Properties

Label 4-24e3-1.1-c1e2-0-6
Degree $4$
Conductor $13824$
Sign $-1$
Analytic cond. $0.881430$
Root an. cond. $0.968940$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 4·5-s + 9-s + 4·15-s − 8·19-s − 16·23-s + 2·25-s − 27-s + 12·29-s + 8·43-s − 4·45-s − 14·49-s − 4·53-s + 8·57-s − 8·67-s + 16·69-s + 16·71-s + 20·73-s − 2·75-s + 81-s − 12·87-s + 32·95-s + 4·97-s − 36·101-s + 64·115-s − 6·121-s + 28·125-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.78·5-s + 1/3·9-s + 1.03·15-s − 1.83·19-s − 3.33·23-s + 2/5·25-s − 0.192·27-s + 2.22·29-s + 1.21·43-s − 0.596·45-s − 2·49-s − 0.549·53-s + 1.05·57-s − 0.977·67-s + 1.92·69-s + 1.89·71-s + 2.34·73-s − 0.230·75-s + 1/9·81-s − 1.28·87-s + 3.28·95-s + 0.406·97-s − 3.58·101-s + 5.96·115-s − 0.545·121-s + 2.50·125-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 13824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13824 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(13824\)    =    \(2^{9} \cdot 3^{3}\)
Sign: $-1$
Analytic conductor: \(0.881430\)
Root analytic conductor: \(0.968940\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{13824} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 13824,\ (\ :1/2, 1/2),\ -1)\)

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

−10.98603906202358244044014470993, −10.54464146795679939621303070304, −9.964671915727173343349228861429, −9.381546175761683500969667585237, −8.240414794588996598510176234846, −8.098990694093691505068092710868, −7.87582021380809768630528089037, −6.70029239224408692327848930668, −6.42897107072744719896577758454, −5.67110067949068956351494574607, −4.48990443653583983179988972356, −4.25303028692796488061253338187, −3.59090248724659668634540115790, −2.18788103243241609268384022128, 0, 2.18788103243241609268384022128, 3.59090248724659668634540115790, 4.25303028692796488061253338187, 4.48990443653583983179988972356, 5.67110067949068956351494574607, 6.42897107072744719896577758454, 6.70029239224408692327848930668, 7.87582021380809768630528089037, 8.098990694093691505068092710868, 8.240414794588996598510176234846, 9.381546175761683500969667585237, 9.964671915727173343349228861429, 10.54464146795679939621303070304, 10.98603906202358244044014470993

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