Properties

Label 4-52272-1.1-c1e2-0-20
Degree $4$
Conductor $52272$
Sign $-1$
Analytic cond. $3.33290$
Root an. cond. $1.35115$
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  + 3-s − 4·7-s + 9-s − 4·13-s − 12·19-s − 4·21-s − 6·25-s + 27-s − 16·31-s + 20·37-s − 4·39-s − 4·43-s − 2·49-s − 12·57-s + 20·61-s − 4·63-s + 24·67-s + 12·73-s − 6·75-s − 4·79-s + 81-s + 16·91-s − 16·93-s − 4·97-s + 8·103-s − 20·109-s + 20·111-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.51·7-s + 1/3·9-s − 1.10·13-s − 2.75·19-s − 0.872·21-s − 6/5·25-s + 0.192·27-s − 2.87·31-s + 3.28·37-s − 0.640·39-s − 0.609·43-s − 2/7·49-s − 1.58·57-s + 2.56·61-s − 0.503·63-s + 2.93·67-s + 1.40·73-s − 0.692·75-s − 0.450·79-s + 1/9·81-s + 1.67·91-s − 1.65·93-s − 0.406·97-s + 0.788·103-s − 1.91·109-s + 1.89·111-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 52272 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 52272 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(52272\)    =    \(2^{4} \cdot 3^{3} \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(3.33290\)
Root analytic conductor: \(1.35115\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 52272,\ (\ :1/2, 1/2),\ -1)\)

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$C_1$ \( 1 - T \)
11$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
7$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
19$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
31$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
43$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
97$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
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

−9.796917351475791676316160488949, −9.413820553443583654003786378045, −8.932199088171823379479580947914, −8.071554028904627902844738284977, −7.996057650348538062805293058353, −7.05430755180880074285547369845, −6.70453643193197308588262041684, −6.20097915233518496322100006686, −5.57273521417705379772496968279, −4.75118710346056158176317605927, −3.79814948526845934439056144160, −3.79294235641559942977546839028, −2.46483616139600209242493840583, −2.21252199435586523776406167130, 0, 2.21252199435586523776406167130, 2.46483616139600209242493840583, 3.79294235641559942977546839028, 3.79814948526845934439056144160, 4.75118710346056158176317605927, 5.57273521417705379772496968279, 6.20097915233518496322100006686, 6.70453643193197308588262041684, 7.05430755180880074285547369845, 7.996057650348538062805293058353, 8.071554028904627902844738284977, 8.932199088171823379479580947914, 9.413820553443583654003786378045, 9.796917351475791676316160488949

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