# Properties

 Degree 2 Conductor $2 \cdot 3^{2} \cdot 5^{2}$ Sign $0.886 - 0.461i$ Motivic weight 1 Primitive yes Self-dual no Analytic rank 0

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## Dirichlet series

 L(s)  = 1 + (−0.965 + 0.258i)2-s + (1.22 − 1.22i)3-s + (0.866 − 0.499i)4-s + (−0.866 + 1.49i)6-s + (1.22 + 4.57i)7-s + (−0.707 + 0.707i)8-s − 2.99i·9-s + (3 + 1.73i)11-s + (0.448 − 1.67i)12-s + (−1.22 + 4.57i)13-s + (−2.36 − 4.09i)14-s + (0.500 − 0.866i)16-s + (0.776 + 2.89i)18-s + 3.19i·19-s + (7.09 + 4.09i)21-s + (−3.34 − 0.896i)22-s + ⋯
 L(s)  = 1 + (−0.683 + 0.183i)2-s + (0.707 − 0.707i)3-s + (0.433 − 0.249i)4-s + (−0.353 + 0.612i)6-s + (0.462 + 1.72i)7-s + (−0.249 + 0.249i)8-s − 0.999i·9-s + (0.904 + 0.522i)11-s + (0.129 − 0.482i)12-s + (−0.339 + 1.26i)13-s + (−0.632 − 1.09i)14-s + (0.125 − 0.216i)16-s + (0.183 + 0.683i)18-s + 0.733i·19-s + (1.54 + 0.894i)21-s + (−0.713 − 0.191i)22-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.886 - 0.461i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.886 - 0.461i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$450$$    =    $$2 \cdot 3^{2} \cdot 5^{2}$$ $$\varepsilon$$ = $0.886 - 0.461i$ motivic weight = $$1$$ character : $\chi_{450} (443, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 450,\ (\ :1/2),\ 0.886 - 0.461i)$$ $$L(1)$$ $$\approx$$ $$1.32661 + 0.324792i$$ $$L(\frac12)$$ $$\approx$$ $$1.32661 + 0.324792i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;3,\;5\}$,$$F_p(T)$$ is a polynomial of degree 2. If $p \in \{2,\;3,\;5\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1 + (0.965 - 0.258i)T$$
3 $$1 + (-1.22 + 1.22i)T$$
5 $$1$$
good7 $$1 + (-1.22 - 4.57i)T + (-6.06 + 3.5i)T^{2}$$
11 $$1 + (-3 - 1.73i)T + (5.5 + 9.52i)T^{2}$$
13 $$1 + (1.22 - 4.57i)T + (-11.2 - 6.5i)T^{2}$$
17 $$1 + 17iT^{2}$$
19 $$1 - 3.19iT - 19T^{2}$$
23 $$1 + (2.12 + 0.568i)T + (19.9 + 11.5i)T^{2}$$
29 $$1 + (-5.36 + 9.29i)T + (-14.5 - 25.1i)T^{2}$$
31 $$1 + (0.0980 + 0.169i)T + (-15.5 + 26.8i)T^{2}$$
37 $$1 + (-5.79 + 5.79i)T - 37iT^{2}$$
41 $$1 + (1.5 - 0.866i)T + (20.5 - 35.5i)T^{2}$$
43 $$1 + (-0.448 + 0.120i)T + (37.2 - 21.5i)T^{2}$$
47 $$1 + (-5.79 + 1.55i)T + (40.7 - 23.5i)T^{2}$$
53 $$1 + (5.79 - 5.79i)T - 53iT^{2}$$
59 $$1 + (2.76 + 4.79i)T + (-29.5 + 51.0i)T^{2}$$
61 $$1 + (-2 + 3.46i)T + (-30.5 - 52.8i)T^{2}$$
67 $$1 + (-5.34 - 1.43i)T + (58.0 + 33.5i)T^{2}$$
71 $$1 - 7.26iT - 71T^{2}$$
73 $$1 + (3.67 + 3.67i)T + 73iT^{2}$$
79 $$1 + (-8.66 - 5i)T + (39.5 + 68.4i)T^{2}$$
83 $$1 + (4.45 + 16.6i)T + (-71.8 + 41.5i)T^{2}$$
89 $$1 + 8.66T + 89T^{2}$$
97 $$1 + (-0.688 - 2.56i)T + (-84.0 + 48.5i)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

−11.46853783225992598242280996268, −9.747749674057188352070701436691, −9.244022547657620350280246816698, −8.488248452178607480773618381695, −7.72051686749188825945507746885, −6.58109383933437927308925966046, −5.89807137174929348587409681904, −4.27263304044983162287177812681, −2.49843011870983795322708962396, −1.75092288025863658103510584705, 1.11328187380056148250192053040, 2.98827104967234392344874016957, 3.93826200667636714244217401464, 5.00655968302684992990315883904, 6.70478880134960809713769029850, 7.65145542356140913712208226309, 8.319607046894964549788182581022, 9.286346332189886810226004788709, 10.24698770054042260667926710033, 10.66702125207365385270671051669