Properties

Label 4-18432-1.1-c1e2-0-8
Degree $4$
Conductor $18432$
Sign $-1$
Analytic cond. $1.17524$
Root an. cond. $1.04119$
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  − 4·5-s − 3·9-s + 2·25-s − 20·29-s + 12·45-s − 14·49-s + 28·53-s − 12·73-s + 9·81-s + 36·97-s − 4·101-s − 22·121-s + 28·125-s + 127-s + 131-s + 137-s + 139-s + 80·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-s + 173-s + 179-s + 181-s + ⋯
L(s)  = 1  − 1.78·5-s − 9-s + 2/5·25-s − 3.71·29-s + 1.78·45-s − 2·49-s + 3.84·53-s − 1.40·73-s + 81-s + 3.65·97-s − 0.398·101-s − 2·121-s + 2.50·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 6.64·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.769·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 18432 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18432 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(18432\)    =    \(2^{11} \cdot 3^{2}\)
Sign: $-1$
Analytic conductor: \(1.17524\)
Root analytic conductor: \(1.04119\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 18432,\ (\ :1/2, 1/2),\ -1)\)

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$C_2$ \( 1 + p T^{2} \)
good5$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
7$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
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$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
43$C_2$ \( ( 1 + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
67$C_2$ \( ( 1 + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2$ \( ( 1 + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 18 T + p T^{2} )^{2} \)
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

−10.90769214371221130983350005998, −10.23383425889508895415100406682, −9.468761705895534248392912886381, −8.955386231165229198073332132052, −8.439450125780651634799093893382, −7.77199473906097062385997282225, −7.52766744709985624839545780760, −6.93944627141196287077129229697, −5.87146418848833687506982135026, −5.54216083419874931368270995905, −4.56897038264081606692544495668, −3.67478222653086463350186782835, −3.55808048734848453068190211023, −2.19360926670427462073525005533, 0, 2.19360926670427462073525005533, 3.55808048734848453068190211023, 3.67478222653086463350186782835, 4.56897038264081606692544495668, 5.54216083419874931368270995905, 5.87146418848833687506982135026, 6.93944627141196287077129229697, 7.52766744709985624839545780760, 7.77199473906097062385997282225, 8.439450125780651634799093893382, 8.955386231165229198073332132052, 9.468761705895534248392912886381, 10.23383425889508895415100406682, 10.90769214371221130983350005998

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