Properties

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

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 110·5-s − 296·11-s + 4.44e3·19-s + 8.97e3·25-s − 540·29-s + 4.09e3·31-s + 4.79e3·41-s + 8.65e3·49-s + 3.25e4·55-s + 7.94e4·59-s − 8.45e4·61-s − 8.49e3·71-s + 7.05e4·79-s − 1.70e5·89-s − 4.88e5·95-s + 8.59e3·101-s + 7.19e4·109-s − 2.56e5·121-s − 6.43e5·125-s + 127-s + 131-s + 137-s + 139-s + 5.94e4·145-s + 149-s + 151-s − 4.50e5·155-s + ⋯
L(s)  = 1  − 1.96·5-s − 0.737·11-s + 2.82·19-s + 2.87·25-s − 0.119·29-s + 0.765·31-s + 0.445·41-s + 0.514·49-s + 1.45·55-s + 2.97·59-s − 2.91·61-s − 0.200·71-s + 1.27·79-s − 2.28·89-s − 5.55·95-s + 0.0838·101-s + 0.580·109-s − 1.59·121-s − 3.68·125-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + 0.234·145-s + 3.69e−6·149-s + 3.56e−6·151-s − 1.50·155-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 518400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 518400 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(518400\)    =    \(2^{8} \cdot 3^{4} \cdot 5^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(5\)
character  :  induced by $\chi_{720} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((4,\ 518400,\ (\ :5/2, 5/2),\ 1)\)
\(L(3)\)  \(\approx\)  \(1.262164682\)
\(L(\frac12)\)  \(\approx\)  \(1.262164682\)
\(L(\frac{7}{2})\)   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$C_2$ \( 1 + 22 p T + p^{5} T^{2} \)
good7$C_2^2$ \( 1 - 8650 T^{2} + p^{10} T^{4} \)
11$C_2$ \( ( 1 + 148 T + p^{5} T^{2} )^{2} \)
13$C_2^2$ \( 1 - 274730 T^{2} + p^{10} T^{4} \)
17$C_2^2$ \( 1 + 1354590 T^{2} + p^{10} T^{4} \)
19$C_2$ \( ( 1 - 2220 T + p^{5} T^{2} )^{2} \)
23$C_2^2$ \( 1 - 11320170 T^{2} + p^{10} T^{4} \)
29$C_2$ \( ( 1 + 270 T + p^{5} T^{2} )^{2} \)
31$C_2$ \( ( 1 - 2048 T + p^{5} T^{2} )^{2} \)
37$C_2^2$ \( 1 - 119573530 T^{2} + p^{10} T^{4} \)
41$C_2$ \( ( 1 - 2398 T + p^{5} T^{2} )^{2} \)
43$C_2^2$ \( 1 - 288754450 T^{2} + p^{10} T^{4} \)
47$C_2^2$ \( 1 - 344584890 T^{2} + p^{10} T^{4} \)
53$C_2^2$ \( 1 - 827605690 T^{2} + p^{10} T^{4} \)
59$C_2$ \( ( 1 - 39740 T + p^{5} T^{2} )^{2} \)
61$C_2$ \( ( 1 + 42298 T + p^{5} T^{2} )^{2} \)
67$C_2^2$ \( 1 - 1669968610 T^{2} + p^{10} T^{4} \)
71$C_2$ \( ( 1 + 4248 T + p^{5} T^{2} )^{2} \)
73$C_2^2$ \( 1 - 3239892370 T^{2} + p^{10} T^{4} \)
79$C_2$ \( ( 1 - 35280 T + p^{5} T^{2} )^{2} \)
83$C_2^2$ \( 1 - 7103795010 T^{2} + p^{10} T^{4} \)
89$C_2$ \( ( 1 + 85210 T + p^{5} T^{2} )^{2} \)
97$C_2^2$ \( 1 - 7720618690 T^{2} + p^{10} T^{4} \)
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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

−9.997237747398641560083973834876, −9.355668036513258762201909828199, −8.985464609204727567999621514017, −8.423186770586250797940167920172, −8.077400479934443942902778531018, −7.58788739672034986071780418237, −7.47366335870589045784942961799, −7.00001335816922293298812854567, −6.52255924616614531058194645684, −5.58798640295237629456379136168, −5.48695229653567532681706656070, −4.65162063057916416811697579632, −4.59151064934415568329109083736, −3.62508818867320182130947131949, −3.55316429666574274638435600366, −2.86044218469418759283312630677, −2.50743321589217937955076141410, −1.28083330032161261943830772669, −0.940046955472601381627138948055, −0.30352029737504893863687685317, 0.30352029737504893863687685317, 0.940046955472601381627138948055, 1.28083330032161261943830772669, 2.50743321589217937955076141410, 2.86044218469418759283312630677, 3.55316429666574274638435600366, 3.62508818867320182130947131949, 4.59151064934415568329109083736, 4.65162063057916416811697579632, 5.48695229653567532681706656070, 5.58798640295237629456379136168, 6.52255924616614531058194645684, 7.00001335816922293298812854567, 7.47366335870589045784942961799, 7.58788739672034986071780418237, 8.077400479934443942902778531018, 8.423186770586250797940167920172, 8.985464609204727567999621514017, 9.355668036513258762201909828199, 9.997237747398641560083973834876

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