Properties

Label 4-1200e2-1.1-c1e2-0-7
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

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 9-s − 12·11-s + 10·19-s + 12·29-s + 2·31-s + 13·49-s − 12·59-s − 26·61-s + 16·79-s + 81-s + 12·99-s − 24·101-s + 14·109-s + 86·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s − 10·171-s + 173-s + 179-s + ⋯
L(s)  = 1  − 1/3·9-s − 3.61·11-s + 2.29·19-s + 2.22·29-s + 0.359·31-s + 13/7·49-s − 1.56·59-s − 3.32·61-s + 1.80·79-s + 1/9·81-s + 1.20·99-s − 2.38·101-s + 1.34·109-s + 7.81·121-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 + 1/13·169-s − 0.764·171-s + 0.0760·173-s + 0.0747·179-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: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1440000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.177827106\)
\(L(\frac12)\) \(\approx\) \(1.177827106\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
5 \( 1 \)
good7$C_2^2$ \( 1 - 13 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 85 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 13 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 13 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 142 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 145 T^{2} + p^{2} T^{4} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.959717629901749504190177564788, −9.742529921245462153067943482561, −9.026110966715894208987583814102, −8.794207683349364235624498809072, −8.027814440861803343804966104436, −7.967030258758887959038005237918, −7.56017366500586457688626847357, −7.36044079011193432919257285284, −6.64003420502952692194388130617, −6.01183768748722130748896607551, −5.72557191323100581696881148215, −5.19477180863916879687016083153, −4.84624624232527391956809797003, −4.72373992798063128931570292863, −3.69177777094725348577589227745, −3.01295475985222491424249264213, −2.72711960451348385144025008890, −2.54407075065313153693300716550, −1.39129501132030225555070081083, −0.47598048900205289374967925284, 0.47598048900205289374967925284, 1.39129501132030225555070081083, 2.54407075065313153693300716550, 2.72711960451348385144025008890, 3.01295475985222491424249264213, 3.69177777094725348577589227745, 4.72373992798063128931570292863, 4.84624624232527391956809797003, 5.19477180863916879687016083153, 5.72557191323100581696881148215, 6.01183768748722130748896607551, 6.64003420502952692194388130617, 7.36044079011193432919257285284, 7.56017366500586457688626847357, 7.967030258758887959038005237918, 8.027814440861803343804966104436, 8.794207683349364235624498809072, 9.026110966715894208987583814102, 9.742529921245462153067943482561, 9.959717629901749504190177564788

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