# Properties

 Degree 4 Conductor $2^{7} \cdot 7^{2}$ Sign $1$ Motivic weight 1 Primitive no Self-dual yes Analytic rank 0

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 4·5-s − 6·9-s + 4·13-s − 12·17-s + 2·25-s + 12·29-s − 4·37-s + 4·41-s − 24·45-s + 49-s + 12·53-s − 12·61-s + 16·65-s + 20·73-s + 27·81-s − 48·85-s − 12·89-s − 12·97-s + 4·101-s − 20·109-s + 4·113-s − 24·117-s − 6·121-s − 28·125-s + 127-s + 131-s + 137-s + ⋯
 L(s)  = 1 + 1.78·5-s − 2·9-s + 1.10·13-s − 2.91·17-s + 2/5·25-s + 2.22·29-s − 0.657·37-s + 0.624·41-s − 3.57·45-s + 1/7·49-s + 1.64·53-s − 1.53·61-s + 1.98·65-s + 2.34·73-s + 3·81-s − 5.20·85-s − 1.27·89-s − 1.21·97-s + 0.398·101-s − 1.91·109-s + 0.376·113-s − 2.21·117-s − 0.545·121-s − 2.50·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 6272 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 6272 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$4$$ $$N$$ = $$6272$$    =    $$2^{7} \cdot 7^{2}$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : $\chi_{6272} (1, \cdot )$ Sato-Tate : $\mathrm{SU}(2)$ primitive : no self-dual : yes analytic rank = 0 Selberg data = $(4,\ 6272,\ (\ :1/2, 1/2),\ 1)$ $L(1)$ $\approx$ $1.004073929$ $L(\frac12)$ $\approx$ $1.004073929$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{2,\;7\}$, $F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4$with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2 $$1$$
7$C_1$$\times$$C_1$ $$( 1 - T )( 1 + T )$$
good3$C_2$ $$( 1 + p T^{2} )^{2}$$
5$C_2$ $$( 1 - 2 T + p T^{2} )^{2}$$
11$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
13$C_2$ $$( 1 - 2 T + p T^{2} )^{2}$$
17$C_2$ $$( 1 + 6 T + p T^{2} )^{2}$$
19$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
23$C_2$ $$( 1 + p T^{2} )^{2}$$
29$C_2$ $$( 1 - 6 T + p T^{2} )^{2}$$
31$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
37$C_2$ $$( 1 + 2 T + p T^{2} )^{2}$$
41$C_2$ $$( 1 - 2 T + p T^{2} )^{2}$$
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$C_2$ $$( 1 - 6 T + p T^{2} )^{2}$$
59$C_2$ $$( 1 + p T^{2} )^{2}$$
61$C_2$ $$( 1 + 6 T + p T^{2} )^{2}$$
67$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
71$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
73$C_2$ $$( 1 - 10 T + p T^{2} )^{2}$$
79$C_2$ $$( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} )$$
83$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
89$C_2$ $$( 1 + 6 T + p T^{2} )^{2}$$
97$C_2$ $$( 1 + 6 T + p T^{2} )^{2}$$
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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

−11.84146297682028690633889717654, −11.40509490101706296629658789662, −10.63587746243635306398043641619, −10.60224244605800499935670328540, −9.457349313423188294666539437989, −9.117894034638208548514808203742, −8.560607806232820418165457845354, −8.188420417684287628984093773695, −6.67949194679031572918581545842, −6.42262084813563967264199691637, −5.82613210142107824134972732669, −5.22316409435338364257345412913, −4.18465932715971087267449515138, −2.79183800612725741835263061431, −2.14556791802447766709825303506, 2.14556791802447766709825303506, 2.79183800612725741835263061431, 4.18465932715971087267449515138, 5.22316409435338364257345412913, 5.82613210142107824134972732669, 6.42262084813563967264199691637, 6.67949194679031572918581545842, 8.188420417684287628984093773695, 8.560607806232820418165457845354, 9.117894034638208548514808203742, 9.457349313423188294666539437989, 10.60224244605800499935670328540, 10.63587746243635306398043641619, 11.40509490101706296629658789662, 11.84146297682028690633889717654