Properties

Degree 4
Conductor $ 2^{4} \cdot 3^{4} \cdot 7^{4} $
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  − 6·11-s + 4·13-s + 4·19-s − 6·23-s + 5·25-s − 12·29-s − 8·31-s − 2·37-s − 24·41-s − 8·43-s + 12·47-s − 6·53-s + 10·61-s − 8·67-s − 12·71-s + 10·73-s + 4·79-s + 24·83-s + 12·89-s − 20·97-s − 12·101-s − 8·103-s − 6·107-s − 14·109-s + 12·113-s + 11·121-s + 127-s + ⋯
L(s)  = 1  − 1.80·11-s + 1.10·13-s + 0.917·19-s − 1.25·23-s + 25-s − 2.22·29-s − 1.43·31-s − 0.328·37-s − 3.74·41-s − 1.21·43-s + 1.75·47-s − 0.824·53-s + 1.28·61-s − 0.977·67-s − 1.42·71-s + 1.17·73-s + 0.450·79-s + 2.63·83-s + 1.27·89-s − 2.03·97-s − 1.19·101-s − 0.788·103-s − 0.580·107-s − 1.34·109-s + 1.12·113-s + 121-s + 0.0887·127-s + ⋯

Functional equation

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

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(3111696\)    =    \(2^{4} \cdot 3^{4} \cdot 7^{4}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{1764} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((4,\ 3111696,\ (\ :1/2, 1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(0.8159311003\)
\(L(\frac12)\)  \(\approx\)  \(0.8159311003\)
\(L(\frac{3}{2})\)   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 $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 4 T - 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 + 8 T + 33 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 12 T + 97 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 10 T + 39 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 8 T - 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 12 T + 55 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
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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

−9.652352784337030265316072572303, −8.964369395771365941610526912509, −8.753171046328865263242767371009, −8.248022322460251768098155017561, −7.959904059948904348760877048000, −7.53772221691430248269179049477, −7.25557579439848056805794805028, −6.61398539613898211369861163130, −6.45083910272001374724332137794, −5.57110160216401925764191329735, −5.44433134748165665733168695409, −5.24181006739380963742293772326, −4.68954446993148147473050923320, −3.76523940903390407862092132855, −3.71354445295736401136271058640, −3.19111518493485473494497950700, −2.58461641125149024107019875519, −1.83445455045527700701481508668, −1.58216474467959829938415842530, −0.32518957930275582294738842578, 0.32518957930275582294738842578, 1.58216474467959829938415842530, 1.83445455045527700701481508668, 2.58461641125149024107019875519, 3.19111518493485473494497950700, 3.71354445295736401136271058640, 3.76523940903390407862092132855, 4.68954446993148147473050923320, 5.24181006739380963742293772326, 5.44433134748165665733168695409, 5.57110160216401925764191329735, 6.45083910272001374724332137794, 6.61398539613898211369861163130, 7.25557579439848056805794805028, 7.53772221691430248269179049477, 7.959904059948904348760877048000, 8.248022322460251768098155017561, 8.753171046328865263242767371009, 8.964369395771365941610526912509, 9.652352784337030265316072572303

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