Properties

Degree 4
Conductor $ 2^{12} \cdot 5^{2} $
Sign $1$
Motivic weight 5
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 110·5-s + 290·9-s + 296·11-s + 4.44e3·19-s + 8.97e3·25-s − 540·29-s − 4.09e3·31-s − 4.79e3·41-s − 3.19e4·45-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 + 2.50e4·81-s + 1.70e5·89-s − 4.88e5·95-s + 8.58e4·99-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 + ⋯
L(s)  = 1  − 1.96·5-s + 1.19·9-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 − 2.34·45-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 + 0.424·81-s + 2.28·89-s − 5.55·95-s + 0.880·99-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 102400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 102400 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(102400\)    =    \(2^{12} \cdot 5^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(5\)
character  :  induced by $\chi_{320} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((4,\ 102400,\ (\ :5/2, 5/2),\ 1)\)
\(L(3)\)  \(\approx\)  \(2.396140765\)
\(L(\frac12)\)  \(\approx\)  \(2.396140765\)
\(L(\frac{7}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;5\}$,\(F_p(T)\) is a polynomial of degree 4. If $p \in \{2,\;5\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
5$C_2$ \( 1 + 22 p T + p^{5} T^{2} \)
good3$C_2^2$ \( 1 - 290 T^{2} + p^{10} T^{4} \)
7$C_2^2$ \( 1 - 8650 T^{2} + p^{10} T^{4} \)
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

−11.40152723005750657822842957554, −10.40798133918283463611814069135, −10.34757264904187364532269775074, −9.362097390708126835512903973996, −9.341510265879589143710613731559, −8.684629844601267694369339341083, −7.976905079686492485291398375542, −7.56174395117571142419891896144, −7.41206681584715330809595035596, −6.85627125866067399252101557287, −6.39042117936015941783695381724, −5.22660877959376333887606380084, −5.18043803815000671230890681206, −4.11918762105717064503094414208, −4.09000221554575593486135417789, −3.31417070250116321552865347618, −2.96737626266581221367272840611, −1.64816353048125030239897637389, −1.06214537871068794523823957956, −0.50331916655101393402041627034, 0.50331916655101393402041627034, 1.06214537871068794523823957956, 1.64816353048125030239897637389, 2.96737626266581221367272840611, 3.31417070250116321552865347618, 4.09000221554575593486135417789, 4.11918762105717064503094414208, 5.18043803815000671230890681206, 5.22660877959376333887606380084, 6.39042117936015941783695381724, 6.85627125866067399252101557287, 7.41206681584715330809595035596, 7.56174395117571142419891896144, 7.976905079686492485291398375542, 8.684629844601267694369339341083, 9.341510265879589143710613731559, 9.362097390708126835512903973996, 10.34757264904187364532269775074, 10.40798133918283463611814069135, 11.40152723005750657822842957554

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