# Properties

 Degree 2 Conductor $2^{2} \cdot 17 \cdot 59$ Sign $0.842 + 0.539i$ Motivic weight 1 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 − 2.81i·3-s + 1.77i·5-s − 1.01i·7-s − 4.92·9-s + 1.98i·11-s + 5.96·13-s + 4.99·15-s + (−3.47 − 2.22i)17-s + 1.12·19-s − 2.85·21-s + 3.54i·23-s + 1.85·25-s + 5.42i·27-s − 3.47i·29-s + 9.40i·31-s + ⋯
 L(s)  = 1 − 1.62i·3-s + 0.793i·5-s − 0.382i·7-s − 1.64·9-s + 0.597i·11-s + 1.65·13-s + 1.28·15-s + (−0.842 − 0.539i)17-s + 0.258·19-s − 0.622·21-s + 0.738i·23-s + 0.370·25-s + 1.04i·27-s − 0.645i·29-s + 1.68i·31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.842 + 0.539i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.842 + 0.539i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$4012$$    =    $$2^{2} \cdot 17 \cdot 59$$ $$\varepsilon$$ = $0.842 + 0.539i$ motivic weight = $$1$$ character : $\chi_{4012} (237, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 4012,\ (\ :1/2),\ 0.842 + 0.539i)$ $L(1)$ $\approx$ $1.904636254$ $L(\frac12)$ $\approx$ $1.904636254$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;17,\;59\}$,$$F_p(T)$$ is a polynomial of degree 2. If $p \in \{2,\;17,\;59\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1$$
17 $$1 + (3.47 + 2.22i)T$$
59 $$1 - T$$
good3 $$1 + 2.81iT - 3T^{2}$$
5 $$1 - 1.77iT - 5T^{2}$$
7 $$1 + 1.01iT - 7T^{2}$$
11 $$1 - 1.98iT - 11T^{2}$$
13 $$1 - 5.96T + 13T^{2}$$
19 $$1 - 1.12T + 19T^{2}$$
23 $$1 - 3.54iT - 23T^{2}$$
29 $$1 + 3.47iT - 29T^{2}$$
31 $$1 - 9.40iT - 31T^{2}$$
37 $$1 - 4.75iT - 37T^{2}$$
41 $$1 - 10.0iT - 41T^{2}$$
43 $$1 + 2.36T + 43T^{2}$$
47 $$1 + 5.00T + 47T^{2}$$
53 $$1 - 11.5T + 53T^{2}$$
61 $$1 - 7.35iT - 61T^{2}$$
67 $$1 - 9.95T + 67T^{2}$$
71 $$1 - 1.41iT - 71T^{2}$$
73 $$1 + 13.9iT - 73T^{2}$$
79 $$1 - 1.05iT - 79T^{2}$$
83 $$1 + 6.70T + 83T^{2}$$
89 $$1 + 16.1T + 89T^{2}$$
97 $$1 - 10.4iT - 97T^{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

−8.297749481030067730540112015589, −7.40418254195725642405161314800, −6.91519807569765027875480083879, −6.48792378225729521022138446746, −5.74107900230731531389597129476, −4.63236576433520516613719856514, −3.48477225263720395113391351857, −2.74997374464709983126727504021, −1.72892790435761139961811952917, −0.962458549213228929918093416170, 0.69225803796983201781551175190, 2.19713671991718761117693011239, 3.41661495323062711101720613507, 3.98038708953789368166486913500, 4.60533097112735564269209176921, 5.58615534224275423020804327033, 5.85403472606905874270471450728, 6.94501700095657045902560825476, 8.428119944980499195923153553966, 8.556677264684644790082808601191