Properties

Degree 4
Conductor $ 2^{6} \cdot 5^{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  − 2·5-s − 2·9-s + 4·13-s − 12·17-s + 3·25-s + 12·29-s + 4·37-s + 12·41-s + 4·45-s − 10·49-s − 12·53-s + 4·61-s − 8·65-s + 4·73-s − 5·81-s + 24·85-s − 12·89-s + 4·97-s + 12·101-s + 4·109-s − 12·113-s − 8·117-s − 22·121-s − 4·125-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  − 0.894·5-s − 2/3·9-s + 1.10·13-s − 2.91·17-s + 3/5·25-s + 2.22·29-s + 0.657·37-s + 1.87·41-s + 0.596·45-s − 1.42·49-s − 1.64·53-s + 0.512·61-s − 0.992·65-s + 0.468·73-s − 5/9·81-s + 2.60·85-s − 1.27·89-s + 0.406·97-s + 1.19·101-s + 0.383·109-s − 1.12·113-s − 0.739·117-s − 2·121-s − 0.357·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(1600\)    =    \(2^{6} \cdot 5^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{1600} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 1600,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $0.5352579714$
$L(\frac12)$  $\approx$  $0.5352579714$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;5\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;5\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2 \( 1 \)
5$C_1$ \( ( 1 + T )^{2} \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
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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

−13.57191933746624476808470409800, −13.00236641577412261230757825497, −12.41085901163162733003481941315, −11.50854453196989171707939391081, −11.10973086408237495845181170606, −10.83473950301065015207539180976, −9.683565298540966130897522415203, −8.740293324354028093320034492248, −8.552217550781204179646223792184, −7.71273110823499542846308181544, −6.57891116465648258947670054106, −6.27087624192875571051851265258, −4.78130792717525308450176413839, −4.12250433368686324236236368171, −2.76929890617261215013507568311, 2.76929890617261215013507568311, 4.12250433368686324236236368171, 4.78130792717525308450176413839, 6.27087624192875571051851265258, 6.57891116465648258947670054106, 7.71273110823499542846308181544, 8.552217550781204179646223792184, 8.740293324354028093320034492248, 9.683565298540966130897522415203, 10.83473950301065015207539180976, 11.10973086408237495845181170606, 11.50854453196989171707939391081, 12.41085901163162733003481941315, 13.00236641577412261230757825497, 13.57191933746624476808470409800

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