Properties

Label 4-320e2-1.1-c1e2-0-12
Degree $4$
Conductor $102400$
Sign $1$
Analytic cond. $6.52911$
Root an. cond. $1.59850$
Motivic weight $1$
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  − 2·5-s + 6·9-s − 25-s + 20·29-s − 20·41-s − 12·45-s + 14·49-s − 20·61-s + 27·81-s + 20·89-s + 4·101-s + 12·109-s − 22·121-s + 12·125-s + 127-s + 131-s + 137-s + 139-s − 40·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-s + 173-s + 179-s + ⋯
L(s)  = 1  − 0.894·5-s + 2·9-s − 1/5·25-s + 3.71·29-s − 3.12·41-s − 1.78·45-s + 2·49-s − 2.56·61-s + 3·81-s + 2.11·89-s + 0.398·101-s + 1.14·109-s − 2·121-s + 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 3.32·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.769·169-s + 0.0760·173-s + 0.0747·179-s + ⋯

Functional equation

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

Invariants

Degree: \(4\)
Conductor: \(102400\)    =    \(2^{12} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(6.52911\)
Root analytic conductor: \(1.59850\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 102400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.537338284\)
\(L(\frac12)\) \(\approx\) \(1.537338284\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + 2 T + p T^{2} \)
good3$C_2$ \( ( 1 - p T^{2} )^{2} \)
7$C_2$ \( ( 1 - p T^{2} )^{2} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
41$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - p T^{2} )^{2} \)
47$C_2$ \( ( 1 - p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \)
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

−11.84408354461813868935945368764, −11.74444734168646895307638647682, −10.66335378443347160423268801462, −10.39603958736884734475203446268, −10.24746998899160334246831870530, −9.662897648802038270966265082053, −8.964854998704914846361387367472, −8.583811000670341614167614238718, −7.900066297661956208511967360682, −7.71380422747132694341431362028, −6.91429802711173902600523257175, −6.75582600729857951182517827055, −6.20057686725751914499571541580, −5.20800393949219296852719967035, −4.52579293529095231292682208738, −4.47609537034312021662712926526, −3.62020807869565141328442857275, −3.03186958667187543938212924882, −1.92886127396355264597428082082, −0.989944096178904175251613966692, 0.989944096178904175251613966692, 1.92886127396355264597428082082, 3.03186958667187543938212924882, 3.62020807869565141328442857275, 4.47609537034312021662712926526, 4.52579293529095231292682208738, 5.20800393949219296852719967035, 6.20057686725751914499571541580, 6.75582600729857951182517827055, 6.91429802711173902600523257175, 7.71380422747132694341431362028, 7.900066297661956208511967360682, 8.583811000670341614167614238718, 8.964854998704914846361387367472, 9.662897648802038270966265082053, 10.24746998899160334246831870530, 10.39603958736884734475203446268, 10.66335378443347160423268801462, 11.74444734168646895307638647682, 11.84408354461813868935945368764

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