# Properties

 Label 4-768e2-1.1-c3e2-0-10 Degree $4$ Conductor $589824$ Sign $1$ Analytic cond. $2053.31$ Root an. cond. $6.73152$ Motivic weight $3$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 6·3-s + 27·9-s − 96·11-s + 108·17-s + 8·19-s − 238·25-s − 108·27-s + 576·33-s + 588·41-s − 376·43-s − 98·49-s − 648·51-s − 48·57-s − 504·59-s − 1.25e3·67-s + 2.01e3·73-s + 1.42e3·75-s + 405·81-s + 1.44e3·83-s + 2.96e3·89-s + 3.64e3·97-s − 2.59e3·99-s + 2.37e3·107-s − 780·113-s + 4.25e3·121-s − 3.52e3·123-s + 127-s + ⋯
 L(s)  = 1 − 1.15·3-s + 9-s − 2.63·11-s + 1.54·17-s + 0.0965·19-s − 1.90·25-s − 0.769·27-s + 3.03·33-s + 2.23·41-s − 1.33·43-s − 2/7·49-s − 1.77·51-s − 0.111·57-s − 1.11·59-s − 2.29·67-s + 3.22·73-s + 2.19·75-s + 5/9·81-s + 1.90·83-s + 3.53·89-s + 3.81·97-s − 2.63·99-s + 2.14·107-s − 0.649·113-s + 3.19·121-s − 2.58·123-s + 0.000698·127-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 589824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 589824 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$589824$$    =    $$2^{16} \cdot 3^{2}$$ Sign: $1$ Analytic conductor: $$2053.31$$ Root analytic conductor: $$6.73152$$ Motivic weight: $$3$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(4,\ 589824,\ (\ :3/2, 3/2),\ 1)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$0.9814087874$$ $$L(\frac12)$$ $$\approx$$ $$0.9814087874$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$\Gal(F_p)$$F_p(T)$
bad2 $$1$$
3$C_1$ $$( 1 + p T )^{2}$$
good5$C_2^2$ $$1 + 238 T^{2} + p^{6} T^{4}$$
7$C_2^2$ $$1 + 2 p^{2} T^{2} + p^{6} T^{4}$$
11$C_2$ $$( 1 + 48 T + p^{3} T^{2} )^{2}$$
13$C_2^2$ $$1 + 2666 T^{2} + p^{6} T^{4}$$
17$C_2$ $$( 1 - 54 T + p^{3} T^{2} )^{2}$$
19$C_2$ $$( 1 - 4 T + p^{3} T^{2} )^{2}$$
23$C_2^2$ $$1 - 5666 T^{2} + p^{6} T^{4}$$
29$C_2^2$ $$1 + 22270 T^{2} + p^{6} T^{4}$$
31$C_2^2$ $$1 + 56114 T^{2} + p^{6} T^{4}$$
37$C_2^2$ $$1 - 4726 T^{2} + p^{6} T^{4}$$
41$C_2$ $$( 1 - 294 T + p^{3} T^{2} )^{2}$$
43$C_2$ $$( 1 + 188 T + p^{3} T^{2} )^{2}$$
47$C_2^2$ $$1 - 48146 T^{2} + p^{6} T^{4}$$
53$C_2^2$ $$1 - 256946 T^{2} + p^{6} T^{4}$$
59$C_2$ $$( 1 + 252 T + p^{3} T^{2} )^{2}$$
61$C_2^2$ $$1 + 445850 T^{2} + p^{6} T^{4}$$
67$C_2$ $$( 1 + 628 T + p^{3} T^{2} )^{2}$$
71$C_2^2$ $$1 + 715774 T^{2} + p^{6} T^{4}$$
73$C_2$ $$( 1 - 1006 T + p^{3} T^{2} )^{2}$$
79$C_2^2$ $$1 - 811150 T^{2} + p^{6} T^{4}$$
83$C_2$ $$( 1 - 720 T + p^{3} T^{2} )^{2}$$
89$C_2$ $$( 1 - 1482 T + p^{3} T^{2} )^{2}$$
97$C_2$ $$( 1 - 1822 T + p^{3} T^{2} )^{2}$$
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$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$

## Imaginary part of the first few zeros on the critical line

−10.23769701594474768655671283090, −9.914727856306761399836281529361, −9.431466408089082362370388150270, −9.006663934073312732513676519477, −8.116529215936313770241247597135, −7.84640976633026993037644784671, −7.52829506010071625537216753328, −7.46358609558734544415614344403, −6.28723739653473530039479463943, −6.21292005604433094456829808534, −5.66868319501735855968916605829, −5.26706013733793090386031089710, −4.87714819642146302850777356992, −4.50453743016072788566029207783, −3.41023000613132208474580890528, −3.39948599498746099996945883740, −2.30133441456184581917067709897, −2.00271603941077803915569238748, −0.878320722141562171371848981036, −0.37220561486953340513308425231, 0.37220561486953340513308425231, 0.878320722141562171371848981036, 2.00271603941077803915569238748, 2.30133441456184581917067709897, 3.39948599498746099996945883740, 3.41023000613132208474580890528, 4.50453743016072788566029207783, 4.87714819642146302850777356992, 5.26706013733793090386031089710, 5.66868319501735855968916605829, 6.21292005604433094456829808534, 6.28723739653473530039479463943, 7.46358609558734544415614344403, 7.52829506010071625537216753328, 7.84640976633026993037644784671, 8.116529215936313770241247597135, 9.006663934073312732513676519477, 9.431466408089082362370388150270, 9.914727856306761399836281529361, 10.23769701594474768655671283090