Properties

Degree $4$
Conductor $108000$
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 + 5-s + 6-s − 8·7-s + 8-s + 9-s + 10-s + 12-s − 8·14-s + 15-s + 16-s − 12·17-s + 18-s + 20-s − 8·21-s + 24-s + 25-s + 27-s − 8·28-s + 30-s + 32-s − 12·34-s − 8·35-s + 36-s + 40-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 3.02·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.288·12-s − 2.13·14-s + 0.258·15-s + 1/4·16-s − 2.91·17-s + 0.235·18-s + 0.223·20-s − 1.74·21-s + 0.204·24-s + 1/5·25-s + 0.192·27-s − 1.51·28-s + 0.182·30-s + 0.176·32-s − 2.05·34-s − 1.35·35-s + 1/6·36-s + 0.158·40-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(108000\)    =    \(2^{5} \cdot 3^{3} \cdot 5^{3}\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{108000} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 108000,\ (\ :1/2, 1/2),\ -1)\)

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$C_1$ \( 1 - T \)
good7$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
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$C_2$ \( ( 1 + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
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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

−9.276841907117526756164869085518, −8.811879039450229959581280002555, −8.644229019078172602684577487278, −7.53023481714424969283045906095, −6.97121259233254998318201926571, −6.73106470746564203527567325446, −6.11397905422548756742635703073, −6.10897773314734163440451778623, −5.03186235211519720784151262088, −4.32687879913691587957737883725, −3.82839635062702052518767893213, −3.10714236804866821433728477742, −2.74628747875934588873509462774, −2.01462355044529552405323097963, 0, 2.01462355044529552405323097963, 2.74628747875934588873509462774, 3.10714236804866821433728477742, 3.82839635062702052518767893213, 4.32687879913691587957737883725, 5.03186235211519720784151262088, 6.10897773314734163440451778623, 6.11397905422548756742635703073, 6.73106470746564203527567325446, 6.97121259233254998318201926571, 7.53023481714424969283045906095, 8.644229019078172602684577487278, 8.811879039450229959581280002555, 9.276841907117526756164869085518

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