Properties

Degree 4
Conductor $ 2^{2} \cdot 3^{2} \cdot 23^{3} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 4-s + 8·5-s − 3·9-s + 4·11-s − 4·13-s + 16-s − 4·17-s + 8·20-s + 23-s + 38·25-s − 3·36-s + 4·44-s − 24·45-s + 2·49-s − 4·52-s − 8·53-s + 32·55-s + 64-s − 32·65-s − 4·68-s + 12·73-s + 8·80-s + 9·81-s + 28·83-s − 32·85-s − 12·89-s + 92-s + ⋯
L(s)  = 1  + 1/2·4-s + 3.57·5-s − 9-s + 1.20·11-s − 1.10·13-s + 1/4·16-s − 0.970·17-s + 1.78·20-s + 0.208·23-s + 38/5·25-s − 1/2·36-s + 0.603·44-s − 3.57·45-s + 2/7·49-s − 0.554·52-s − 1.09·53-s + 4.31·55-s + 1/8·64-s − 3.96·65-s − 0.485·68-s + 1.40·73-s + 0.894·80-s + 81-s + 3.07·83-s − 3.47·85-s − 1.27·89-s + 0.104·92-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 438012 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 438012 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(438012\)    =    \(2^{2} \cdot 3^{2} \cdot 23^{3}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{438012} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 438012,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $4.206707818$
$L(\frac12)$  $\approx$  $4.206707818$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;23\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;3,\;23\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
3$C_2$ \( 1 + p T^{2} \)
23$C_1$ \( 1 - T \)
good5$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−8.971935452602447588346290189052, −8.300196560171154480855065910688, −7.62735421295774645335638863042, −6.69441913641732873591204403077, −6.69265944578033613099294080985, −6.35362788354985461499174366441, −5.84432417959727908710252168629, −5.35356885744180803495724283233, −5.12924034341750180180238442817, −4.47212152404542749707456940527, −3.42573528437023625765419608105, −2.63232737474552247848840602025, −2.38695715760664154900264131006, −1.87977103445283065955695773340, −1.21768411761699883032703079663, 1.21768411761699883032703079663, 1.87977103445283065955695773340, 2.38695715760664154900264131006, 2.63232737474552247848840602025, 3.42573528437023625765419608105, 4.47212152404542749707456940527, 5.12924034341750180180238442817, 5.35356885744180803495724283233, 5.84432417959727908710252168629, 6.35362788354985461499174366441, 6.69265944578033613099294080985, 6.69441913641732873591204403077, 7.62735421295774645335638863042, 8.300196560171154480855065910688, 8.971935452602447588346290189052

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