Properties

Degree 4
Conductor $ 2^{4} \cdot 3^{4} \cdot 7^{4} $
Sign $1$
Motivic weight 0
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·25-s + 2·37-s + 2·43-s − 2·67-s − 2·79-s + 2·109-s − 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯
L(s)  = 1  + 2·25-s + 2·37-s + 2·43-s − 2·67-s − 2·79-s + 2·109-s − 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3111696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3111696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(3111696\)    =    \(2^{4} \cdot 3^{4} \cdot 7^{4}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  induced by $\chi_{1764} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((4,\ 3111696,\ (\ :0, 0),\ 1)\)
\(L(\frac{1}{2})\)  \(\approx\)  \(1.250796336\)
\(L(\frac12)\)  \(\approx\)  \(1.250796336\)
\(L(1)\)   not available
\(L(1)\)   not available

Euler product

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

−9.546241743492637231301743564817, −9.257921809794012541669276835668, −8.865857949801841562228746163395, −8.694103321015126369281764338354, −7.990575680104474115606634308158, −7.79251351648400939042685222173, −7.22425857804538120847758920325, −7.08962852102827203835123985182, −6.34061604149370236202950670022, −6.19813868057511161059301623153, −5.67759528043369136618398569354, −5.24536026167022527772398936435, −4.61772661579888641664611156362, −4.43807658020438802710731254167, −3.91280339917285512320307417121, −3.26571334090267834472061865824, −2.63217218719953035275204927225, −2.57510274776060613681995637119, −1.50487015914138607533452316670, −0.942561592259345294468296927486, 0.942561592259345294468296927486, 1.50487015914138607533452316670, 2.57510274776060613681995637119, 2.63217218719953035275204927225, 3.26571334090267834472061865824, 3.91280339917285512320307417121, 4.43807658020438802710731254167, 4.61772661579888641664611156362, 5.24536026167022527772398936435, 5.67759528043369136618398569354, 6.19813868057511161059301623153, 6.34061604149370236202950670022, 7.08962852102827203835123985182, 7.22425857804538120847758920325, 7.79251351648400939042685222173, 7.990575680104474115606634308158, 8.694103321015126369281764338354, 8.865857949801841562228746163395, 9.257921809794012541669276835668, 9.546241743492637231301743564817

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