Properties

Label 4-1200e2-1.1-c1e2-0-19
Degree $4$
Conductor $1440000$
Sign $1$
Analytic cond. $91.8156$
Root an. cond. $3.09548$
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  − 3·9-s + 10·13-s + 20·37-s + 11·49-s − 26·61-s − 20·73-s + 9·81-s + 10·97-s + 38·109-s − 30·117-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 49·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯
L(s)  = 1  − 9-s + 2.77·13-s + 3.28·37-s + 11/7·49-s − 3.32·61-s − 2.34·73-s + 81-s + 1.01·97-s + 3.63·109-s − 2.77·117-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 + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 3.76·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1440000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1440000\)    =    \(2^{8} \cdot 3^{2} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(91.8156\)
Root analytic conductor: \(3.09548\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1200} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1440000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.370977287\)
\(L(\frac12)\) \(\approx\) \(2.370977287\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3$C_2$ \( 1 + p T^{2} \)
5 \( 1 \)
good7$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - p T^{2} )^{2} \)
19$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - p T^{2} )^{2} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 13 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
83$C_2$ \( ( 1 + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
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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

−10.04192331602531944883277762109, −9.235249485033766911680955272731, −9.110824401300649064494489415528, −8.850439930202716412889193890995, −8.351007169137469552578276505397, −7.86426106080415528937829408504, −7.73886512552691845048061528922, −7.06933300342397849113927914571, −6.35857194223539498169326806148, −6.16909240897454492883685038915, −5.80447354023165349599257911722, −5.61799415787528107734284135978, −4.63063445303689785949919255565, −4.35004484194610625156092304974, −3.85588443782433436363421834666, −3.11337740610490125154215786245, −3.02073701919884995268063813039, −2.13886360736245894790926960656, −1.34931575640980451940120621710, −0.73445572320676177890159626893, 0.73445572320676177890159626893, 1.34931575640980451940120621710, 2.13886360736245894790926960656, 3.02073701919884995268063813039, 3.11337740610490125154215786245, 3.85588443782433436363421834666, 4.35004484194610625156092304974, 4.63063445303689785949919255565, 5.61799415787528107734284135978, 5.80447354023165349599257911722, 6.16909240897454492883685038915, 6.35857194223539498169326806148, 7.06933300342397849113927914571, 7.73886512552691845048061528922, 7.86426106080415528937829408504, 8.351007169137469552578276505397, 8.850439930202716412889193890995, 9.110824401300649064494489415528, 9.235249485033766911680955272731, 10.04192331602531944883277762109

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