# Properties

 Degree 4 Conductor $2^{2} \cdot 3^{2} \cdot 7^{2} \cdot 11^{2}$ Sign $-1$ Motivic weight 1 Primitive no Self-dual yes Analytic rank 1

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 2·2-s − 2·3-s + 3·4-s − 4·6-s + 4·8-s + 9-s − 6·12-s + 5·16-s − 12·17-s + 2·18-s − 8·24-s − 10·25-s + 4·27-s + 12·29-s − 8·31-s + 6·32-s − 24·34-s + 3·36-s + 4·37-s − 12·41-s − 10·48-s + 49-s − 20·50-s + 24·51-s + 8·54-s + 24·58-s − 16·62-s + ⋯
 L(s)  = 1 + 1.41·2-s − 1.15·3-s + 3/2·4-s − 1.63·6-s + 1.41·8-s + 1/3·9-s − 1.73·12-s + 5/4·16-s − 2.91·17-s + 0.471·18-s − 1.63·24-s − 2·25-s + 0.769·27-s + 2.22·29-s − 1.43·31-s + 1.06·32-s − 4.11·34-s + 1/2·36-s + 0.657·37-s − 1.87·41-s − 1.44·48-s + 1/7·49-s − 2.82·50-s + 3.36·51-s + 1.08·54-s + 3.15·58-s − 2.03·62-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 213444 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 213444 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$4$$ $$N$$ = $$213444$$    =    $$2^{2} \cdot 3^{2} \cdot 7^{2} \cdot 11^{2}$$ $$\varepsilon$$ = $-1$ motivic weight = $$1$$ character : $\chi_{213444} (1, \cdot )$ Sato-Tate : $\mathrm{SU}(2)$ primitive : no self-dual : yes analytic rank = 1 Selberg data = $(4,\ 213444,\ (\ :1/2, 1/2),\ -1)$ $L(1)$ $=$ $0$ $L(\frac12)$ $=$ $0$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;3,\;7,\;11\}$,$F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4$with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;3,\;7,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_1$ $$( 1 - T )^{2}$$
3$C_2$ $$1 + 2 T + p T^{2}$$
7$C_1$$\times$$C_1$ $$( 1 - T )( 1 + T )$$
11$C_2$ $$1 + p T^{2}$$
good5$C_2$ $$( 1 + p T^{2} )^{2}$$
13$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
17$C_2$ $$( 1 + 6 T + p T^{2} )^{2}$$
19$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
23$C_2$ $$( 1 + p T^{2} )^{2}$$
29$C_2$ $$( 1 - 6 T + p T^{2} )^{2}$$
31$C_2$ $$( 1 + 4 T + p T^{2} )^{2}$$
37$C_2$ $$( 1 - 2 T + p T^{2} )^{2}$$
41$C_2$ $$( 1 + 6 T + p T^{2} )^{2}$$
43$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
47$C_2$ $$( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )$$
53$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
59$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
61$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
67$C_2$ $$( 1 + 4 T + p T^{2} )^{2}$$
71$C_2$ $$( 1 + p T^{2} )^{2}$$
73$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
79$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
83$C_2$ $$( 1 - 6 T + p T^{2} )^{2}$$
89$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
97$C_2$ $$( 1 + 10 T + p T^{2} )^{2}$$
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\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}

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

−8.888226006934558931566258496787, −8.145906800879012290031802042193, −7.80365099617056766338064266882, −6.82397964191468499058519060303, −6.71310127772875365703449938682, −6.42455331216463290634831658204, −5.76321350617402115136418955935, −5.31921059379060528260176389250, −4.85990751466979898816213085515, −4.19102102405251164216473944262, −4.09052365620160024762211243831, −3.03039761753313177141617611476, −2.39101862347891857041372548490, −1.67120553980965395706477292535, 0, 1.67120553980965395706477292535, 2.39101862347891857041372548490, 3.03039761753313177141617611476, 4.09052365620160024762211243831, 4.19102102405251164216473944262, 4.85990751466979898816213085515, 5.31921059379060528260176389250, 5.76321350617402115136418955935, 6.42455331216463290634831658204, 6.71310127772875365703449938682, 6.82397964191468499058519060303, 7.80365099617056766338064266882, 8.145906800879012290031802042193, 8.888226006934558931566258496787