Properties

Label 4-384e2-1.1-c1e2-0-3
Degree $4$
Conductor $147456$
Sign $1$
Analytic cond. $9.40192$
Root an. cond. $1.75107$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s − 8·7-s + 9-s + 4·13-s + 4·19-s − 16·21-s − 6·25-s − 4·27-s + 20·37-s + 8·39-s + 12·43-s + 34·49-s + 8·57-s + 4·61-s − 8·63-s + 20·67-s + 28·73-s − 12·75-s − 16·79-s − 11·81-s − 32·91-s − 4·97-s − 8·103-s − 12·109-s + 40·111-s + 4·117-s − 18·121-s + ⋯
L(s)  = 1  + 1.15·3-s − 3.02·7-s + 1/3·9-s + 1.10·13-s + 0.917·19-s − 3.49·21-s − 6/5·25-s − 0.769·27-s + 3.28·37-s + 1.28·39-s + 1.82·43-s + 34/7·49-s + 1.05·57-s + 0.512·61-s − 1.00·63-s + 2.44·67-s + 3.27·73-s − 1.38·75-s − 1.80·79-s − 1.22·81-s − 3.35·91-s − 0.406·97-s − 0.788·103-s − 1.14·109-s + 3.79·111-s + 0.369·117-s − 1.63·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147456 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147456 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(147456\)    =    \(2^{14} \cdot 3^{2}\)
Sign: $1$
Analytic conductor: \(9.40192\)
Root analytic conductor: \(1.75107\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 147456,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.600644157\)
\(L(\frac12)\) \(\approx\) \(1.600644157\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - 2 T + p T^{2} \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
7$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
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$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
97$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
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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

−9.385355573475711088007535446099, −9.115244373205364915675377664046, −8.173183686627627779120693205031, −8.076512848121596140241215200165, −7.36082202049510326881285758548, −6.78571919296342801026858814862, −6.38329769807791755045956385135, −5.82368326970125389675476631831, −5.64130249043000575312616744005, −4.28451177805084716850337467950, −3.67212844043023358887177272155, −3.56276783709589056726776122484, −2.71947017805277709590872673137, −2.46608091328452001690059553408, −0.795101455562395073664105794433, 0.795101455562395073664105794433, 2.46608091328452001690059553408, 2.71947017805277709590872673137, 3.56276783709589056726776122484, 3.67212844043023358887177272155, 4.28451177805084716850337467950, 5.64130249043000575312616744005, 5.82368326970125389675476631831, 6.38329769807791755045956385135, 6.78571919296342801026858814862, 7.36082202049510326881285758548, 8.076512848121596140241215200165, 8.173183686627627779120693205031, 9.115244373205364915675377664046, 9.385355573475711088007535446099

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