Properties

Label 4-1080e2-1.1-c1e2-0-31
Degree $4$
Conductor $1166400$
Sign $-1$
Analytic cond. $74.3706$
Root an. cond. $2.93663$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 4·7-s − 4·13-s − 10·19-s + 25-s + 14·31-s − 12·37-s + 8·43-s − 2·49-s − 22·61-s + 28·67-s − 24·73-s − 6·79-s − 16·91-s + 32·97-s + 8·103-s − 38·109-s − 6·121-s + 127-s + 131-s − 40·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯
L(s)  = 1  + 1.51·7-s − 1.10·13-s − 2.29·19-s + 1/5·25-s + 2.51·31-s − 1.97·37-s + 1.21·43-s − 2/7·49-s − 2.81·61-s + 3.42·67-s − 2.80·73-s − 0.675·79-s − 1.67·91-s + 3.24·97-s + 0.788·103-s − 3.63·109-s − 0.545·121-s + 0.0887·127-s + 0.0873·131-s − 3.46·133-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1166400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1166400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1166400\)    =    \(2^{6} \cdot 3^{6} \cdot 5^{2}\)
Sign: $-1$
Analytic conductor: \(74.3706\)
Root analytic conductor: \(2.93663\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 1166400,\ (\ :1/2, 1/2),\ -1)\)

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 \( 1 \)
5$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good7$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
19$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
61$C_2$ \( ( 1 + 11 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
89$C_2$ \( ( 1 + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 16 T + p T^{2} )^{2} \)
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

−8.001744768198934904474847883292, −7.35945403171969665462762828441, −7.02981671194631499195835940912, −6.50781265613608301773830140513, −6.10221061334331611396169073880, −5.60401601702349757480000809362, −4.83906285718461367327183444990, −4.67563018669795991922191356223, −4.47121123035280332113441908167, −3.72366077209677482599481271751, −2.97136231353523248199016031137, −2.35555435096008089827209080495, −1.95034099206603296055540986865, −1.21890687718080188092827090519, 0, 1.21890687718080188092827090519, 1.95034099206603296055540986865, 2.35555435096008089827209080495, 2.97136231353523248199016031137, 3.72366077209677482599481271751, 4.47121123035280332113441908167, 4.67563018669795991922191356223, 4.83906285718461367327183444990, 5.60401601702349757480000809362, 6.10221061334331611396169073880, 6.50781265613608301773830140513, 7.02981671194631499195835940912, 7.35945403171969665462762828441, 8.001744768198934904474847883292

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