Properties

Degree 4
Conductor $ 2^{5} \cdot 3^{2} \cdot 11 $
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 + 3-s + 4-s − 6-s − 8-s − 2·9-s + 2·11-s + 12-s − 2·13-s + 16-s + 2·18-s − 2·22-s − 3·23-s − 24-s − 4·25-s + 2·26-s − 5·27-s − 32-s + 2·33-s − 2·36-s + 37-s − 2·39-s + 2·44-s + 3·46-s + 9·47-s + 48-s + 5·49-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.353·8-s − 2/3·9-s + 0.603·11-s + 0.288·12-s − 0.554·13-s + 1/4·16-s + 0.471·18-s − 0.426·22-s − 0.625·23-s − 0.204·24-s − 4/5·25-s + 0.392·26-s − 0.962·27-s − 0.176·32-s + 0.348·33-s − 1/3·36-s + 0.164·37-s − 0.320·39-s + 0.301·44-s + 0.442·46-s + 1.31·47-s + 0.144·48-s + 5/7·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3168 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3168 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(3168\)    =    \(2^{5} \cdot 3^{2} \cdot 11\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{3168} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((4,\ 3168,\ (\ :1/2, 1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(0.6250213940\)
\(L(\frac12)\)  \(\approx\)  \(0.6250213940\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;11\}$,\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;3,\;11\}$, 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$C_2$ \( 1 - T + p T^{2} \)
11$C_1$$\times$$C_2$ \( ( 1 + T )( 1 - 3 T + p T^{2} ) \)
good5$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \)
7$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
23$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
29$C_2^2$ \( 1 - 17 T^{2} + p^{2} T^{4} \)
31$C_2^2$ \( 1 + 28 T^{2} + p^{2} T^{4} \)
37$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
43$C_2^2$ \( 1 + 40 T^{2} + p^{2} T^{4} \)
47$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
71$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
73$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + T + p T^{2} ) \)
79$C_2^2$ \( 1 - 77 T^{2} + p^{2} T^{4} \)
83$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
89$C_2^2$ \( 1 + 73 T^{2} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
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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

−12.65418471580462255718354903878, −11.94945510831953546666205870104, −11.67859104995133782855162915724, −10.89157814259766502301721731237, −10.22551646350884739634709025716, −9.617946753058921074165390870758, −8.993615420670583592832545430956, −8.579257706904438873046700044755, −7.70668237752242463432576097104, −7.33142598722630380941995415168, −6.24329912599518429624292681724, −5.68464252582251042101571546693, −4.40114328389798423754560516310, −3.32354453455858658608472519573, −2.18090952847020073511516588570, 2.18090952847020073511516588570, 3.32354453455858658608472519573, 4.40114328389798423754560516310, 5.68464252582251042101571546693, 6.24329912599518429624292681724, 7.33142598722630380941995415168, 7.70668237752242463432576097104, 8.579257706904438873046700044755, 8.993615420670583592832545430956, 9.617946753058921074165390870758, 10.22551646350884739634709025716, 10.89157814259766502301721731237, 11.67859104995133782855162915724, 11.94945510831953546666205870104, 12.65418471580462255718354903878

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