Properties

Degree 4
Conductor $ 2^{7} \cdot 3^{6} $
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  + 2-s + 4-s + 8-s + 6·11-s + 16-s + 4·19-s + 6·22-s − 25-s + 32-s + 4·38-s + 12·41-s − 20·43-s + 6·44-s − 13·49-s − 50-s − 24·59-s + 64-s + 28·67-s − 14·73-s + 4·76-s + 12·82-s + 6·83-s − 20·86-s + 6·88-s + 36·89-s − 2·97-s − 13·98-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.353·8-s + 1.80·11-s + 1/4·16-s + 0.917·19-s + 1.27·22-s − 1/5·25-s + 0.176·32-s + 0.648·38-s + 1.87·41-s − 3.04·43-s + 0.904·44-s − 1.85·49-s − 0.141·50-s − 3.12·59-s + 1/8·64-s + 3.42·67-s − 1.63·73-s + 0.458·76-s + 1.32·82-s + 0.658·83-s − 2.15·86-s + 0.639·88-s + 3.81·89-s − 0.203·97-s − 1.31·98-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 93312 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 93312 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(93312\)    =    \(2^{7} \cdot 3^{6}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{93312} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 93312,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $2.656428221$
$L(\frac12)$  $\approx$  $2.656428221$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;3\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_1$ \( 1 - T \)
3 \( 1 \)
good5$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
7$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
11$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$ \( ( 1 + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
59$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 18 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
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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

−9.512741326522107701681952133776, −9.300538826904107531692188360431, −8.699511810288277483701861071116, −7.82723843094092867969941089213, −7.82087500680797002813709866504, −6.71508725988629390136097161445, −6.68641118928175950907657181471, −6.10265864740654543835243104495, −5.43217621063437274714591238543, −4.77709333112371596512170551343, −4.34994968190190118318545746782, −3.38549178369413487864105189864, −3.35914372705473359723249826168, −2.05899421145709367031472603742, −1.26980488075185704480426069120, 1.26980488075185704480426069120, 2.05899421145709367031472603742, 3.35914372705473359723249826168, 3.38549178369413487864105189864, 4.34994968190190118318545746782, 4.77709333112371596512170551343, 5.43217621063437274714591238543, 6.10265864740654543835243104495, 6.68641118928175950907657181471, 6.71508725988629390136097161445, 7.82087500680797002813709866504, 7.82723843094092867969941089213, 8.699511810288277483701861071116, 9.300538826904107531692188360431, 9.512741326522107701681952133776

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