Properties

Degree 4
Conductor $ 2^{8} \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

\[\begin{aligned}\Lambda(s)=\mathstrut & 160000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160000 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(160000\)    =    \(2^{8} \cdot 5^{4}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(5\)
character  :  induced by $\chi_{400} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((4,\ 160000,\ (\ :5/2, 5/2),\ 1)\)
\(L(3)\)  \(\approx\)  \(0.08858534915\)
\(L(\frac12)\)  \(\approx\)  \(0.08858534915\)
\(L(\frac{7}{2})\)   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

−11.14180002977974230286346796683, −10.22667512986123092027863206468, −9.534422627379939472138700865798, −9.393117155538108805360097274862, −8.832618319384428701695857030777, −8.639848101825814351483785904487, −7.931649351082568095211837178263, −7.20787018801093003844439942436, −7.01030291017772773675187342935, −6.46359565226431338626738630584, −6.01921398360381519634105701257, −5.40689053955335448399949534211, −4.91454334050867966667843613230, −4.01079209902903751165275080001, −3.71088382271684895858455575310, −3.52942750425916199681839269115, −2.24357545680527458964554055139, −1.64306845570803581816906740911, −1.46392406625239292460273680500, −0.06969172079413995811069384016, 0.06969172079413995811069384016, 1.46392406625239292460273680500, 1.64306845570803581816906740911, 2.24357545680527458964554055139, 3.52942750425916199681839269115, 3.71088382271684895858455575310, 4.01079209902903751165275080001, 4.91454334050867966667843613230, 5.40689053955335448399949534211, 6.01921398360381519634105701257, 6.46359565226431338626738630584, 7.01030291017772773675187342935, 7.20787018801093003844439942436, 7.931649351082568095211837178263, 8.639848101825814351483785904487, 8.832618319384428701695857030777, 9.393117155538108805360097274862, 9.534422627379939472138700865798, 10.22667512986123092027863206468, 11.14180002977974230286346796683

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