Properties

Label 4-40e4-1.1-c1e2-0-41
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  − 4·3-s + 6·9-s + 4·27-s − 12·41-s − 20·43-s + 14·49-s − 28·67-s − 37·81-s − 36·83-s − 36·89-s − 12·107-s − 14·121-s + 48·123-s + 127-s + 80·129-s + 131-s + 137-s + 139-s − 56·147-s + 149-s + 151-s + 157-s + 163-s + 167-s − 26·169-s + 173-s + 179-s + ⋯
L(s)  = 1  − 2.30·3-s + 2·9-s + 0.769·27-s − 1.87·41-s − 3.04·43-s + 2·49-s − 3.42·67-s − 4.11·81-s − 3.95·83-s − 3.81·89-s − 1.16·107-s − 1.27·121-s + 4.32·123-s + 0.0887·127-s + 7.04·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 4.61·147-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 2·169-s + 0.0760·173-s + 0.0747·179-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: induced by $\chi_{1600} (1, \cdot )$
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$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
7$C_2$ \( ( 1 - p T^{2} )^{2} \)
11$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + p T^{2} )^{2} \)
17$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 - p T^{2} )^{2} \)
29$C_2$ \( ( 1 - p T^{2} )^{2} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - p T^{2} )^{2} \)
53$C_2$ \( ( 1 + p T^{2} )^{2} \)
59$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - p T^{2} )^{2} \)
67$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 142 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2$ \( ( 1 + 18 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + 18 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 94 T^{2} + p^{2} T^{4} \)
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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

−9.269512082871218195265542415710, −8.632998009671044337676015389991, −8.493264023253021968128737306529, −8.167837006198653615517623175890, −7.19053170722770689998093942716, −7.12083300677956764268778798250, −6.78432884768124675850943487843, −6.21806256559437420914353141876, −5.86527169376354659790671520497, −5.64006864436831248426242047187, −5.02968443177532664818177209995, −4.98450710694390351225680959151, −4.29779990415778891390719514373, −3.90871129395749100222680498506, −2.93801831066259447993989928945, −2.86502747756074582961215094783, −1.60047304650876806231188647670, −1.29213611856867358002892438412, 0, 0, 1.29213611856867358002892438412, 1.60047304650876806231188647670, 2.86502747756074582961215094783, 2.93801831066259447993989928945, 3.90871129395749100222680498506, 4.29779990415778891390719514373, 4.98450710694390351225680959151, 5.02968443177532664818177209995, 5.64006864436831248426242047187, 5.86527169376354659790671520497, 6.21806256559437420914353141876, 6.78432884768124675850943487843, 7.12083300677956764268778798250, 7.19053170722770689998093942716, 8.167837006198653615517623175890, 8.493264023253021968128737306529, 8.632998009671044337676015389991, 9.269512082871218195265542415710

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