# Properties

 Degree 2 Conductor $2^{2} \cdot 3 \cdot 5^{2} \cdot 7$ Sign $0.447 - 0.894i$ Motivic weight 1 Primitive yes Self-dual no Analytic rank 0

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## Dirichlet series

 L(s)  = 1 − i·3-s + i·7-s − 9-s − 6·11-s − 2i·13-s + 4·19-s + 21-s + 6i·23-s + i·27-s − 6·29-s + 8·31-s + 6i·33-s + 2i·37-s − 2·39-s + 12·41-s + ⋯
 L(s)  = 1 − 0.577i·3-s + 0.377i·7-s − 0.333·9-s − 1.80·11-s − 0.554i·13-s + 0.917·19-s + 0.218·21-s + 1.25i·23-s + 0.192i·27-s − 1.11·29-s + 1.43·31-s + 1.04i·33-s + 0.328i·37-s − 0.320·39-s + 1.87·41-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$2100$$    =    $$2^{2} \cdot 3 \cdot 5^{2} \cdot 7$$ $$\varepsilon$$ = $0.447 - 0.894i$ motivic weight = $$1$$ character : $\chi_{2100} (1849, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 2100,\ (\ :1/2),\ 0.447 - 0.894i)$$ $$L(1)$$ $$\approx$$ $$1.068786012$$ $$L(\frac12)$$ $$\approx$$ $$1.068786012$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;3,\;5,\;7\}$,$$F_p(T)$$ is a polynomial of degree 2. If $p \in \{2,\;3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1$$
3 $$1 + iT$$
5 $$1$$
7 $$1 - iT$$
good11 $$1 + 6T + 11T^{2}$$
13 $$1 + 2iT - 13T^{2}$$
17 $$1 - 17T^{2}$$
19 $$1 - 4T + 19T^{2}$$
23 $$1 - 6iT - 23T^{2}$$
29 $$1 + 6T + 29T^{2}$$
31 $$1 - 8T + 31T^{2}$$
37 $$1 - 2iT - 37T^{2}$$
41 $$1 - 12T + 41T^{2}$$
43 $$1 - 4iT - 43T^{2}$$
47 $$1 - 12iT - 47T^{2}$$
53 $$1 - 6iT - 53T^{2}$$
59 $$1 + 59T^{2}$$
61 $$1 + 10T + 61T^{2}$$
67 $$1 - 8iT - 67T^{2}$$
71 $$1 - 6T + 71T^{2}$$
73 $$1 - 10iT - 73T^{2}$$
79 $$1 - 4T + 79T^{2}$$
83 $$1 - 12iT - 83T^{2}$$
89 $$1 + 12T + 89T^{2}$$
97 $$1 + 10iT - 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

−9.334160588088703665723149310247, −8.096362834242939938755733891477, −7.84588758143579943271053413032, −7.09959069637711725499202161259, −5.79532631877383755956047939166, −5.56894161311508485763590873490, −4.50078541102630895590818772348, −3.07240320708764163054184539503, −2.56113527066698518566206620410, −1.16120108529434125444805543616, 0.40506453482796761740976323754, 2.20418094039544649709065316617, 3.07919939410614854098454602045, 4.13766361371917066456946450974, 4.94175526113973912105672302566, 5.61235911729251425840469039932, 6.62069975807823456338094688079, 7.58239115349956445927609861427, 8.112979180081501232426812543219, 9.056382081793747333227466982444