Properties

Degree $4$
Conductor $540000$
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-s + 3-s + 4-s − 6-s − 8-s + 9-s + 12-s − 4·13-s + 16-s − 18-s − 24-s + 4·26-s + 27-s − 32-s + 36-s − 4·37-s − 4·39-s + 48-s + 2·49-s − 4·52-s − 54-s − 20·61-s + 64-s − 72-s − 4·73-s + 4·74-s + 4·78-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.353·8-s + 1/3·9-s + 0.288·12-s − 1.10·13-s + 1/4·16-s − 0.235·18-s − 0.204·24-s + 0.784·26-s + 0.192·27-s − 0.176·32-s + 1/6·36-s − 0.657·37-s − 0.640·39-s + 0.144·48-s + 2/7·49-s − 0.554·52-s − 0.136·54-s − 2.56·61-s + 1/8·64-s − 0.117·72-s − 0.468·73-s + 0.464·74-s + 0.452·78-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 540000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 540000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(540000\)    =    \(2^{5} \cdot 3^{3} \cdot 5^{4}\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{540000} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 540000,\ (\ :1/2, 1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_1$ \( 1 + T \)
3$C_1$ \( 1 - T \)
5 \( 1 \)
good7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \)
97$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.128521468164053694071078641328, −7.84831740970949093070141074865, −7.53023481714424969283045906095, −6.97121259233254998318201926571, −6.58767052470959898247985953468, −6.10897773314734163440451778623, −5.37349128799592068356886197367, −5.03186235211519720784151262088, −4.36655234059998826349058539516, −3.82839635062702052518767893213, −3.10714236804866821433728477742, −2.65454187476862090428654397793, −2.01462355044529552405323097963, −1.28283795797463795719739258116, 0, 1.28283795797463795719739258116, 2.01462355044529552405323097963, 2.65454187476862090428654397793, 3.10714236804866821433728477742, 3.82839635062702052518767893213, 4.36655234059998826349058539516, 5.03186235211519720784151262088, 5.37349128799592068356886197367, 6.10897773314734163440451778623, 6.58767052470959898247985953468, 6.97121259233254998318201926571, 7.53023481714424969283045906095, 7.84831740970949093070141074865, 8.128521468164053694071078641328

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