# Properties

 Degree 4 Conductor $2^{2} \cdot 3^{2} \cdot 7^{3}$ Sign $1$ Motivic weight 1 Primitive no Self-dual yes Analytic rank 0

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 2·3-s + 4-s + 7-s + 9-s − 2·12-s + 16-s + 12·17-s − 2·21-s − 10·25-s + 4·27-s + 28-s + 36-s + 4·37-s + 12·41-s + 16·43-s − 24·47-s − 2·48-s + 49-s − 24·51-s − 12·59-s + 63-s + 64-s − 8·67-s + 12·68-s + 20·75-s + 16·79-s − 11·81-s + ⋯
 L(s)  = 1 − 1.15·3-s + 1/2·4-s + 0.377·7-s + 1/3·9-s − 0.577·12-s + 1/4·16-s + 2.91·17-s − 0.436·21-s − 2·25-s + 0.769·27-s + 0.188·28-s + 1/6·36-s + 0.657·37-s + 1.87·41-s + 2.43·43-s − 3.50·47-s − 0.288·48-s + 1/7·49-s − 3.36·51-s − 1.56·59-s + 0.125·63-s + 1/8·64-s − 0.977·67-s + 1.45·68-s + 2.30·75-s + 1.80·79-s − 1.22·81-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 12348 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 12348 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$4$$ $$N$$ = $$12348$$    =    $$2^{2} \cdot 3^{2} \cdot 7^{3}$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : $\chi_{12348} (1, \cdot )$ Sato-Tate : $\mathrm{SU}(2)$ primitive : no self-dual : yes analytic rank = 0 Selberg data = $(4,\ 12348,\ (\ :1/2, 1/2),\ 1)$ $L(1)$ $\approx$ $0.8566614515$ $L(\frac12)$ $\approx$ $0.8566614515$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{2,\;3,\;7\}$, $F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4$with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;3,\;7\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2$C_1$$\times$$C_1$ $$( 1 - T )( 1 + T )$$
3$C_2$ $$1 + 2 T + p T^{2}$$
7$C_1$ $$1 - T$$
good5$C_2$ $$( 1 + p T^{2} )^{2}$$
11$C_2$ $$( 1 + p T^{2} )^{2}$$
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$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
31$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
37$C_2$ $$( 1 - 2 T + p T^{2} )^{2}$$
41$C_2$ $$( 1 - 6 T + p T^{2} )^{2}$$
43$C_2$ $$( 1 - 8 T + p T^{2} )^{2}$$
47$C_2$ $$( 1 + 12 T + p T^{2} )^{2}$$
53$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
59$C_2$ $$( 1 + 6 T + p T^{2} )^{2}$$
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$C_2$ $$( 1 - 8 T + p T^{2} )^{2}$$
83$C_2$ $$( 1 + 6 T + p T^{2} )^{2}$$
89$C_2$ $$( 1 + 6 T + p T^{2} )^{2}$$
97$C_2$ $$( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )$$
show less
\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

−11.23136141438460806904855801814, −11.03381616069283788346556339454, −10.27339907123230780461643050534, −9.765547119459919407856461234632, −9.393998538973797562285424645673, −8.245010328271296459357780366393, −7.64010214491199898083268392854, −7.57571100088867902110310233811, −6.37349457460642388728354325328, −5.88407913223931048395780461484, −5.57928681742950427486583645839, −4.74683922157615295493069282209, −3.79391663707115759548696099652, −2.83833445427180997893704571146, −1.33827299261372059142982522511, 1.33827299261372059142982522511, 2.83833445427180997893704571146, 3.79391663707115759548696099652, 4.74683922157615295493069282209, 5.57928681742950427486583645839, 5.88407913223931048395780461484, 6.37349457460642388728354325328, 7.57571100088867902110310233811, 7.64010214491199898083268392854, 8.245010328271296459357780366393, 9.393998538973797562285424645673, 9.765547119459919407856461234632, 10.27339907123230780461643050534, 11.03381616069283788346556339454, 11.23136141438460806904855801814