Properties

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

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·2-s + 12·4-s + 4·5-s − 6·7-s − 32·8-s − 16·10-s + 21·11-s + 22·13-s + 24·14-s + 80·16-s + 22·17-s + 48·20-s − 84·22-s − 42·23-s + 25·25-s − 88·26-s − 72·28-s + 34·29-s + 12·31-s − 192·32-s − 88·34-s − 24·35-s − 32·37-s − 128·40-s + 13·41-s − 87·43-s + 252·44-s + ⋯
L(s)  = 1  − 2·2-s + 3·4-s + 4/5·5-s − 6/7·7-s − 4·8-s − 8/5·10-s + 1.90·11-s + 1.69·13-s + 12/7·14-s + 5·16-s + 1.29·17-s + 12/5·20-s − 3.81·22-s − 1.82·23-s + 25-s − 3.38·26-s − 2.57·28-s + 1.17·29-s + 0.387·31-s − 6·32-s − 2.58·34-s − 0.685·35-s − 0.864·37-s − 3.19·40-s + 0.317·41-s − 2.02·43-s + 5.72·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 11664 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11664 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(11664\)    =    \(2^{4} \cdot 3^{6}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(2\)
character  :  induced by $\chi_{108} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((4,\ 11664,\ (\ :1, 1),\ 1)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.877877\)
\(L(\frac12)\)  \(\approx\)  \(0.877877\)
\(L(2)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3\}$,\(F_p(T)\) 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$C_1$ \( ( 1 + p T )^{2} \)
3 \( 1 \)
good5$C_2^2$ \( 1 - 4 T - 9 T^{2} - 4 p^{2} T^{3} + p^{4} T^{4} \)
7$C_2^2$ \( 1 + 6 T + 61 T^{2} + 6 p^{2} T^{3} + p^{4} T^{4} \)
11$C_2^2$ \( 1 - 21 T + 268 T^{2} - 21 p^{2} T^{3} + p^{4} T^{4} \)
13$C_2$ \( ( 1 - 23 T + p^{2} T^{2} )( 1 + T + p^{2} T^{2} ) \)
17$C_2$ \( ( 1 - 11 T + p^{2} T^{2} )^{2} \)
19$C_2^2$ \( 1 - 479 T^{2} + p^{4} T^{4} \)
23$C_2^2$ \( 1 + 42 T + 1117 T^{2} + 42 p^{2} T^{3} + p^{4} T^{4} \)
29$C_2^2$ \( 1 - 34 T + 315 T^{2} - 34 p^{2} T^{3} + p^{4} T^{4} \)
31$C_2^2$ \( 1 - 12 T + 1009 T^{2} - 12 p^{2} T^{3} + p^{4} T^{4} \)
37$C_2$ \( ( 1 + 16 T + p^{2} T^{2} )^{2} \)
41$C_2^2$ \( 1 - 13 T - 1512 T^{2} - 13 p^{2} T^{3} + p^{4} T^{4} \)
43$C_2^2$ \( 1 + 87 T + 4372 T^{2} + 87 p^{2} T^{3} + p^{4} T^{4} \)
47$C_2^2$ \( 1 + 6 T + 2221 T^{2} + 6 p^{2} T^{3} + p^{4} T^{4} \)
53$C_2$ \( ( 1 + 52 T + p^{2} T^{2} )^{2} \)
59$C_2^2$ \( 1 - 93 T + 6364 T^{2} - 93 p^{2} T^{3} + p^{4} T^{4} \)
61$C_2^2$ \( 1 - 16 T - 3465 T^{2} - 16 p^{2} T^{3} + p^{4} T^{4} \)
67$C_1$$\times$$C_2$ \( ( 1 - p T )^{2}( 1 - p T + p^{2} T^{2} ) \)
71$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
73$C_2$ \( ( 1 + 25 T + p^{2} T^{2} )^{2} \)
79$C_2^2$ \( 1 - 48 T + 7009 T^{2} - 48 p^{2} T^{3} + p^{4} T^{4} \)
83$C_2^2$ \( 1 + 60 T + 8089 T^{2} + 60 p^{2} T^{3} + p^{4} T^{4} \)
89$C_2$ \( ( 1 - 2 T + p^{2} T^{2} )^{2} \)
97$C_2^2$ \( 1 - 43 T - 7560 T^{2} - 43 p^{2} T^{3} + p^{4} T^{4} \)
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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

−13.85961218737605431131420894618, −13.09562163463974244992776810633, −12.20499939245017695472547043173, −12.17943238780265407766427289870, −11.40299044144793240646202092613, −11.01480917134874881744645439968, −10.11747961441074300386019301745, −9.961368820786361898919390258136, −9.550397450611721853186796576782, −8.934177262867916437164762452338, −8.238224340695612330926027781716, −8.191583940616612312323452113217, −6.82709696882981068985607989581, −6.59683339993252342435269938996, −6.22553180476418329971461814382, −5.54923994748377296817078999355, −3.71305706960788304913441990174, −3.20438836984961025217929012962, −1.80570243621156538987813081825, −1.06738850104828374950707415430, 1.06738850104828374950707415430, 1.80570243621156538987813081825, 3.20438836984961025217929012962, 3.71305706960788304913441990174, 5.54923994748377296817078999355, 6.22553180476418329971461814382, 6.59683339993252342435269938996, 6.82709696882981068985607989581, 8.191583940616612312323452113217, 8.238224340695612330926027781716, 8.934177262867916437164762452338, 9.550397450611721853186796576782, 9.961368820786361898919390258136, 10.11747961441074300386019301745, 11.01480917134874881744645439968, 11.40299044144793240646202092613, 12.17943238780265407766427289870, 12.20499939245017695472547043173, 13.09562163463974244992776810633, 13.85961218737605431131420894618

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