Properties

Degree $4$
Conductor $230400$
Sign $-1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s − 2·5-s + 3·9-s − 4·13-s − 4·15-s − 25-s + 4·27-s − 16·31-s + 12·37-s − 8·39-s − 12·41-s − 8·43-s − 6·45-s − 14·49-s − 4·53-s + 8·65-s + 8·67-s − 16·71-s − 2·75-s + 16·79-s + 5·81-s + 8·83-s − 12·89-s − 32·93-s + 24·107-s + 24·111-s − 12·117-s + ⋯
L(s)  = 1  + 1.15·3-s − 0.894·5-s + 9-s − 1.10·13-s − 1.03·15-s − 1/5·25-s + 0.769·27-s − 2.87·31-s + 1.97·37-s − 1.28·39-s − 1.87·41-s − 1.21·43-s − 0.894·45-s − 2·49-s − 0.549·53-s + 0.992·65-s + 0.977·67-s − 1.89·71-s − 0.230·75-s + 1.80·79-s + 5/9·81-s + 0.878·83-s − 1.27·89-s − 3.31·93-s + 2.32·107-s + 2.27·111-s − 1.10·117-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(230400\)    =    \(2^{10} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{230400} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 230400,\ (\ :1/2, 1/2),\ -1)\)

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 \( 1 \)
3$C_1$ \( ( 1 - T )^{2} \)
5$C_2$ \( 1 + 2 T + p T^{2} \)
good7$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
show more
show less
   \(L(s) = \displaystyle\prod_{\mathfrak{p}\ \mathrm{bad}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s})^{-1} \prod_{\mathfrak{p}\ \mathrm{good}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s} + (N\mathfrak{p})^{-2s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.681842007730582707537242091206, −8.173282349834764667604075089133, −7.953704755279657121335115603175, −7.26958585488936859387169650645, −7.25136384633542617765663547930, −6.50923348756050362748698725466, −5.87866003990815751467424375923, −5.01357758335939619070346679818, −4.79956865275231776293703222232, −4.02593153952443544832427859826, −3.48889300562368399939117276327, −3.14826068230388029805668446658, −2.23734692694206799493488443258, −1.66440253308003017822578074122, 0, 1.66440253308003017822578074122, 2.23734692694206799493488443258, 3.14826068230388029805668446658, 3.48889300562368399939117276327, 4.02593153952443544832427859826, 4.79956865275231776293703222232, 5.01357758335939619070346679818, 5.87866003990815751467424375923, 6.50923348756050362748698725466, 7.25136384633542617765663547930, 7.26958585488936859387169650645, 7.953704755279657121335115603175, 8.173282349834764667604075089133, 8.681842007730582707537242091206

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