Properties

Label 4-40e4-1.1-c1e2-0-22
Degree $4$
Conductor $2560000$
Sign $1$
Analytic cond. $163.227$
Root an. cond. $3.57436$
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  + 2·3-s − 2·7-s + 2·9-s − 2·13-s − 2·17-s + 8·19-s − 4·21-s + 10·23-s + 6·27-s − 10·37-s − 4·39-s + 12·41-s − 6·43-s + 14·47-s + 2·49-s − 4·51-s − 2·53-s + 16·57-s + 8·59-s − 4·61-s − 4·63-s + 14·67-s + 20·69-s − 18·73-s + 16·79-s + 11·81-s + 10·83-s + ⋯
L(s)  = 1  + 1.15·3-s − 0.755·7-s + 2/3·9-s − 0.554·13-s − 0.485·17-s + 1.83·19-s − 0.872·21-s + 2.08·23-s + 1.15·27-s − 1.64·37-s − 0.640·39-s + 1.87·41-s − 0.914·43-s + 2.04·47-s + 2/7·49-s − 0.560·51-s − 0.274·53-s + 2.11·57-s + 1.04·59-s − 0.512·61-s − 0.503·63-s + 1.71·67-s + 2.40·69-s − 2.10·73-s + 1.80·79-s + 11/9·81-s + 1.09·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2560000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2560000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2560000\)    =    \(2^{12} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(163.227\)
Root analytic conductor: \(3.57436\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2560000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.541791827\)
\(L(\frac12)\) \(\approx\) \(3.541791827\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good3$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
7$C_2^2$ \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2^2$ \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
31$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 - 106 T^{2} + p^{2} T^{4} \)
73$C_2^2$ \( 1 + 18 T + 162 T^{2} + 18 p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
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

−9.399138532315601929631861252584, −9.243072371392499420253651797293, −8.797798566465001675462334830021, −8.683378625186750792913895541591, −7.908526189653864853575777906300, −7.69670304685218610623931435606, −7.16307409618371448895878095344, −6.93727411485892768383615336976, −6.62869100103402877147323772212, −5.95393472857215121955830003603, −5.33786599831243196001657380320, −5.15604000074851863198251272913, −4.59793702833499293242457194665, −4.01837887188466131351765351014, −3.34403631815961901670637454084, −3.28253317423061125992855079629, −2.55911487843505104698930491476, −2.39312566231043210505475636148, −1.33689939001090117967721838301, −0.73496640570089924081057845686, 0.73496640570089924081057845686, 1.33689939001090117967721838301, 2.39312566231043210505475636148, 2.55911487843505104698930491476, 3.28253317423061125992855079629, 3.34403631815961901670637454084, 4.01837887188466131351765351014, 4.59793702833499293242457194665, 5.15604000074851863198251272913, 5.33786599831243196001657380320, 5.95393472857215121955830003603, 6.62869100103402877147323772212, 6.93727411485892768383615336976, 7.16307409618371448895878095344, 7.69670304685218610623931435606, 7.908526189653864853575777906300, 8.683378625186750792913895541591, 8.797798566465001675462334830021, 9.243072371392499420253651797293, 9.399138532315601929631861252584

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