Properties

Label 4-60e4-1.1-c0e2-0-10
Degree $4$
Conductor $12960000$
Sign $1$
Analytic cond. $3.22789$
Root an. cond. $1.34038$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 3·7-s + 3·21-s + 3·23-s − 27-s − 29-s − 41-s − 3·47-s + 5·49-s − 61-s + 3·67-s + 3·69-s − 81-s − 3·83-s − 87-s + 2·89-s − 2·101-s − 2·109-s − 121-s − 123-s + 127-s + 131-s + 137-s + 139-s − 3·141-s + 5·147-s + 149-s + ⋯
L(s)  = 1  + 3-s + 3·7-s + 3·21-s + 3·23-s − 27-s − 29-s − 41-s − 3·47-s + 5·49-s − 61-s + 3·67-s + 3·69-s − 81-s − 3·83-s − 87-s + 2·89-s − 2·101-s − 2·109-s − 121-s − 123-s + 127-s + 131-s + 137-s + 139-s − 3·141-s + 5·147-s + 149-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12960000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12960000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(12960000\)    =    \(2^{8} \cdot 3^{4} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(3.22789\)
Root analytic conductor: \(1.34038\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 12960000,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.190132412\)
\(L(\frac12)\) \(\approx\) \(3.190132412\)
\(L(1)\) 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$C_2$ \( 1 - T + T^{2} \)
5 \( 1 \)
good7$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 - T + T^{2} ) \)
11$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
13$C_2^2$ \( 1 - T^{2} + T^{4} \)
17$C_2$ \( ( 1 + T^{2} )^{2} \)
19$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
23$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 - T + T^{2} ) \)
29$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 - T + T^{2} ) \)
31$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
37$C_2$ \( ( 1 + T^{2} )^{2} \)
41$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 - T + T^{2} ) \)
43$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
47$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T + T^{2} ) \)
53$C_2$ \( ( 1 + T^{2} )^{2} \)
59$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
61$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 - T + T^{2} ) \)
67$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 - T + T^{2} ) \)
71$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
73$C_2$ \( ( 1 + T^{2} )^{2} \)
79$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
83$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T + T^{2} ) \)
89$C_2$ \( ( 1 - T + T^{2} )^{2} \)
97$C_2^2$ \( 1 - T^{2} + T^{4} \)
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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

−8.774233293235374318073716510000, −8.413552785979424476797980298620, −8.223318241315047015748702234177, −7.939884712043278586036290269757, −7.64955022906691997211082746268, −7.19968733023872853066613802129, −6.74106956231082083138956396413, −6.60735342854946417288903520780, −5.51852554434213728838964364638, −5.50827504382956451802312524405, −5.08713894358954790474357448754, −4.77111005575150577678909005913, −4.45266255159392665606585548180, −3.87043298703629266043180909715, −3.40271583100355970322259024002, −2.92890576232569559654620344112, −2.51315371053266114050122915054, −1.79692073878321632397685117391, −1.63846770763424277429347827945, −1.10428701198058591233431240327, 1.10428701198058591233431240327, 1.63846770763424277429347827945, 1.79692073878321632397685117391, 2.51315371053266114050122915054, 2.92890576232569559654620344112, 3.40271583100355970322259024002, 3.87043298703629266043180909715, 4.45266255159392665606585548180, 4.77111005575150577678909005913, 5.08713894358954790474357448754, 5.50827504382956451802312524405, 5.51852554434213728838964364638, 6.60735342854946417288903520780, 6.74106956231082083138956396413, 7.19968733023872853066613802129, 7.64955022906691997211082746268, 7.939884712043278586036290269757, 8.223318241315047015748702234177, 8.413552785979424476797980298620, 8.774233293235374318073716510000

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