# Properties

 Degree 2 Conductor $2^{4} \cdot 3^{2} \cdot 5^{2}$ Sign $-0.382 - 0.923i$ Motivic weight 0 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (−0.707 − 0.707i)2-s + 1.00i·4-s + (0.707 − 0.707i)8-s − 1.00·16-s − 1.41·17-s + (−1 + i)19-s − 1.41·23-s + 2i·31-s + (0.707 + 0.707i)32-s + (1.00 + 1.00i)34-s + 1.41·38-s + (1.00 + 1.00i)46-s − 1.41i·47-s − 49-s + (−1 − i)61-s + (1.41 − 1.41i)62-s + ⋯
 L(s)  = 1 + (−0.707 − 0.707i)2-s + 1.00i·4-s + (0.707 − 0.707i)8-s − 1.00·16-s − 1.41·17-s + (−1 + i)19-s − 1.41·23-s + 2i·31-s + (0.707 + 0.707i)32-s + (1.00 + 1.00i)34-s + 1.41·38-s + (1.00 + 1.00i)46-s − 1.41i·47-s − 49-s + (−1 − i)61-s + (1.41 − 1.41i)62-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$3600$$    =    $$2^{4} \cdot 3^{2} \cdot 5^{2}$$ $$\varepsilon$$ = $-0.382 - 0.923i$ motivic weight = $$0$$ character : $\chi_{3600} (451, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 3600,\ (\ :0),\ -0.382 - 0.923i)$$ $$L(\frac{1}{2})$$ $$\approx$$ $$0.2434917459$$ $$L(\frac12)$$ $$\approx$$ $$0.2434917459$$ $$L(1)$$ 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 + (0.707 + 0.707i)T$$
3 $$1$$
5 $$1$$
good7 $$1 + T^{2}$$
11 $$1 + iT^{2}$$
13 $$1 + iT^{2}$$
17 $$1 + 1.41T + T^{2}$$
19 $$1 + (1 - i)T - iT^{2}$$
23 $$1 + 1.41T + T^{2}$$
29 $$1 + iT^{2}$$
31 $$1 - 2iT - T^{2}$$
37 $$1 - iT^{2}$$
41 $$1 - T^{2}$$
43 $$1 + iT^{2}$$
47 $$1 + 1.41iT - T^{2}$$
53 $$1 - iT^{2}$$
59 $$1 + iT^{2}$$
61 $$1 + (1 + i)T + iT^{2}$$
67 $$1 - iT^{2}$$
71 $$1 + T^{2}$$
73 $$1 - T^{2}$$
79 $$1 - T^{2}$$
83 $$1 + (1.41 - 1.41i)T - iT^{2}$$
89 $$1 - T^{2}$$
97 $$1 + 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

−8.807908588071774236629396179495, −8.476538488618976638541710143482, −7.71909858512125339816937232839, −6.78853433220247096546344185676, −6.24597591718127930628764576941, −4.98217247665878702418684643061, −4.14603874487716110791227026129, −3.44094805437287590450927038824, −2.30813523539259161066549742251, −1.61437694517079147150296135734, 0.16671518267732937658998879965, 1.81600961662093426441045057405, 2.60723553073464479934983299064, 4.24353879066118819682560675171, 4.61350983078824726940189166123, 5.87879829589252799161627287048, 6.26604891605964461487472289660, 7.07475716280327681707089289740, 7.81080650170841406206213795148, 8.479955180473945963497386086152