{2}$$
67 $$1 - 11.3iT - 67T^{2}$$
71 $$1 + (65.5 - 27.1i)T^{2}$$
73 $$1 + (-27.9 - 67.4i)T^{2}$$
79 $$1 + (-9.03 - 1.79i)T + (72.9 + 30.2i)T^{2}$$
83 $$1 + (2.50 - 6.05i)T + (-58.6 - 58.6i)T^{2}$$
89 $$1 - 89iT^{2}$$
97 $$1 + (-1.58 + 2.37i)T + (-37.1 - 89.6i)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

−10.73580007837636831079779815375, −10.07269735903560581797440595277, −8.608800791518867945384301759302, −8.303506466371751615017087613733, −7.17532130982921236183237540132, −6.51643652807600702783285577667, −5.30108663664572023393021741098, −4.32108625254821952779104924156, −2.83027011566362021345788607150, −2.27237243837536742143792983319, 0.44033068918331144713624285261, 2.43579154902726021476492979937, 3.01721708468101531013998040064, 5.09957847238062074751101362596, 5.32937969484098975246824400751, 6.55093474340671184324090532934, 7.44295563965590973947249908106, 8.198706890359962021276411764356, 9.427578834247746463050709065274, 10.21016670825225218775024689672