Properties

Label 4-800e2-1.1-c1e2-0-52
Degree $4$
Conductor $640000$
Sign $1$
Analytic cond. $40.8069$
Root an. cond. $2.52745$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Related objects

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 6·9-s − 12·13-s − 4·17-s − 20·29-s + 4·37-s + 20·41-s − 14·49-s − 28·53-s − 20·61-s + 12·73-s + 27·81-s + 20·89-s − 36·97-s − 4·101-s + 12·109-s + 28·113-s + 72·117-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 24·153-s + 157-s + 163-s + ⋯
L(s)  = 1  − 2·9-s − 3.32·13-s − 0.970·17-s − 3.71·29-s + 0.657·37-s + 3.12·41-s − 2·49-s − 3.84·53-s − 2.56·61-s + 1.40·73-s + 3·81-s + 2.11·89-s − 3.65·97-s − 0.398·101-s + 1.14·109-s + 2.63·113-s + 6.65·117-s − 2·121-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 + 1.94·153-s + 0.0798·157-s + 0.0783·163-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 640000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(640000\)    =    \(2^{10} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(40.8069\)
Root analytic conductor: \(2.52745\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{640000} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 640000,\ (\ :1/2, 1/2),\ 1)\)

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$C_2$ \( ( 1 + p T^{2} )^{2} \)
7$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2$ \( ( 1 + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 18 T + p T^{2} )^{2} \)
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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

−12.92623728289713341830407643628, −12.30526715927341231407569907858, −12.30526715927341231407569907858, −11.38174335680955231447991435552, −11.38174335680955231447991435552, −10.84453522425202397161354104633, −10.84453522425202397161354104633, −9.617316516935557446365175892726, −9.617316516935557446365175892726, −9.206579341791145473157585005601, −9.206579341791145473157585005601, −8.025782606763976976877695514876, −8.025782606763976976877695514876, −7.38383189330491254934984805094, −7.38383189330491254934984805094, −6.27997548582365804353754428543, −6.27997548582365804353754428543, −5.37327377173791634385808352079, −5.37327377173791634385808352079, −4.46989807490289628768060263585, −4.46989807490289628768060263585, −3.13241440937412885242101141742, −3.13241440937412885242101141742, −2.12458086750050120053975349819, −2.12458086750050120053975349819, 0, 0, 2.12458086750050120053975349819, 2.12458086750050120053975349819, 3.13241440937412885242101141742, 3.13241440937412885242101141742, 4.46989807490289628768060263585, 4.46989807490289628768060263585, 5.37327377173791634385808352079, 5.37327377173791634385808352079, 6.27997548582365804353754428543, 6.27997548582365804353754428543, 7.38383189330491254934984805094, 7.38383189330491254934984805094, 8.025782606763976976877695514876, 8.025782606763976976877695514876, 9.206579341791145473157585005601, 9.206579341791145473157585005601, 9.617316516935557446365175892726, 9.617316516935557446365175892726, 10.84453522425202397161354104633, 10.84453522425202397161354104633, 11.38174335680955231447991435552, 11.38174335680955231447991435552, 12.30526715927341231407569907858, 12.30526715927341231407569907858, 12.92623728289713341830407643628

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