Properties

Degree 4
Conductor $ 2^{4} \cdot 3^{6} $
Sign $1$
Motivic weight 4
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 10·7-s − 68·13-s − 128·19-s + 1.16e3·25-s − 1.39e3·31-s − 1.49e3·37-s + 5.23e3·43-s − 4.72e3·49-s + 1.28e4·61-s − 1.04e4·67-s − 9.03e3·73-s + 1.50e4·79-s − 680·91-s + 2.11e4·97-s − 1.16e4·103-s − 1.00e4·109-s + 1.55e4·121-s + 127-s + 131-s − 1.28e3·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯
L(s)  = 1  + 0.204·7-s − 0.402·13-s − 0.354·19-s + 1.87·25-s − 1.45·31-s − 1.09·37-s + 2.83·43-s − 1.96·49-s + 3.44·61-s − 2.32·67-s − 1.69·73-s + 2.40·79-s − 0.0821·91-s + 2.24·97-s − 1.09·103-s − 0.845·109-s + 1.06·121-s + 6.20e−5·127-s + 5.82e−5·131-s − 0.0723·133-s + 5.32e−5·137-s + 5.17e−5·139-s + 4.50e−5·149-s + 4.38e−5·151-s + 4.05e−5·157-s + 3.76e−5·163-s + 3.58e−5·167-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 11664 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11664 ^{s/2} \, \Gamma_{\C}(s+2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(11664\)    =    \(2^{4} \cdot 3^{6}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(4\)
character  :  induced by $\chi_{108} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((4,\ 11664,\ (\ :2, 2),\ 1)\)
\(L(\frac{5}{2})\)  \(\approx\)  \(2.01467\)
\(L(\frac12)\)  \(\approx\)  \(2.01467\)
\(L(3)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3\}$,\(F_p(T)\) is a polynomial of degree 4. 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 \( 1 \)
3 \( 1 \)
good5$C_2^2$ \( 1 - 1169 T^{2} + p^{8} T^{4} \)
7$C_2$ \( ( 1 - 5 T + p^{4} T^{2} )^{2} \)
11$C_2^2$ \( 1 - 15593 T^{2} + p^{8} T^{4} \)
13$C_2$ \( ( 1 + 34 T + p^{4} T^{2} )^{2} \)
17$C_2^2$ \( 1 + 35458 T^{2} + p^{8} T^{4} \)
19$C_2$ \( ( 1 + 64 T + p^{4} T^{2} )^{2} \)
23$C_2^2$ \( 1 - 185138 T^{2} + p^{8} T^{4} \)
29$C_2^2$ \( 1 - 286718 T^{2} + p^{8} T^{4} \)
31$C_2$ \( ( 1 + 697 T + p^{4} T^{2} )^{2} \)
37$C_2$ \( ( 1 + 748 T + p^{4} T^{2} )^{2} \)
41$C_2^2$ \( 1 - 5183666 T^{2} + p^{8} T^{4} \)
43$C_2$ \( ( 1 - 2618 T + p^{4} T^{2} )^{2} \)
47$C_2^2$ \( 1 - 2758046 T^{2} + p^{8} T^{4} \)
53$C_2^2$ \( 1 - 14633921 T^{2} + p^{8} T^{4} \)
59$C_2^2$ \( 1 + 9567874 T^{2} + p^{8} T^{4} \)
61$C_2$ \( ( 1 - 6404 T + p^{4} T^{2} )^{2} \)
67$C_2$ \( ( 1 + 5218 T + p^{4} T^{2} )^{2} \)
71$C_2^2$ \( 1 - 7658462 T^{2} + p^{8} T^{4} \)
73$C_2$ \( ( 1 + 4519 T + p^{4} T^{2} )^{2} \)
79$C_2$ \( ( 1 - 7502 T + p^{4} T^{2} )^{2} \)
83$C_2^2$ \( 1 - 64875281 T^{2} + p^{8} T^{4} \)
89$C_2^2$ \( 1 - 46736606 T^{2} + p^{8} T^{4} \)
97$C_2$ \( ( 1 - 10571 T + p^{4} T^{2} )^{2} \)
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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

−13.09186791230056260538149114567, −12.73343990958904782252015097480, −12.33158767946432614357884831771, −11.67421060643095213104317458864, −11.03801273603851871844645563982, −10.70007531978504615614967744152, −10.16218332862058325924040514815, −9.379555639978207914235101469964, −8.957264686496322542703138345278, −8.441811738788672930349170847216, −7.65873495575248042996558133761, −7.15524687000509825256436040542, −6.58311622632324317087726498629, −5.76164196829385482110821900163, −5.12205023863242180547277956679, −4.47663938633016616264341739330, −3.61446601895199997040693208283, −2.74604489179466577702223320029, −1.81074364198028364053065974691, −0.64334920823501279540036670758, 0.64334920823501279540036670758, 1.81074364198028364053065974691, 2.74604489179466577702223320029, 3.61446601895199997040693208283, 4.47663938633016616264341739330, 5.12205023863242180547277956679, 5.76164196829385482110821900163, 6.58311622632324317087726498629, 7.15524687000509825256436040542, 7.65873495575248042996558133761, 8.441811738788672930349170847216, 8.957264686496322542703138345278, 9.379555639978207914235101469964, 10.16218332862058325924040514815, 10.70007531978504615614967744152, 11.03801273603851871844645563982, 11.67421060643095213104317458864, 12.33158767946432614357884831771, 12.73343990958904782252015097480, 13.09186791230056260538149114567

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