Properties

Degree 4
Conductor $ 2^{6} \cdot 3^{2} \cdot 7 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 4-s + 3·7-s − 3·8-s − 9-s + 3·14-s − 16-s − 4·17-s − 18-s − 4·23-s − 6·25-s − 3·28-s + 4·31-s + 5·32-s − 4·34-s + 36-s − 8·41-s − 4·46-s + 12·47-s + 6·49-s − 6·50-s − 9·56-s + 4·62-s − 3·63-s + 7·64-s + 4·68-s + 3·72-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s + 1.13·7-s − 1.06·8-s − 1/3·9-s + 0.801·14-s − 1/4·16-s − 0.970·17-s − 0.235·18-s − 0.834·23-s − 6/5·25-s − 0.566·28-s + 0.718·31-s + 0.883·32-s − 0.685·34-s + 1/6·36-s − 1.24·41-s − 0.589·46-s + 1.75·47-s + 6/7·49-s − 0.848·50-s − 1.20·56-s + 0.508·62-s − 0.377·63-s + 7/8·64-s + 0.485·68-s + 0.353·72-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(4032\)    =    \(2^{6} \cdot 3^{2} \cdot 7\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{4032} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 4032,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $0.9678252925$
$L(\frac12)$  $\approx$  $0.9678252925$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;7\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;3,\;7\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2$C_2$ \( 1 - T + p T^{2} \)
3$C_2$ \( 1 + T^{2} \)
7$C_1$$\times$$C_2$ \( ( 1 + T )( 1 - 4 T + p T^{2} ) \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
11$V_4$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
13$V_4$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
23$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
29$C_2$ \( ( 1 - p T^{2} )^{2} \)
31$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
41$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
43$V_4$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
47$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \)
53$V_4$ \( 1 + 38 T^{2} + p^{2} T^{4} \)
59$V_4$ \( 1 - 54 T^{2} + p^{2} T^{4} \)
61$V_4$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
67$V_4$ \( 1 - 26 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
79$C_2$$\times$$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 - 4 T + p T^{2} ) \)
83$V_4$ \( 1 - 78 T^{2} + p^{2} T^{4} \)
89$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
97$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
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\[\begin{aligned} L(s) = \prod_{\mathfrak{p}\ \mathrm{bad}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s})^{-1} \prod_{\mathfrak{p}\ \mathrm{good}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s} + (N\mathfrak{p})^{-2s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−12.36021754353015496421482418180, −11.98566448029874371175619227026, −11.52475335651447367015294906093, −10.83089289495071548852543718100, −10.17157672444718663560350386527, −9.408398385496674260626968138199, −8.780510706819587469543522642198, −8.225807961009441439699575191924, −7.64732326813100286750821949387, −6.57789503000444893301550976642, −5.85944151284085004982177107902, −5.13436599806522948692536302643, −4.43740873400926585273602040722, −3.70641959270425262073508709657, −2.28562209272240743597100597091, 2.28562209272240743597100597091, 3.70641959270425262073508709657, 4.43740873400926585273602040722, 5.13436599806522948692536302643, 5.85944151284085004982177107902, 6.57789503000444893301550976642, 7.64732326813100286750821949387, 8.225807961009441439699575191924, 8.780510706819587469543522642198, 9.408398385496674260626968138199, 10.17157672444718663560350386527, 10.83089289495071548852543718100, 11.52475335651447367015294906093, 11.98566448029874371175619227026, 12.36021754353015496421482418180

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