↑

Properties

Label 4-160e2-1.1-c0e2-0-0
Degree $4$
Conductor $25600$
Sign $1$
Analytic cond. $0.00637608$
Root an. cond. $0.282578$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2·13-s − 2·17-s − 25-s + 2·37-s + 2·53-s + 2·73-s − 81-s + 2·97-s − 4·101-s − 2·113-s − 2·121-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 + 191-s + 193-s + 197-s + ⋯
L(s)  = 1  − 2·13-s − 2·17-s − 25-s + 2·37-s + 2·53-s + 2·73-s − 81-s + 2·97-s − 4·101-s − 2·113-s − 2·121-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 + 191-s + 193-s + 197-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 25600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(25600\)    =    \(2^{10} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(0.00637608\)
Root analytic conductor: \(0.282578\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 25600,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4041864365\)
\(L(\frac12)\) \(\approx\) \(0.4041864365\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
5$C_2$ \( 1 + T^{2} \)
good3$C_2^2$ \( 1 + T^{4} \)
7$C_2^2$ \( 1 + T^{4} \)
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_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
31$C_2$ \( ( 1 + T^{2} )^{2} \)
37$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
41$C_2$ \( ( 1 + T^{2} )^{2} \)
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_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
97$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.22226723029430935142271601052, −13.06560846434277772522137725342, −12.21353085036213570284905205661, −12.05397578143127107551365370655, −11.29170416241625925518280239708, −11.07925748798546145881732505980, −10.24071687327993332863047005091, −9.913649929430436329933245040040, −9.221074973098363193163920528040, −9.047836141025918656673687999356, −7.980117240652363769239370864423, −7.85863292749122721691212523124, −6.83939292138785200432351293434, −6.80059945319875688719833291978, −5.80140199949532992853905517256, −5.19020953810386242049906005307, −4.43966521530663007674469412826, −4.03087968401882776601714179342, −2.66147261116974021335730422679, −2.21888865321118726163530306768, 2.21888865321118726163530306768, 2.66147261116974021335730422679, 4.03087968401882776601714179342, 4.43966521530663007674469412826, 5.19020953810386242049906005307, 5.80140199949532992853905517256, 6.80059945319875688719833291978, 6.83939292138785200432351293434, 7.85863292749122721691212523124, 7.980117240652363769239370864423, 9.047836141025918656673687999356, 9.221074973098363193163920528040, 9.913649929430436329933245040040, 10.24071687327993332863047005091, 11.07925748798546145881732505980, 11.29170416241625925518280239708, 12.05397578143127107551365370655, 12.21353085036213570284905205661, 13.06560846434277772522137725342, 13.22226723029430935142271601052

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