Properties

Degree 4
Conductor $ 2^{12} \cdot 3^{4} \cdot 5^{2} $
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  + 2·5-s − 2·13-s + 2·17-s + 3·25-s − 2·37-s + 4·41-s + 2·53-s − 4·65-s − 2·73-s + 4·85-s − 2·97-s − 2·113-s − 2·121-s + 4·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + ⋯
L(s)  = 1  + 2·5-s − 2·13-s + 2·17-s + 3·25-s − 2·37-s + 4·41-s + 2·53-s − 4·65-s − 2·73-s + 4·85-s − 2·97-s − 2·113-s − 2·121-s + 4·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8294400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8294400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(8294400\)    =    \(2^{12} \cdot 3^{4} \cdot 5^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  induced by $\chi_{2880} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 8294400,\ (\ :0, 0),\ 1)$
$L(\frac{1}{2})$  $\approx$  $2.113065027$
$L(\frac12)$  $\approx$  $2.113065027$
$L(1)$   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 \( 1 \)
5$C_1$ \( ( 1 - T )^{2} \)
good7$C_2^2$ \( 1 + T^{4} \)
11$C_2$ \( ( 1 + T^{2} )^{2} \)
13$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
17$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
19$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
23$C_2^2$ \( 1 + T^{4} \)
29$C_2$ \( ( 1 + T^{2} )^{2} \)
31$C_2$ \( ( 1 + T^{2} )^{2} \)
37$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
41$C_1$ \( ( 1 - T )^{4} \)
43$C_2^2$ \( 1 + T^{4} \)
47$C_2^2$ \( 1 + T^{4} \)
53$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
59$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
61$C_2$ \( ( 1 + T^{2} )^{2} \)
67$C_2^2$ \( 1 + T^{4} \)
71$C_2$ \( ( 1 + T^{2} )^{2} \)
73$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
79$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
83$C_2^2$ \( 1 + T^{4} \)
89$C_2$ \( ( 1 + T^{2} )^{2} \)
97$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
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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

−9.122985483421329651393304358192, −9.054808555652830358511866688992, −8.451572014982720513571784719359, −7.947055717383514059880694882193, −7.53179085325822748549171207671, −7.28399161595257855237613723379, −6.87958905364385779778160339889, −6.53701648709589852388108456791, −5.82704323853452741440798708245, −5.70793900477979294766613176962, −5.33107753825434546417942632052, −5.21468584215457646832547000997, −4.37805761102425759304775548008, −4.27348818929452709643991685152, −3.36024568570201550238938503975, −2.93801762967853872878877683254, −2.43350896458390204821321676454, −2.28211909805043134554833880301, −1.46392526469955693743567413879, −1.00589148276618140205811909963, 1.00589148276618140205811909963, 1.46392526469955693743567413879, 2.28211909805043134554833880301, 2.43350896458390204821321676454, 2.93801762967853872878877683254, 3.36024568570201550238938503975, 4.27348818929452709643991685152, 4.37805761102425759304775548008, 5.21468584215457646832547000997, 5.33107753825434546417942632052, 5.70793900477979294766613176962, 5.82704323853452741440798708245, 6.53701648709589852388108456791, 6.87958905364385779778160339889, 7.28399161595257855237613723379, 7.53179085325822748549171207671, 7.947055717383514059880694882193, 8.451572014982720513571784719359, 9.054808555652830358511866688992, 9.122985483421329651393304358192

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