# Properties

 Degree 2 Conductor $2^{4} \cdot 3^{2} \cdot 5^{2}$ Sign $-0.391 - 0.920i$ Motivic weight 0 Primitive yes Self-dual no Analytic rank 0

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## Dirichlet series

 L(s)  = 1 + (−1 + i)13-s + (−1.41 + 1.41i)17-s − 1.41·29-s + (1 + i)37-s + 1.41i·41-s + i·49-s + (−1.41 − 1.41i)53-s + (1 − i)73-s + 1.41·89-s + (1 + i)97-s + 1.41i·101-s + ⋯
 L(s)  = 1 + (−1 + i)13-s + (−1.41 + 1.41i)17-s − 1.41·29-s + (1 + i)37-s + 1.41i·41-s + i·49-s + (−1.41 − 1.41i)53-s + (1 − i)73-s + 1.41·89-s + (1 + i)97-s + 1.41i·101-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.391 - 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.391 - 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$3600$$    =    $$2^{4} \cdot 3^{2} \cdot 5^{2}$$ $$\varepsilon$$ = $-0.391 - 0.920i$ motivic weight = $$0$$ character : $\chi_{3600} (1007, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 3600,\ (\ :0),\ -0.391 - 0.920i)$$ $$L(\frac{1}{2})$$ $$\approx$$ $$0.7407791707$$ $$L(\frac12)$$ $$\approx$$ $$0.7407791707$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;3,\;5\}$,$$F_p(T)$$ 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$$
3 $$1$$
5 $$1$$
good7 $$1 - iT^{2}$$
11 $$1 + T^{2}$$
13 $$1 + (1 - i)T - iT^{2}$$
17 $$1 + (1.41 - 1.41i)T - iT^{2}$$
19 $$1 + T^{2}$$
23 $$1 - iT^{2}$$
29 $$1 + 1.41T + T^{2}$$
31 $$1 - T^{2}$$
37 $$1 + (-1 - i)T + iT^{2}$$
41 $$1 - 1.41iT - T^{2}$$
43 $$1 + iT^{2}$$
47 $$1 + iT^{2}$$
53 $$1 + (1.41 + 1.41i)T + iT^{2}$$
59 $$1 - T^{2}$$
61 $$1 + T^{2}$$
67 $$1 - iT^{2}$$
71 $$1 + T^{2}$$
73 $$1 + (-1 + i)T - iT^{2}$$
79 $$1 + T^{2}$$
83 $$1 - iT^{2}$$
89 $$1 - 1.41T + T^{2}$$
97 $$1 + (-1 - i)T + iT^{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

−9.103922071014337572811379018402, −8.142141148309820052391565959894, −7.58603639358810626738074282252, −6.50876609373685622225389472634, −6.29596244471888632877575886245, −5.01615814203792437299528726007, −4.43822268827050100980762420462, −3.60535296727699148717555715829, −2.41360974141554376186838932514, −1.65719099457495338257020987190, 0.39850318103113549641196065899, 2.10734300108169582526941406054, 2.79469070822809598425280377879, 3.86226711840103339051260172007, 4.78952659669949514086963591717, 5.39124956076113367634690634621, 6.25443319809882618780658313562, 7.32181432205187540716255600752, 7.46839342528560548135297605897, 8.567248892908303044708224629718