Properties

Label 4-384e2-1.1-c1e2-0-26
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 9-s + 12·13-s + 12·17-s − 10·25-s + 8·29-s + 4·37-s − 4·41-s − 10·49-s − 24·53-s + 4·61-s − 20·73-s + 81-s + 4·89-s − 12·97-s + 8·101-s − 4·109-s − 28·113-s + 12·117-s − 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 12·153-s + 157-s + ⋯
L(s)  = 1  + 1/3·9-s + 3.32·13-s + 2.91·17-s − 2·25-s + 1.48·29-s + 0.657·37-s − 0.624·41-s − 1.42·49-s − 3.29·53-s + 0.512·61-s − 2.34·73-s + 1/9·81-s + 0.423·89-s − 1.21·97-s + 0.796·101-s − 0.383·109-s − 2.63·113-s + 1.10·117-s − 0.545·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.970·153-s + 0.0798·157-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: $\chi_{147456} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 147456,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.146294034\)
\(L(\frac12)\) \(\approx\) \(2.146294034\)
\(L(\frac{3}{2})\) 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$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good5$C_2$ \( ( 1 + p T^{2} )^{2} \)
7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
53$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
73$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
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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

−9.464663439718228165986962411065, −8.658903060264415544055202217864, −8.272861522300808832842531458007, −7.83756825226352882885863464519, −7.70036027517747780731408099423, −6.52243750033071865852983380426, −6.40139715881748782453956361829, −5.75568326693151721019291253980, −5.52928834758807112839354938677, −4.57667813854686016923499181308, −3.94434944725762226728078180666, −3.36330701716196318281397292070, −3.12662782741322079682173901720, −1.50413098873776121963751941726, −1.27109672072212401682622254955, 1.27109672072212401682622254955, 1.50413098873776121963751941726, 3.12662782741322079682173901720, 3.36330701716196318281397292070, 3.94434944725762226728078180666, 4.57667813854686016923499181308, 5.52928834758807112839354938677, 5.75568326693151721019291253980, 6.40139715881748782453956361829, 6.52243750033071865852983380426, 7.70036027517747780731408099423, 7.83756825226352882885863464519, 8.272861522300808832842531458007, 8.658903060264415544055202217864, 9.464663439718228165986962411065

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