Properties

Label 4-24e4-1.1-c1e2-0-23
Degree $4$
Conductor $331776$
Sign $1$
Analytic cond. $21.1543$
Root an. cond. $2.14461$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 8·11-s + 4·13-s − 8·23-s − 2·25-s + 4·37-s + 8·47-s + 2·49-s + 4·61-s − 8·71-s + 12·73-s + 8·83-s + 12·97-s + 16·107-s − 12·109-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 32·143-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + ⋯
L(s)  = 1  + 2.41·11-s + 1.10·13-s − 1.66·23-s − 2/5·25-s + 0.657·37-s + 1.16·47-s + 2/7·49-s + 0.512·61-s − 0.949·71-s + 1.40·73-s + 0.878·83-s + 1.21·97-s + 1.54·107-s − 1.14·109-s + 2.36·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 2.67·143-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2/13·169-s + 0.0760·173-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 331776 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 331776 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(331776\)    =    \(2^{12} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(21.1543\)
Root analytic conductor: \(2.14461\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 331776,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.246784987\)
\(L(\frac12)\) \(\approx\) \(2.246784987\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good5$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
13$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
23$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \)
29$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
31$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
37$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
43$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \)
47$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \)
53$C_2^2$ \( 1 - 78 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 + 54 T^{2} + p^{2} T^{4} \)
71$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \)
73$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
79$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \)
83$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
89$C_2^2$ \( 1 + 82 T^{2} + p^{2} T^{4} \)
97$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.860293623907220205425142464779, −8.359928029079382740491318287777, −7.87442979793193114050909529796, −7.40651282791721830848811722212, −6.71605745042668346866195445671, −6.42006005986117181844411854997, −6.01521810197558518898996737262, −5.63465979451754059328128876241, −4.76723819984032771797850442201, −4.06104331164239983333717797274, −3.91493062249936280240153026092, −3.42556204045955455355073800536, −2.37750486710140961706976126001, −1.66681127077403182119774979601, −0.947635099091523370166355272990, 0.947635099091523370166355272990, 1.66681127077403182119774979601, 2.37750486710140961706976126001, 3.42556204045955455355073800536, 3.91493062249936280240153026092, 4.06104331164239983333717797274, 4.76723819984032771797850442201, 5.63465979451754059328128876241, 6.01521810197558518898996737262, 6.42006005986117181844411854997, 6.71605745042668346866195445671, 7.40651282791721830848811722212, 7.87442979793193114050909529796, 8.359928029079382740491318287777, 8.860293623907220205425142464779

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