Properties

Label 4-86400-1.1-c1e2-0-28
Degree $4$
Conductor $86400$
Sign $-1$
Analytic cond. $5.50893$
Root an. cond. $1.53202$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 2·5-s + 9-s − 8·11-s − 2·15-s − 4·17-s − 25-s − 27-s + 8·33-s + 8·43-s + 2·45-s − 14·49-s + 4·51-s + 4·53-s − 16·55-s − 8·59-s − 4·61-s − 8·67-s − 16·71-s + 75-s + 81-s − 8·85-s − 8·99-s + 32·103-s − 4·109-s − 36·113-s + 26·121-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.894·5-s + 1/3·9-s − 2.41·11-s − 0.516·15-s − 0.970·17-s − 1/5·25-s − 0.192·27-s + 1.39·33-s + 1.21·43-s + 0.298·45-s − 2·49-s + 0.560·51-s + 0.549·53-s − 2.15·55-s − 1.04·59-s − 0.512·61-s − 0.977·67-s − 1.89·71-s + 0.115·75-s + 1/9·81-s − 0.867·85-s − 0.804·99-s + 3.15·103-s − 0.383·109-s − 3.38·113-s + 2.36·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 86400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 86400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(86400\)    =    \(2^{7} \cdot 3^{3} \cdot 5^{2}\)
Sign: $-1$
Analytic conductor: \(5.50893\)
Root analytic conductor: \(1.53202\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{86400} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 86400,\ (\ :1/2, 1/2),\ -1)\)

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 \( 1 \)
3$C_1$ \( 1 + T \)
5$C_2$ \( 1 - 2 T + p T^{2} \)
good7$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
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$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
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$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
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

−9.449285690166437837164957292815, −9.065170023749915638598404079220, −8.428095002482665926481658846183, −7.70640093924588702961218917702, −7.62403207119791885504595168937, −6.81554110451771320335088635337, −6.14969218484336320739052245813, −5.85324000516580696107361587496, −5.20634536916068220796571779353, −4.83326855247570334999188527992, −4.18465656456029023241423917214, −3.04557269583849472425830805016, −2.51659494204614894722398853037, −1.71961790068640789274174212358, 0, 1.71961790068640789274174212358, 2.51659494204614894722398853037, 3.04557269583849472425830805016, 4.18465656456029023241423917214, 4.83326855247570334999188527992, 5.20634536916068220796571779353, 5.85324000516580696107361587496, 6.14969218484336320739052245813, 6.81554110451771320335088635337, 7.62403207119791885504595168937, 7.70640093924588702961218917702, 8.428095002482665926481658846183, 9.065170023749915638598404079220, 9.449285690166437837164957292815

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