Properties

Label 4-600e2-1.1-c1e2-0-4
Degree $4$
Conductor $360000$
Sign $1$
Analytic cond. $22.9539$
Root an. cond. $2.18884$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 2·4-s − 3·9-s + 8·11-s − 4·16-s + 6·18-s − 4·19-s − 16·22-s + 8·32-s − 6·36-s + 8·38-s + 12·43-s + 16·44-s − 6·49-s − 8·64-s + 12·67-s − 4·73-s − 8·76-s + 9·81-s − 4·83-s − 24·86-s + 16·89-s − 12·97-s + 12·98-s − 24·99-s − 4·107-s − 8·113-s + ⋯
L(s)  = 1  − 1.41·2-s + 4-s − 9-s + 2.41·11-s − 16-s + 1.41·18-s − 0.917·19-s − 3.41·22-s + 1.41·32-s − 36-s + 1.29·38-s + 1.82·43-s + 2.41·44-s − 6/7·49-s − 64-s + 1.46·67-s − 0.468·73-s − 0.917·76-s + 81-s − 0.439·83-s − 2.58·86-s + 1.69·89-s − 1.21·97-s + 1.21·98-s − 2.41·99-s − 0.386·107-s − 0.752·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(360000\)    =    \(2^{6} \cdot 3^{2} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(22.9539\)
Root analytic conductor: \(2.18884\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 360000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8272479599\)
\(L(\frac12)\) \(\approx\) \(0.8272479599\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + p T + p T^{2} \)
3$C_2$ \( 1 + p T^{2} \)
5 \( 1 \)
good7$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
11$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
19$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
29$C_2^2$ \( 1 - 38 T^{2} + p^{2} T^{4} \)
31$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \)
37$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \)
47$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 74 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
67$C_2$$\times$$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
73$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
79$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
83$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \)
89$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 6 T + p T^{2} ) \)
97$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
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

−8.804400760277296935278472799595, −8.380018784779441788640689352242, −7.998490972581732874153624166283, −7.45198882082106711256292593384, −6.81099659872539151796830733341, −6.58469828728654140933948790347, −6.13034010270847416309161660219, −5.58081699436130103163546295653, −4.79134339715399583259674675515, −4.15149539549220881464426199091, −3.83364358408795496225066049548, −2.96937441874436385115731334599, −2.19325361375638815380037028298, −1.52366117420926900974558058918, −0.68289980570026390974489222967, 0.68289980570026390974489222967, 1.52366117420926900974558058918, 2.19325361375638815380037028298, 2.96937441874436385115731334599, 3.83364358408795496225066049548, 4.15149539549220881464426199091, 4.79134339715399583259674675515, 5.58081699436130103163546295653, 6.13034010270847416309161660219, 6.58469828728654140933948790347, 6.81099659872539151796830733341, 7.45198882082106711256292593384, 7.998490972581732874153624166283, 8.380018784779441788640689352242, 8.804400760277296935278472799595

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