Properties

Label 4-2001e2-1.1-c1e2-0-2
Degree $4$
Conductor $4004001$
Sign $1$
Analytic cond. $255.298$
Root an. cond. $3.99725$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s − 4·4-s − 4·7-s + 3·9-s − 8·12-s + 2·13-s + 12·16-s + 2·17-s − 2·19-s − 8·21-s − 2·23-s − 4·25-s + 4·27-s + 16·28-s + 2·29-s − 16·31-s − 12·36-s + 10·37-s + 4·39-s − 4·41-s − 10·43-s − 20·47-s + 24·48-s + 4·49-s + 4·51-s − 8·52-s − 4·53-s + ⋯
L(s)  = 1  + 1.15·3-s − 2·4-s − 1.51·7-s + 9-s − 2.30·12-s + 0.554·13-s + 3·16-s + 0.485·17-s − 0.458·19-s − 1.74·21-s − 0.417·23-s − 4/5·25-s + 0.769·27-s + 3.02·28-s + 0.371·29-s − 2.87·31-s − 2·36-s + 1.64·37-s + 0.640·39-s − 0.624·41-s − 1.52·43-s − 2.91·47-s + 3.46·48-s + 4/7·49-s + 0.560·51-s − 1.10·52-s − 0.549·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4004001 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004001 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(4004001\)    =    \(3^{2} \cdot 23^{2} \cdot 29^{2}\)
Sign: $1$
Analytic conductor: \(255.298\)
Root analytic conductor: \(3.99725\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 4004001,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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)$
bad3$C_1$ \( ( 1 - T )^{2} \)
23$C_1$ \( ( 1 + T )^{2} \)
29$C_1$ \( ( 1 - T )^{2} \)
good2$C_2$ \( ( 1 + p T^{2} )^{2} \)
5$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \)
7$D_{4}$ \( 1 + 4 T + 12 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
17$D_{4}$ \( 1 - 2 T + 29 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
31$D_{4}$ \( 1 + 16 T + 120 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - 10 T + 75 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 4 T + 32 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 10 T + 87 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 20 T + 4 p T^{2} + 20 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
59$D_{4}$ \( 1 + 6 T + 121 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 16 T + 162 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 8 T + 126 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 10 T + 113 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 4 T + 4 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 2 T + 135 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 20 T + 260 T^{2} + 20 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 6 T + 181 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 24 T + 314 T^{2} - 24 p T^{3} + p^{2} T^{4} \)
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

−9.104795992674507996002777626039, −8.675678596954291857478276909862, −8.209453235055108870077734210692, −7.956326453341084321704128478652, −7.65654790965138032112514291698, −7.15386939190466272148671191760, −6.45808048593907849488503444264, −6.23736341909999860013623991370, −5.73980940392516242861162408648, −5.33961672601415239137363710270, −4.61346084806988000078946641270, −4.51904068327314592633435541565, −3.75858411052598056594396779379, −3.63093431891875533086220189659, −3.13760818701733725459935368523, −2.94010847927804657570018077431, −1.68640674939053920681755660529, −1.48482094009180195888644385117, 0, 0, 1.48482094009180195888644385117, 1.68640674939053920681755660529, 2.94010847927804657570018077431, 3.13760818701733725459935368523, 3.63093431891875533086220189659, 3.75858411052598056594396779379, 4.51904068327314592633435541565, 4.61346084806988000078946641270, 5.33961672601415239137363710270, 5.73980940392516242861162408648, 6.23736341909999860013623991370, 6.45808048593907849488503444264, 7.15386939190466272148671191760, 7.65654790965138032112514291698, 7.956326453341084321704128478652, 8.209453235055108870077734210692, 8.675678596954291857478276909862, 9.104795992674507996002777626039

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