# Properties

 Degree 4 Conductor $3^{2} \cdot 5^{3}$ Sign $1$ Motivic weight 1 Primitive no Self-dual yes Analytic rank 0

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 3·4-s + 5-s + 9-s − 8·11-s + 5·16-s + 8·19-s − 3·20-s + 25-s − 4·29-s − 3·36-s + 20·41-s + 24·44-s + 45-s − 14·49-s − 8·55-s − 8·59-s − 4·61-s − 3·64-s − 16·71-s − 24·76-s + 5·80-s + 81-s − 12·89-s + 8·95-s − 8·99-s − 3·100-s + 12·101-s + ⋯
 L(s)  = 1 − 3/2·4-s + 0.447·5-s + 1/3·9-s − 2.41·11-s + 5/4·16-s + 1.83·19-s − 0.670·20-s + 1/5·25-s − 0.742·29-s − 1/2·36-s + 3.12·41-s + 3.61·44-s + 0.149·45-s − 2·49-s − 1.07·55-s − 1.04·59-s − 0.512·61-s − 3/8·64-s − 1.89·71-s − 2.75·76-s + 0.559·80-s + 1/9·81-s − 1.27·89-s + 0.820·95-s − 0.804·99-s − 0.299·100-s + 1.19·101-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 1125 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 1125 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$4$$ $$N$$ = $$1125$$    =    $$3^{2} \cdot 5^{3}$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : $\chi_{1125} (1, \cdot )$ Sato-Tate : $\mathrm{SU}(2)$ primitive : no self-dual : yes analytic rank = 0 Selberg data = $(4,\ 1125,\ (\ :1/2, 1/2),\ 1)$ $L(1)$ $\approx$ $0.4386469691$ $L(\frac12)$ $\approx$ $0.4386469691$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{3,\;5\}$, $F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4$with $b_p = a_p^2 - a_{p^2}$. If $p \in \{3,\;5\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad3$C_1$$\times$$C_1$ $$( 1 - T )( 1 + T )$$
5$C_1$ $$1 - T$$
good2$C_2$ $$( 1 - T + p T^{2} )( 1 + T + p T^{2} )$$
7$C_2$ $$( 1 + p T^{2} )^{2}$$
11$C_2$ $$( 1 + 4 T + p T^{2} )^{2}$$
13$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
17$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
19$C_2$ $$( 1 - 4 T + p T^{2} )^{2}$$
23$C_2$ $$( 1 + p T^{2} )^{2}$$
29$C_2$ $$( 1 + 2 T + p T^{2} )^{2}$$
31$C_2$ $$( 1 + p T^{2} )^{2}$$
37$C_2$ $$( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )$$
41$C_2$ $$( 1 - 10 T + p T^{2} )^{2}$$
43$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
47$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
53$C_2$ $$( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )$$
59$C_2$ $$( 1 + 4 T + p T^{2} )^{2}$$
61$C_2$ $$( 1 + 2 T + p T^{2} )^{2}$$
67$C_2$ $$( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )$$
71$C_2$ $$( 1 + 8 T + p T^{2} )^{2}$$
73$C_2$ $$( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )$$
79$C_2$ $$( 1 + p T^{2} )^{2}$$
83$C_2$ $$( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )$$
89$C_2$ $$( 1 + 6 T + p T^{2} )^{2}$$
97$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
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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

−14.00692585049802430183598730465, −13.37224542302974911325065415564, −12.89741011334273074809178538613, −12.64617876135106218873747419153, −11.42796116649064334018778191223, −10.67892245123374144028858824682, −9.965958896202437987484594622667, −9.523451675812284265792380668213, −8.843956595114680827452383410006, −7.68003844385765985083943135134, −7.66488013441745380243230842523, −5.86497653842965752491328535183, −5.23920392624592057055772361749, −4.52160444884874669934496061646, −3.01549784140686353642765162463, 3.01549784140686353642765162463, 4.52160444884874669934496061646, 5.23920392624592057055772361749, 5.86497653842965752491328535183, 7.66488013441745380243230842523, 7.68003844385765985083943135134, 8.843956595114680827452383410006, 9.523451675812284265792380668213, 9.965958896202437987484594622667, 10.67892245123374144028858824682, 11.42796116649064334018778191223, 12.64617876135106218873747419153, 12.89741011334273074809178538613, 13.37224542302974911325065415564, 14.00692585049802430183598730465