Properties

Label 4-560e2-1.1-c1e2-0-36
Degree $4$
Conductor $313600$
Sign $-1$
Analytic cond. $19.9954$
Root an. cond. $2.11462$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s + 2·7-s − 2·9-s + 4·13-s + 3·25-s − 8·31-s − 4·35-s − 20·43-s + 4·45-s − 12·47-s − 3·49-s + 4·61-s − 4·63-s − 8·65-s + 4·67-s − 5·81-s + 8·91-s + 12·101-s + 28·103-s − 12·107-s − 12·113-s − 8·117-s − 22·121-s − 4·125-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  − 0.894·5-s + 0.755·7-s − 2/3·9-s + 1.10·13-s + 3/5·25-s − 1.43·31-s − 0.676·35-s − 3.04·43-s + 0.596·45-s − 1.75·47-s − 3/7·49-s + 0.512·61-s − 0.503·63-s − 0.992·65-s + 0.488·67-s − 5/9·81-s + 0.838·91-s + 1.19·101-s + 2.75·103-s − 1.16·107-s − 1.12·113-s − 0.739·117-s − 2·121-s − 0.357·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 313600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 313600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(313600\)    =    \(2^{8} \cdot 5^{2} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(19.9954\)
Root analytic conductor: \(2.11462\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 313600,\ (\ :1/2, 1/2),\ -1)\)

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 \( 1 \)
5$C_1$ \( ( 1 + T )^{2} \)
7$C_2$ \( 1 - 2 T + p T^{2} \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
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$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
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$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.552217550781204179646223792184, −8.274191454688856203377872859552, −7.67724524894114537171979145145, −7.31157654449653438132198826839, −6.57891116465648258947670054106, −6.36392101906843607751379404756, −5.60452081656466550382223684619, −5.07009055292225152793438727510, −4.78130792717525308450176413839, −3.95271496302851434324941322577, −3.49047115245004503734053909847, −3.10876796327263031806431254214, −2.02979685108826286576114913510, −1.36690026055352676588947736551, 0, 1.36690026055352676588947736551, 2.02979685108826286576114913510, 3.10876796327263031806431254214, 3.49047115245004503734053909847, 3.95271496302851434324941322577, 4.78130792717525308450176413839, 5.07009055292225152793438727510, 5.60452081656466550382223684619, 6.36392101906843607751379404756, 6.57891116465648258947670054106, 7.31157654449653438132198826839, 7.67724524894114537171979145145, 8.274191454688856203377872859552, 8.552217550781204179646223792184

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