Properties

Degree 4
Conductor $ 2^{4} \cdot 5^{2} \cdot 11^{2} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 4-s + 2·5-s − 3·8-s − 6·9-s + 2·10-s + 4·13-s − 16-s + 12·17-s − 6·18-s − 2·20-s + 3·25-s + 4·26-s + 12·29-s + 5·32-s + 12·34-s + 6·36-s − 4·37-s − 6·40-s + 4·41-s − 12·45-s − 14·49-s + 3·50-s − 4·52-s − 4·53-s + 12·58-s − 20·61-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s + 0.894·5-s − 1.06·8-s − 2·9-s + 0.632·10-s + 1.10·13-s − 1/4·16-s + 2.91·17-s − 1.41·18-s − 0.447·20-s + 3/5·25-s + 0.784·26-s + 2.22·29-s + 0.883·32-s + 2.05·34-s + 36-s − 0.657·37-s − 0.948·40-s + 0.624·41-s − 1.78·45-s − 2·49-s + 0.424·50-s − 0.554·52-s − 0.549·53-s + 1.57·58-s − 2.56·61-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 48400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 48400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(48400\)    =    \(2^{4} \cdot 5^{2} \cdot 11^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{48400} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 48400,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $1.770622110$
$L(\frac12)$  $\approx$  $1.770622110$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;5,\;11\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;5,\;11\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2$C_2$ \( 1 - T + p T^{2} \)
5$C_1$ \( ( 1 - T )^{2} \)
11$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good3$C_2$ \( ( 1 + p T^{2} )^{2} \)
7$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
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$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
53$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
61$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
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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

−10.07588915028781859505118646541, −9.376036497146997809466572379928, −9.365267708430525132762162914792, −8.402071812243779349662792300606, −8.248642768475844318802620430882, −7.78041223670494609407513138154, −6.49908708730306794178412805091, −6.16821463973303244938343944630, −5.83555924118533390298503875284, −5.09766059609940450216711100862, −4.97617253141282559040022801022, −3.60799086718076388676488423337, −3.25423179226253980082861279896, −2.69303075235727048947048138558, −1.13555459814448309624758433433, 1.13555459814448309624758433433, 2.69303075235727048947048138558, 3.25423179226253980082861279896, 3.60799086718076388676488423337, 4.97617253141282559040022801022, 5.09766059609940450216711100862, 5.83555924118533390298503875284, 6.16821463973303244938343944630, 6.49908708730306794178412805091, 7.78041223670494609407513138154, 8.248642768475844318802620430882, 8.402071812243779349662792300606, 9.365267708430525132762162914792, 9.376036497146997809466572379928, 10.07588915028781859505118646541

Graph of the $Z$-function along the critical line