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 + 4·11-s + 4·12-s − 4·16-s − 4·17-s − 6·18-s − 10·19-s − 8·22-s + 4·27-s + 8·32-s + 8·33-s + 8·34-s + 6·36-s + 20·38-s − 16·41-s − 2·43-s + 8·44-s − 8·48-s − 5·49-s − 8·51-s − 8·54-s − 20·57-s − 20·59-s + ⋯
L(s)  = 1  − 1.41·2-s + 1.15·3-s + 4-s − 1.63·6-s + 9-s + 1.20·11-s + 1.15·12-s − 16-s − 0.970·17-s − 1.41·18-s − 2.29·19-s − 1.70·22-s + 0.769·27-s + 1.41·32-s + 1.39·33-s + 1.37·34-s + 36-s + 3.24·38-s − 2.49·41-s − 0.304·43-s + 1.20·44-s − 1.15·48-s − 5/7·49-s − 1.12·51-s − 1.08·54-s − 2.64·57-s − 2.60·59-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 $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
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$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
17$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
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$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
59$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 + 17 T + p T^{2} )^{2} \)
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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, −8.040086974208892692137412150811, −7.01259548848434300802989966251, −6.77287251257420576381216593643, −6.63552338173317415872093243656, −5.90404919386938898421911646081, −4.85692283621813202383375265351, −4.48018036005395779311676580836, −3.96102438209774458569462189574, −3.36957507180644347787302261531, −2.51201353173268436104607760745, −1.90319955654708104595670502735, −1.48067503408916751090348731726, 0, 1.48067503408916751090348731726, 1.90319955654708104595670502735, 2.51201353173268436104607760745, 3.36957507180644347787302261531, 3.96102438209774458569462189574, 4.48018036005395779311676580836, 4.85692283621813202383375265351, 5.90404919386938898421911646081, 6.63552338173317415872093243656, 6.77287251257420576381216593643, 7.01259548848434300802989966251, 8.040086974208892692137412150811, 8.114757177440921419069207299350, 8.655074250265874594654667643843

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