Properties

Label 4-3e6-1.1-c1e2-0-0
Degree $4$
Conductor $729$
Sign $1$
Analytic cond. $0.0464816$
Root an. cond. $0.464323$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

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Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·4-s − 2·7-s + 10·13-s + 12·16-s − 14·19-s − 10·25-s + 8·28-s − 8·31-s + 22·37-s + 16·43-s − 11·49-s − 40·52-s − 2·61-s − 32·64-s + 10·67-s − 14·73-s + 56·76-s + 34·79-s − 20·91-s − 38·97-s + 40·100-s − 26·103-s + 4·109-s − 24·112-s − 22·121-s + 32·124-s + 127-s + ⋯
L(s)  = 1  − 2·4-s − 0.755·7-s + 2.77·13-s + 3·16-s − 3.21·19-s − 2·25-s + 1.51·28-s − 1.43·31-s + 3.61·37-s + 2.43·43-s − 1.57·49-s − 5.54·52-s − 0.256·61-s − 4·64-s + 1.22·67-s − 1.63·73-s + 6.42·76-s + 3.82·79-s − 2.09·91-s − 3.85·97-s + 4·100-s − 2.56·103-s + 0.383·109-s − 2.26·112-s − 2·121-s + 2.87·124-s + 0.0887·127-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(729\)    =    \(3^{6}\)
Sign: $1$
Analytic conductor: \(0.0464816\)
Root analytic conductor: \(0.464323\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{729} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 729,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3467791637\)
\(L(\frac12)\) \(\approx\) \(0.3467791637\)
\(L(\frac{3}{2})\) 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)$
bad3 \( 1 \)
good2$C_2$ \( ( 1 + p T^{2} )^{2} \)
5$C_2$ \( ( 1 + p T^{2} )^{2} \)
7$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 17 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 19 T + p T^{2} )^{2} \)
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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

−21.00705820570875902116997986320, −19.50763388930887128039872729916, −19.50763388930887128039872729916, −18.55924110538682425020987188449, −18.55924110538682425020987188449, −17.54958210256512249523818407718, −17.54958210256512249523818407718, −16.30001713525188163931408346208, −16.30001713525188163931408346208, −14.89211076740924954992271086519, −14.89211076740924954992271086519, −13.58211369833951574831601484907, −13.58211369833951574831601484907, −12.71563949069013251763955822657, −12.71563949069013251763955822657, −10.90872829298908742564232271224, −10.90872829298908742564232271224, −9.429199208210393651255146412199, −9.429199208210393651255146412199, −8.217650367462526737991465554229, −8.217650367462526737991465554229, −6.04893540000987436402649531985, −6.04893540000987436402649531985, −4.04304401379743272242521882180, −4.04304401379743272242521882180, 4.04304401379743272242521882180, 4.04304401379743272242521882180, 6.04893540000987436402649531985, 6.04893540000987436402649531985, 8.217650367462526737991465554229, 8.217650367462526737991465554229, 9.429199208210393651255146412199, 9.429199208210393651255146412199, 10.90872829298908742564232271224, 10.90872829298908742564232271224, 12.71563949069013251763955822657, 12.71563949069013251763955822657, 13.58211369833951574831601484907, 13.58211369833951574831601484907, 14.89211076740924954992271086519, 14.89211076740924954992271086519, 16.30001713525188163931408346208, 16.30001713525188163931408346208, 17.54958210256512249523818407718, 17.54958210256512249523818407718, 18.55924110538682425020987188449, 18.55924110538682425020987188449, 19.50763388930887128039872729916, 19.50763388930887128039872729916, 21.00705820570875902116997986320

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