Properties

Degree 4
Conductor $ 2^{8} \cdot 3^{4} \cdot 5^{4} $
Sign $1$
Motivic weight 0
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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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

\( d \)  =  \(4\)
\( N \)  =  \(12960000\)    =    \(2^{8} \cdot 3^{4} \cdot 5^{4}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  induced by $\chi_{3600} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((4,\ 12960000,\ (\ :0, 0),\ 1)\)
\(L(\frac{1}{2})\)  \(\approx\)  \(0.07437147082\)
\(L(\frac12)\)  \(\approx\)  \(0.07437147082\)
\(L(1)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5\}$,\(F_p(T)\) is a polynomial of degree 4. If $p \in \{2,\;3,\;5\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$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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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−8.950471305527200165913420314449, −8.830543883062758526579946750916, −7.950821820276303306155377821785, −7.84801008363821895872030989119, −7.35014736276083701131312028986, −6.80518579417157308514338345250, −6.69321995569481288978389517780, −6.09038764979851478767533676633, −6.04059374890192725873992573803, −5.80180359799048099296536788634, −5.40602217646731782991194303894, −4.73977578504509466980451232513, −4.16922672851381335766982729892, −3.89103169893303500081729982636, −3.43693181128580221651685283601, −3.16050658357450261029861473460, −2.41442422883984642557358974120, −2.24532460033465538263627394744, −1.20214410967519935251523772018, −0.17670312809240730142134776671, 0.17670312809240730142134776671, 1.20214410967519935251523772018, 2.24532460033465538263627394744, 2.41442422883984642557358974120, 3.16050658357450261029861473460, 3.43693181128580221651685283601, 3.89103169893303500081729982636, 4.16922672851381335766982729892, 4.73977578504509466980451232513, 5.40602217646731782991194303894, 5.80180359799048099296536788634, 6.04059374890192725873992573803, 6.09038764979851478767533676633, 6.69321995569481288978389517780, 6.80518579417157308514338345250, 7.35014736276083701131312028986, 7.84801008363821895872030989119, 7.950821820276303306155377821785, 8.830543883062758526579946750916, 8.950471305527200165913420314449

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