Properties

Label 4-1400e2-1.1-c1e2-0-5
Degree $4$
Conductor $1960000$
Sign $1$
Analytic cond. $124.971$
Root an. cond. $3.34350$
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  + 3·3-s − 4·7-s + 3·9-s + 11-s − 4·13-s + 3·17-s − 5·19-s − 12·21-s − 3·23-s − 12·29-s + 31-s + 3·33-s − 5·37-s − 12·39-s − 20·41-s + 8·43-s + 47-s + 9·49-s + 9·51-s − 9·53-s − 15·57-s − 3·59-s − 3·61-s − 12·63-s + 11·67-s − 9·69-s + 32·71-s + ⋯
L(s)  = 1  + 1.73·3-s − 1.51·7-s + 9-s + 0.301·11-s − 1.10·13-s + 0.727·17-s − 1.14·19-s − 2.61·21-s − 0.625·23-s − 2.22·29-s + 0.179·31-s + 0.522·33-s − 0.821·37-s − 1.92·39-s − 3.12·41-s + 1.21·43-s + 0.145·47-s + 9/7·49-s + 1.26·51-s − 1.23·53-s − 1.98·57-s − 0.390·59-s − 0.384·61-s − 1.51·63-s + 1.34·67-s − 1.08·69-s + 3.79·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1960000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1960000\)    =    \(2^{6} \cdot 5^{4} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(124.971\)
Root analytic conductor: \(3.34350\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1960000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.727315140\)
\(L(\frac12)\) \(\approx\) \(1.727315140\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7$C_2$ \( 1 + 4 T + p T^{2} \)
good3$C_2$ \( ( 1 - p T + p T^{2} )( 1 + p T^{2} ) \)
11$C_2^2$ \( 1 - T - 10 T^{2} - p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 5 T + 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 3 T - 14 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 - T - 30 T^{2} - p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 5 T - 12 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - T - 46 T^{2} - p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 9 T + 28 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 3 T - 50 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 3 T - 52 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 16 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
79$C_2^2$ \( 1 - 11 T + 42 T^{2} - 11 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 9 T - 8 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
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

−9.951557107883136220901478614631, −9.384173760551883886338485093543, −9.116127967281315344394440667656, −8.452417284738415844048676469056, −8.314955947288639081695278425182, −7.77954381182709700534669919593, −7.53918369702528869890902434783, −6.79667888119964279304563455634, −6.73143181364307256976399491170, −6.21638233515775495439081897598, −5.62109851521756496915958159966, −5.14011500336487292009151597330, −4.67334725040166923800437325202, −3.74285229862067030346367920367, −3.60525821180646879202693484188, −3.38982081492977341361608835567, −2.70338721237779144279009942853, −2.01271593918107366333146534594, −2.01147723082133800464218375439, −0.43985898638767071116077523231, 0.43985898638767071116077523231, 2.01147723082133800464218375439, 2.01271593918107366333146534594, 2.70338721237779144279009942853, 3.38982081492977341361608835567, 3.60525821180646879202693484188, 3.74285229862067030346367920367, 4.67334725040166923800437325202, 5.14011500336487292009151597330, 5.62109851521756496915958159966, 6.21638233515775495439081897598, 6.73143181364307256976399491170, 6.79667888119964279304563455634, 7.53918369702528869890902434783, 7.77954381182709700534669919593, 8.314955947288639081695278425182, 8.452417284738415844048676469056, 9.116127967281315344394440667656, 9.384173760551883886338485093543, 9.951557107883136220901478614631

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