# Properties

 Degree 4 Conductor $2^{6} \cdot 3^{2} \cdot 5^{4}$ 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 + 2·4-s − 4·6-s + 3·9-s + 4·12-s − 2·13-s − 4·16-s − 6·18-s + 4·26-s + 4·27-s − 6·31-s + 8·32-s + 6·36-s − 4·37-s − 4·39-s − 16·41-s − 2·43-s − 8·48-s − 5·49-s − 4·52-s + 8·53-s − 8·54-s + 12·62-s − 8·64-s + 6·67-s − 16·71-s + ⋯
 L(s)  = 1 − 1.41·2-s + 1.15·3-s + 4-s − 1.63·6-s + 9-s + 1.15·12-s − 0.554·13-s − 16-s − 1.41·18-s + 0.784·26-s + 0.769·27-s − 1.07·31-s + 1.41·32-s + 36-s − 0.657·37-s − 0.640·39-s − 2.49·41-s − 0.304·43-s − 1.15·48-s − 5/7·49-s − 0.554·52-s + 1.09·53-s − 1.08·54-s + 1.52·62-s − 64-s + 0.733·67-s − 1.89·71-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 360000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 360000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$4$$ $$N$$ = $$360000$$    =    $$2^{6} \cdot 3^{2} \cdot 5^{4}$$ $$\varepsilon$$ = $-1$ motivic weight = $$1$$ character : $\chi_{360000} (1, \cdot )$ Sato-Tate : $\mathrm{SU}(2)$ primitive : no self-dual : yes analytic rank = $$1$$ Selberg data = $$(4,\ 360000,\ (\ :1/2, 1/2),\ -1)$$ $$L(1)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;3,\;5\}$,$F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4$with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;3,\;5\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_2$ $$1 + p T + p T^{2}$$
3$C_1$ $$( 1 - T )^{2}$$
5 $$1$$
good7$C_2$ $$( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} )$$
11$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
13$C_2$ $$( 1 + T + p T^{2} )^{2}$$
17$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
19$C_2$ $$( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} )$$
23$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
29$C_2$ $$( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )$$
31$C_2$ $$( 1 + 3 T + p T^{2} )^{2}$$
37$C_2$ $$( 1 + 2 T + p T^{2} )^{2}$$
41$C_2$ $$( 1 + 8 T + p T^{2} )^{2}$$
43$C_2$ $$( 1 + T + p T^{2} )^{2}$$
47$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
53$C_2$ $$( 1 - 4 T + p T^{2} )^{2}$$
59$C_2$ $$( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )$$
61$C_2$ $$( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} )$$
67$C_2$ $$( 1 - 3 T + p T^{2} )^{2}$$
71$C_2$ $$( 1 + 8 T + p T^{2} )^{2}$$
73$C_2$ $$( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} )$$
79$C_2$ $$( 1 + p T^{2} )^{2}$$
83$C_2$ $$( 1 + 6 T + p T^{2} )^{2}$$
89$C_2$ $$( 1 + p T^{2} )^{2}$$
97$C_2$ $$( 1 - 17 T + p T^{2} )( 1 + 17 T + p T^{2} )$$
show less
\begin{aligned}L(s) = \prod_{\mathfrak{p}\ \mathrm{bad}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s})^{-1} \prod_{\mathfrak{p}\ \mathrm{good}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s} + (N\mathfrak{p})^{-2s})^{-1}\end{aligned}

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

−8.655074250265874594654667643843, −8.114757177440921419069207299350, −7.74916178315119539489350856070, −7.30514032256911612413843551960, −6.77287251257420576381216593643, −6.65396759238287778208236419506, −5.58318025222206456893422400511, −5.14493093108369759562477643348, −4.48018036005395779311676580836, −3.89570536316911001261511296620, −3.25692776068793976659434863504, −2.64453939621087097050043751453, −1.90319955654708104595670502735, −1.43286730840512162749271730076, 0, 1.43286730840512162749271730076, 1.90319955654708104595670502735, 2.64453939621087097050043751453, 3.25692776068793976659434863504, 3.89570536316911001261511296620, 4.48018036005395779311676580836, 5.14493093108369759562477643348, 5.58318025222206456893422400511, 6.65396759238287778208236419506, 6.77287251257420576381216593643, 7.30514032256911612413843551960, 7.74916178315119539489350856070, 8.114757177440921419069207299350, 8.655074250265874594654667643843