Properties

Label 4-40e4-1.1-c1e2-0-44
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 $2$

Origins

Origins of factors

Downloads

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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: \(2\)
Selberg data: \((4,\ 2560000,\ (\ :1/2, 1/2),\ 1)\)

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)$
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$C_2^2$ \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
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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

−9.025637204908989211419191621093, −8.948862844657013768046769506814, −8.406837679272210433957835242484, −7.889536571156655130389814405102, −7.63282397342565243643138374584, −7.28854716051763515581253346644, −6.55940039953861065185824892891, −6.39077459417847177487225285081, −5.86575297264700713333873134176, −5.67345537595415015490197509243, −5.11131520097096109500924912143, −4.51785488697542070280998210141, −4.24887415762888992609219846950, −4.05225121294251739274196101591, −3.10534604220966593671916779248, −2.48054409441270760871203712167, −1.78081875858630517063994731620, −1.56730681119818120057759854084, 0, 0, 1.56730681119818120057759854084, 1.78081875858630517063994731620, 2.48054409441270760871203712167, 3.10534604220966593671916779248, 4.05225121294251739274196101591, 4.24887415762888992609219846950, 4.51785488697542070280998210141, 5.11131520097096109500924912143, 5.67345537595415015490197509243, 5.86575297264700713333873134176, 6.39077459417847177487225285081, 6.55940039953861065185824892891, 7.28854716051763515581253346644, 7.63282397342565243643138374584, 7.889536571156655130389814405102, 8.406837679272210433957835242484, 8.948862844657013768046769506814, 9.025637204908989211419191621093

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