# Properties

 Degree 4 Conductor $5^{3} \cdot 11$ Sign $1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 0

# Origins

## Dirichlet series

 L(s)  = 1 − 4-s − 5-s − 2·9-s − 11-s − 3·16-s + 4·19-s + 20-s + 25-s + 4·31-s + 2·36-s + 44-s + 2·45-s + 14·49-s + 55-s − 12·59-s − 8·61-s + 7·64-s − 4·76-s + 4·79-s + 3·80-s − 5·81-s − 12·89-s − 4·95-s + 2·99-s − 100-s − 12·101-s + 4·109-s + ⋯
 L(s)  = 1 − 1/2·4-s − 0.447·5-s − 2/3·9-s − 0.301·11-s − 3/4·16-s + 0.917·19-s + 0.223·20-s + 1/5·25-s + 0.718·31-s + 1/3·36-s + 0.150·44-s + 0.298·45-s + 2·49-s + 0.134·55-s − 1.56·59-s − 1.02·61-s + 7/8·64-s − 0.458·76-s + 0.450·79-s + 0.335·80-s − 5/9·81-s − 1.27·89-s − 0.410·95-s + 0.201·99-s − 0.0999·100-s − 1.19·101-s + 0.383·109-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$4$$ $$N$$ = $$1375$$    =    $$5^{3} \cdot 11$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : $\chi_{1375} (1, \cdot )$ Sato-Tate : $\mathrm{SU}(2)$ primitive : yes self-dual : yes analytic rank = 0 Selberg data = $(4,\ 1375,\ (\ :1/2, 1/2),\ 1)$ $L(1)$ $\approx$ $0.4936014653$ $L(\frac12)$ $\approx$ $0.4936014653$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{5,\;11\}$,$F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4$with $b_p = a_p^2 - a_{p^2}$. If $p \in \{5,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad5$C_1$ $$1 + T$$
11$C_1$$\times$$C_2$ $$( 1 + T )( 1 + p T^{2} )$$
good2$C_2^2$ $$1 + T^{2} + p^{2} T^{4}$$
3$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
7$C_2$ $$( 1 - p T^{2} )^{2}$$
13$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
17$C_2^2$ $$1 + 10 T^{2} + p^{2} T^{4}$$
19$C_2$$\times$$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
23$C_2^2$ $$1 - 14 T^{2} + p^{2} T^{4}$$
29$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
31$C_2$$\times$$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
37$C_2^2$ $$1 - 2 T^{2} + p^{2} T^{4}$$
41$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
43$C_2^2$ $$1 + 58 T^{2} + p^{2} T^{4}$$
47$C_2^2$ $$1 - 14 T^{2} + p^{2} T^{4}$$
53$C_2^2$ $$1 - 50 T^{2} + p^{2} T^{4}$$
59$C_2$$\times$$C_2$ $$( 1 + p T^{2} )( 1 + 12 T + p T^{2} )$$
61$C_2$$\times$$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} )$$
67$C_2$ $$( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} )$$
71$C_2$ $$( 1 + p T^{2} )^{2}$$
73$C_2^2$ $$1 - 38 T^{2} + p^{2} T^{4}$$
79$C_2$$\times$$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
83$C_2^2$ $$1 + 58 T^{2} + p^{2} T^{4}$$
89$C_2$$\times$$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 18 T + p T^{2} )$$
97$C_2^2$ $$1 + 22 T^{2} + p^{2} T^{4}$$
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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

−13.75539545875999856307279939656, −13.40790056481005511778602564463, −12.42556471409184263431561392223, −11.99318278851470879678465377104, −11.27748233626068839689230673523, −10.71705090685439114964224517084, −9.850315198815836374223293791035, −9.142886062799315513511972371368, −8.558267557755620760586313944365, −7.78942950669997960453513423290, −7.04063543577616078388107278171, −5.99080016242421450165909429669, −5.09698455816962605625731143596, −4.17584295292025542328208055314, −2.89395902106255541986149724624, 2.89395902106255541986149724624, 4.17584295292025542328208055314, 5.09698455816962605625731143596, 5.99080016242421450165909429669, 7.04063543577616078388107278171, 7.78942950669997960453513423290, 8.558267557755620760586313944365, 9.142886062799315513511972371368, 9.850315198815836374223293791035, 10.71705090685439114964224517084, 11.27748233626068839689230673523, 11.99318278851470879678465377104, 12.42556471409184263431561392223, 13.40790056481005511778602564463, 13.75539545875999856307279939656