Properties

Degree 4
Conductor $ 2^{6} \cdot 3^{2} \cdot 5^{4} $
Sign $-1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 1

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 2·3-s + 2·4-s − 4·6-s + 3·9-s + 4·12-s − 2·13-s − 4·16-s − 6·18-s + 4·26-s + 4·27-s − 6·31-s + 8·32-s + 6·36-s − 4·37-s − 4·39-s − 16·41-s − 2·43-s − 8·48-s − 5·49-s − 4·52-s + 8·53-s − 8·54-s + 12·62-s − 8·64-s + 6·67-s − 16·71-s + ⋯
L(s)  = 1  − 1.41·2-s + 1.15·3-s + 4-s − 1.63·6-s + 9-s + 1.15·12-s − 0.554·13-s − 16-s − 1.41·18-s + 0.784·26-s + 0.769·27-s − 1.07·31-s + 1.41·32-s + 36-s − 0.657·37-s − 0.640·39-s − 2.49·41-s − 0.304·43-s − 1.15·48-s − 5/7·49-s − 0.554·52-s + 1.09·53-s − 1.08·54-s + 1.52·62-s − 64-s + 0.733·67-s − 1.89·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(360000\)    =    \(2^{6} \cdot 3^{2} \cdot 5^{4}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{360000} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((4,\ 360000,\ (\ :1/2, 1/2),\ -1)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5\}$,\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;3,\;5\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_2$ \( 1 + p T + p T^{2} \)
3$C_1$ \( ( 1 - T )^{2} \)
5 \( 1 \)
good7$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
11$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
13$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
31$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 17 T + p T^{2} ) \)
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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

−8.655074250265874594654667643843, −8.114757177440921419069207299350, −7.74916178315119539489350856070, −7.30514032256911612413843551960, −6.77287251257420576381216593643, −6.65396759238287778208236419506, −5.58318025222206456893422400511, −5.14493093108369759562477643348, −4.48018036005395779311676580836, −3.89570536316911001261511296620, −3.25692776068793976659434863504, −2.64453939621087097050043751453, −1.90319955654708104595670502735, −1.43286730840512162749271730076, 0, 1.43286730840512162749271730076, 1.90319955654708104595670502735, 2.64453939621087097050043751453, 3.25692776068793976659434863504, 3.89570536316911001261511296620, 4.48018036005395779311676580836, 5.14493093108369759562477643348, 5.58318025222206456893422400511, 6.65396759238287778208236419506, 6.77287251257420576381216593643, 7.30514032256911612413843551960, 7.74916178315119539489350856070, 8.114757177440921419069207299350, 8.655074250265874594654667643843

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