Properties

Label 4-40e4-1.1-c1e2-0-5
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·9-s − 4·13-s − 4·17-s − 12·29-s − 20·37-s + 4·41-s − 6·49-s + 12·53-s + 4·61-s + 12·73-s − 5·81-s + 20·89-s − 4·97-s + 4·101-s + 36·109-s − 4·113-s − 8·117-s + 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 8·153-s + 157-s + 163-s + ⋯
L(s)  = 1  + 2/3·9-s − 1.10·13-s − 0.970·17-s − 2.22·29-s − 3.28·37-s + 0.624·41-s − 6/7·49-s + 1.64·53-s + 0.512·61-s + 1.40·73-s − 5/9·81-s + 2.11·89-s − 0.406·97-s + 0.398·101-s + 3.44·109-s − 0.376·113-s − 0.739·117-s + 0.909·121-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 − 0.646·153-s + 0.0798·157-s + 0.0783·163-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

\(L(1)\) \(\approx\) \(1.150149788\)
\(L(\frac12)\) \(\approx\) \(1.150149788\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
7$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 86 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 + 126 T^{2} + p^{2} T^{4} \)
71$C_2^2$ \( 1 + 110 T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
79$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \)
83$C_2^2$ \( 1 + 158 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.703275820484704096939853742859, −9.217213684924187167006777694099, −8.708559081948524404822471887853, −8.639225088524735331753480486892, −7.990811881348407989967704229473, −7.37401096126552261772733669767, −7.23203599502431104428265703229, −7.03318599604345545892302064245, −6.41845644443075369845279756685, −5.95739101623444747499372868575, −5.43463877895105829439051057151, −5.02131327190238839039908654587, −4.73024301599727083002851187900, −4.13521943015852292143199079077, −3.48643355186219516463291962461, −3.46936882159321081901922128882, −2.24984460088202391526922434250, −2.20933294345756036163890187215, −1.52217646853372676400449939557, −0.40311229020551760365318498096, 0.40311229020551760365318498096, 1.52217646853372676400449939557, 2.20933294345756036163890187215, 2.24984460088202391526922434250, 3.46936882159321081901922128882, 3.48643355186219516463291962461, 4.13521943015852292143199079077, 4.73024301599727083002851187900, 5.02131327190238839039908654587, 5.43463877895105829439051057151, 5.95739101623444747499372868575, 6.41845644443075369845279756685, 7.03318599604345545892302064245, 7.23203599502431104428265703229, 7.37401096126552261772733669767, 7.990811881348407989967704229473, 8.639225088524735331753480486892, 8.708559081948524404822471887853, 9.217213684924187167006777694099, 9.703275820484704096939853742859

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