Properties

Label 4-1900e2-1.1-c1e2-0-0
Degree $4$
Conductor $3610000$
Sign $1$
Analytic cond. $230.176$
Root an. cond. $3.89507$
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 + 12·29-s − 16·31-s + 12·41-s + 10·49-s + 24·59-s + 12·61-s + 16·79-s + 27·81-s − 28·89-s − 48·99-s − 28·101-s + 4·109-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 + 2.22·29-s − 2.87·31-s + 1.87·41-s + 10/7·49-s + 3.12·59-s + 1.53·61-s + 1.80·79-s + 3·81-s − 2.96·89-s − 4.82·99-s − 2.78·101-s + 0.383·109-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

\[\begin{aligned}\Lambda(s)=\mathstrut & 3610000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3610000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3610000\)    =    \(2^{4} \cdot 5^{4} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(230.176\)
Root analytic conductor: \(3.89507\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 3610000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.169698042\)
\(L(\frac12)\) \(\approx\) \(2.169698042\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 42 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 42 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \)
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

−9.484368424030761845389022164638, −9.077293093414878039845065090700, −8.454940702545327898411804806210, −8.274552119508155899957419576267, −7.75649600970488012689430024740, −7.47898526201407817200021756921, −7.02282302209655139132229049351, −6.88673264989188868325553981914, −6.32249610706455623820169931528, −5.58196388526139325212439436531, −5.23966529840681822352569847175, −5.22957222326122652244461581748, −4.31132258830550929197656182777, −4.17357629543407708518549710199, −3.72864693903098481959129775750, −2.93964007791977598814731013201, −2.31587808798972415861147701726, −2.23650415450181533851702708922, −1.26536922468714984063166449458, −0.57436921608937453122972389544, 0.57436921608937453122972389544, 1.26536922468714984063166449458, 2.23650415450181533851702708922, 2.31587808798972415861147701726, 2.93964007791977598814731013201, 3.72864693903098481959129775750, 4.17357629543407708518549710199, 4.31132258830550929197656182777, 5.22957222326122652244461581748, 5.23966529840681822352569847175, 5.58196388526139325212439436531, 6.32249610706455623820169931528, 6.88673264989188868325553981914, 7.02282302209655139132229049351, 7.47898526201407817200021756921, 7.75649600970488012689430024740, 8.274552119508155899957419576267, 8.454940702545327898411804806210, 9.077293093414878039845065090700, 9.484368424030761845389022164638

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