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  + 2·2-s + 4·5-s + 6·7-s − 8·8-s + 8·10-s − 21·11-s + 22·13-s + 12·14-s − 16·16-s + 22·17-s − 42·22-s + 42·23-s + 25·25-s + 44·26-s + 34·29-s − 12·31-s + 44·34-s + 24·35-s − 32·37-s − 32·40-s + 13·41-s + 87·43-s + 84·46-s + 6·47-s − 25·49-s + 50·50-s − 104·53-s + ⋯
L(s)  = 1  + 2-s + 4/5·5-s + 6/7·7-s − 8-s + 4/5·10-s − 1.90·11-s + 1.69·13-s + 6/7·14-s − 16-s + 1.29·17-s − 1.90·22-s + 1.82·23-s + 25-s + 1.69·26-s + 1.17·29-s − 0.387·31-s + 1.29·34-s + 0.685·35-s − 0.864·37-s − 4/5·40-s + 0.317·41-s + 2.02·43-s + 1.82·46-s + 6/47·47-s − 0.510·49-s + 50-s − 1.96·53-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  =  \((4,\ 11664,\ (\ :1, 1),\ 1)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(2.93085\)
\(L(\frac12)\)  \(\approx\)  \(2.93085\)
\(L(2)\)   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$C_2$ \( 1 - p T + p^{2} T^{2} \)
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

−13.93850122987775560337118558998, −13.12478370062573265831788423363, −12.92291934826692834874083558352, −12.42948474150826786960092143337, −11.76846080072790180638911620209, −10.90826749306954118387315446407, −10.75845189929520487363540718787, −10.27570757075348754178528398541, −9.203298071683234380438208606820, −9.026514674388931796367953414332, −8.176963678357732761926164500091, −7.78669486520028863702416882409, −6.88879958809555512549124770057, −5.92883519417016020947571069029, −5.71396854540461132195584079105, −4.93828920587932712708747906170, −4.58068332897633590863243645595, −3.23571630612650386065815094918, −2.84990598171486041913618377563, −1.30496350118634431438247537243, 1.30496350118634431438247537243, 2.84990598171486041913618377563, 3.23571630612650386065815094918, 4.58068332897633590863243645595, 4.93828920587932712708747906170, 5.71396854540461132195584079105, 5.92883519417016020947571069029, 6.88879958809555512549124770057, 7.78669486520028863702416882409, 8.176963678357732761926164500091, 9.026514674388931796367953414332, 9.203298071683234380438208606820, 10.27570757075348754178528398541, 10.75845189929520487363540718787, 10.90826749306954118387315446407, 11.76846080072790180638911620209, 12.42948474150826786960092143337, 12.92291934826692834874083558352, 13.12478370062573265831788423363, 13.93850122987775560337118558998

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