Properties

Degree 4
Conductor $ 2^{4} \cdot 5^{4} $
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  + 2·9-s − 960·11-s + 2.40e3·19-s − 1.10e4·29-s + 1.87e4·31-s − 2.87e4·41-s − 1.39e4·49-s + 2.34e4·59-s + 2.64e4·61-s − 5.90e4·71-s − 6.24e4·79-s − 5.90e4·81-s − 2.39e5·89-s − 1.92e3·99-s + 2.02e5·101-s + 6.23e3·109-s + 3.69e5·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 3.55e5·169-s + ⋯
L(s)  = 1  + 0.00823·9-s − 2.39·11-s + 1.53·19-s − 2.44·29-s + 3.49·31-s − 2.67·41-s − 0.827·49-s + 0.878·59-s + 0.908·61-s − 1.39·71-s − 1.12·79-s − 0.999·81-s − 3.19·89-s − 0.0196·99-s + 1.97·101-s + 0.0502·109-s + 2.29·121-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + 3.69e−6·149-s + 3.56e−6·151-s + 3.23e−6·157-s + 2.94e−6·163-s + 2.77e−6·167-s + 0.958·169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(10000\)    =    \(2^{4} \cdot 5^{4}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(5\)
character  :  induced by $\chi_{100} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((4,\ 10000,\ (\ :5/2, 5/2),\ 1)\)
\(L(3)\)  \(\approx\)  \(1.24215\)
\(L(\frac12)\)  \(\approx\)  \(1.24215\)
\(L(\frac{7}{2})\)   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 \( 1 \)
good3$C_2^2$ \( 1 - 2 T^{2} + p^{10} T^{4} \)
7$C_2^2$ \( 1 + 13910 T^{2} + p^{10} T^{4} \)
11$C_2$ \( ( 1 + 480 T + p^{5} T^{2} )^{2} \)
13$C_2^2$ \( 1 - 355702 T^{2} + p^{10} T^{4} \)
17$C_2^2$ \( 1 - 2805118 T^{2} + p^{10} T^{4} \)
19$C_2$ \( ( 1 - 1204 T + p^{5} T^{2} )^{2} \)
23$C_2^2$ \( 1 - 2722090 T^{2} + p^{10} T^{4} \)
29$C_2$ \( ( 1 + 5526 T + p^{5} T^{2} )^{2} \)
31$C_2$ \( ( 1 - 9356 T + p^{5} T^{2} )^{2} \)
37$C_2^2$ \( 1 - 107125990 T^{2} + p^{10} T^{4} \)
41$C_2$ \( ( 1 + 14394 T + p^{5} T^{2} )^{2} \)
43$C_2^2$ \( 1 - 293879986 T^{2} + p^{10} T^{4} \)
47$C_2^2$ \( 1 - 197996698 T^{2} + p^{10} T^{4} \)
53$C_2^2$ \( 1 - 817259110 T^{2} + p^{10} T^{4} \)
59$C_2$ \( ( 1 - 11748 T + p^{5} T^{2} )^{2} \)
61$C_2$ \( ( 1 - 13202 T + p^{5} T^{2} )^{2} \)
67$C_2^2$ \( 1 - 2567032450 T^{2} + p^{10} T^{4} \)
71$C_2$ \( ( 1 + 29532 T + p^{5} T^{2} )^{2} \)
73$C_2^2$ \( 1 - 3010587982 T^{2} + p^{10} T^{4} \)
79$C_2$ \( ( 1 + 31208 T + p^{5} T^{2} )^{2} \)
83$C_2^2$ \( 1 - 6398448130 T^{2} + p^{10} T^{4} \)
89$C_2$ \( ( 1 + 119514 T + p^{5} T^{2} )^{2} \)
97$C_2^2$ \( 1 - 8214543550 T^{2} + p^{10} T^{4} \)
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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

−13.06765048208727941147586477003, −12.94413892549526068153423632625, −12.06029969465593194594557080627, −11.51420899461360142956913198021, −11.23067453634351279752107739075, −10.21215189734531821802357364648, −10.10016586258612539175154861715, −9.673403851168556882213035860148, −8.486876319914660903375291367264, −8.351436312350168455783687219368, −7.54330958615218194979753127202, −7.21395051296656781889049430941, −6.28726574150430824333206229539, −5.34886129743555918672349690512, −5.25920091769356198073809954037, −4.32708726719985766938909321496, −3.15968668002452611948618399455, −2.76210968078179921982665480749, −1.66100540952640138243097922729, −0.42922929739508661423373705015, 0.42922929739508661423373705015, 1.66100540952640138243097922729, 2.76210968078179921982665480749, 3.15968668002452611948618399455, 4.32708726719985766938909321496, 5.25920091769356198073809954037, 5.34886129743555918672349690512, 6.28726574150430824333206229539, 7.21395051296656781889049430941, 7.54330958615218194979753127202, 8.351436312350168455783687219368, 8.486876319914660903375291367264, 9.673403851168556882213035860148, 10.10016586258612539175154861715, 10.21215189734531821802357364648, 11.23067453634351279752107739075, 11.51420899461360142956913198021, 12.06029969465593194594557080627, 12.94413892549526068153423632625, 13.06765048208727941147586477003

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