Properties

Label 4-3800e2-1.1-c1e2-0-5
Degree $4$
Conductor $14440000$
Sign $1$
Analytic cond. $920.706$
Root an. cond. $5.50846$
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  + 6·9-s − 8·11-s + 2·19-s + 4·29-s + 4·41-s + 14·49-s + 8·59-s − 4·61-s + 16·71-s + 16·79-s + 27·81-s + 28·89-s − 48·99-s − 20·101-s + 20·109-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 10·169-s + 12·171-s + ⋯
L(s)  = 1  + 2·9-s − 2.41·11-s + 0.458·19-s + 0.742·29-s + 0.624·41-s + 2·49-s + 1.04·59-s − 0.512·61-s + 1.89·71-s + 1.80·79-s + 3·81-s + 2.96·89-s − 4.82·99-s − 1.99·101-s + 1.91·109-s + 2.36·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-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 + 0.917·171-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 14440000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14440000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(14440000\)    =    \(2^{6} \cdot 5^{4} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(920.706\)
Root analytic conductor: \(5.50846\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 14440000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.240053747\)
\(L(\frac12)\) \(\approx\) \(3.240053747\)
\(L(\frac{3}{2})\) 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 \( 1 \)
19$C_1$ \( ( 1 - T )^{2} \)
good3$C_2$ \( ( 1 - p T^{2} )^{2} \)
7$C_2$ \( ( 1 - p T^{2} )^{2} \)
11$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 142 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 150 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
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

−8.508553586204081158508369390371, −8.282863763175222891834331344165, −7.83020251693616215678729509286, −7.54903438877418717676603420192, −7.32378912793155189945924058663, −7.01591283215782530277463636810, −6.42966035251266168333488596484, −6.21209353945122681868230499151, −5.54375321471243224689918990874, −5.25405959809490831572889143135, −4.89411199117978251148315187504, −4.66331988457416655776782236872, −3.98656084744732598759965771930, −3.83164515058154759077597532364, −3.13540596757981450058861795088, −2.72819593184818726856591699654, −2.04618870146160148589825248941, −2.04159280703731431639476203844, −0.922003547097038685831122276390, −0.66631612904177261118759649419, 0.66631612904177261118759649419, 0.922003547097038685831122276390, 2.04159280703731431639476203844, 2.04618870146160148589825248941, 2.72819593184818726856591699654, 3.13540596757981450058861795088, 3.83164515058154759077597532364, 3.98656084744732598759965771930, 4.66331988457416655776782236872, 4.89411199117978251148315187504, 5.25405959809490831572889143135, 5.54375321471243224689918990874, 6.21209353945122681868230499151, 6.42966035251266168333488596484, 7.01591283215782530277463636810, 7.32378912793155189945924058663, 7.54903438877418717676603420192, 7.83020251693616215678729509286, 8.282863763175222891834331344165, 8.508553586204081158508369390371

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