Properties

Degree $4$
Conductor $1728$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 9-s − 4·13-s − 8·19-s − 6·25-s − 27-s + 16·31-s + 12·37-s + 4·39-s + 8·43-s − 14·49-s + 8·57-s − 4·61-s − 8·67-s + 20·73-s + 6·75-s − 16·79-s + 81-s − 16·93-s + 4·97-s + 32·103-s − 4·109-s − 12·111-s − 4·117-s − 6·121-s + 127-s − 8·129-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/3·9-s − 1.10·13-s − 1.83·19-s − 6/5·25-s − 0.192·27-s + 2.87·31-s + 1.97·37-s + 0.640·39-s + 1.21·43-s − 2·49-s + 1.05·57-s − 0.512·61-s − 0.977·67-s + 2.34·73-s + 0.692·75-s − 1.80·79-s + 1/9·81-s − 1.65·93-s + 0.406·97-s + 3.15·103-s − 0.383·109-s − 1.13·111-s − 0.369·117-s − 0.545·121-s + 0.0887·127-s − 0.704·129-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{1728} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1728,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5247171448\)
\(L(\frac12)\) \(\approx\) \(0.5247171448\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
7$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
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$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
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

−13.38110032732463692857243615568, −12.79661702509897009228746961274, −12.31725686074733747032880351584, −11.58204746144152243175303945055, −11.17432181718975973992131540327, −10.16546244400809085298891139382, −9.964671915727173343349228861429, −9.065170023749915638598404079220, −8.098990694093691505068092710868, −7.62403207119791885504595168937, −6.42897107072744719896577758454, −6.14969218484336320739052245813, −4.83326855247570334999188527992, −4.25303028692796488061253338187, −2.51659494204614894722398853037, 2.51659494204614894722398853037, 4.25303028692796488061253338187, 4.83326855247570334999188527992, 6.14969218484336320739052245813, 6.42897107072744719896577758454, 7.62403207119791885504595168937, 8.098990694093691505068092710868, 9.065170023749915638598404079220, 9.964671915727173343349228861429, 10.16546244400809085298891139382, 11.17432181718975973992131540327, 11.58204746144152243175303945055, 12.31725686074733747032880351584, 12.79661702509897009228746961274, 13.38110032732463692857243615568

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