Properties

Degree 4
Conductor $ 2^{4} \cdot 3^{6} $
Sign $1$
Motivic weight 5
Primitive no
Self-dual yes
Analytic rank 2

Origins

Origins of factors

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 32·7-s − 486·11-s + 208·13-s − 1.94e3·17-s − 632·19-s − 6.80e3·23-s − 2.92e3·25-s − 1.16e4·29-s + 3.32e3·31-s − 9.95e3·37-s − 1.36e4·41-s + 4.96e3·43-s − 1.84e4·47-s − 2.95e3·49-s + 1.16e4·53-s − 1.94e3·59-s + 8.17e3·61-s + 9.00e4·67-s + 4.47e4·71-s − 1.21e5·73-s + 1.55e4·77-s + 2.87e4·79-s + 1.50e4·83-s + 1.78e5·89-s − 6.65e3·91-s + 8.89e4·97-s + 2.02e5·101-s + ⋯
L(s)  = 1  − 0.246·7-s − 1.21·11-s + 0.341·13-s − 1.63·17-s − 0.401·19-s − 2.68·23-s − 0.937·25-s − 2.57·29-s + 0.621·31-s − 1.19·37-s − 1.26·41-s + 0.409·43-s − 1.21·47-s − 0.175·49-s + 0.570·53-s − 0.0727·59-s + 0.281·61-s + 2.45·67-s + 1.05·71-s − 2.66·73-s + 0.298·77-s + 0.518·79-s + 0.240·83-s + 2.39·89-s − 0.0842·91-s + 0.959·97-s + 1.97·101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 11664 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11664 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(11664\)    =    \(2^{4} \cdot 3^{6}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(5\)
character  :  induced by $\chi_{108} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(2\)
Selberg data  =  \((4,\ 11664,\ (\ :5/2, 5/2),\ 1)\)
\(L(3)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{7}{2})\)   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 \( 1 \)
3 \( 1 \)
good5$C_2^2$ \( 1 + 2929 T^{2} + p^{10} T^{4} \)
7$D_{4}$ \( 1 + 32 T + 3981 T^{2} + 32 p^{5} T^{3} + p^{10} T^{4} \)
11$D_{4}$ \( 1 + 486 T + 168607 T^{2} + 486 p^{5} T^{3} + p^{10} T^{4} \)
13$D_{4}$ \( 1 - 16 p T + 275178 T^{2} - 16 p^{6} T^{3} + p^{10} T^{4} \)
17$D_{4}$ \( 1 + 1944 T + 2456098 T^{2} + 1944 p^{5} T^{3} + p^{10} T^{4} \)
19$D_{4}$ \( 1 + 632 T + 4932498 T^{2} + 632 p^{5} T^{3} + p^{10} T^{4} \)
23$D_{4}$ \( 1 + 6804 T + 24393154 T^{2} + 6804 p^{5} T^{3} + p^{10} T^{4} \)
29$D_{4}$ \( 1 + 11664 T + 70238998 T^{2} + 11664 p^{5} T^{3} + p^{10} T^{4} \)
31$D_{4}$ \( 1 - 3328 T + 38238117 T^{2} - 3328 p^{5} T^{3} + p^{10} T^{4} \)
37$D_{4}$ \( 1 + 9956 T + 151512798 T^{2} + 9956 p^{5} T^{3} + p^{10} T^{4} \)
41$D_{4}$ \( 1 + 13608 T + 266834974 T^{2} + 13608 p^{5} T^{3} + p^{10} T^{4} \)
43$D_{4}$ \( 1 - 4960 T + 213010962 T^{2} - 4960 p^{5} T^{3} + p^{10} T^{4} \)
47$D_{4}$ \( 1 + 18468 T + 312496354 T^{2} + 18468 p^{5} T^{3} + p^{10} T^{4} \)
53$D_{4}$ \( 1 - 11664 T + 484234009 T^{2} - 11664 p^{5} T^{3} + p^{10} T^{4} \)
59$D_{4}$ \( 1 + 1944 T + 961283686 T^{2} + 1944 p^{5} T^{3} + p^{10} T^{4} \)
61$D_{4}$ \( 1 - 8176 T - 15702054 T^{2} - 8176 p^{5} T^{3} + p^{10} T^{4} \)
67$D_{4}$ \( 1 - 90064 T + 4055628738 T^{2} - 90064 p^{5} T^{3} + p^{10} T^{4} \)
71$D_{4}$ \( 1 - 44712 T + 2046094414 T^{2} - 44712 p^{5} T^{3} + p^{10} T^{4} \)
73$D_{4}$ \( 1 + 121214 T + 6975764499 T^{2} + 121214 p^{5} T^{3} + p^{10} T^{4} \)
79$D_{4}$ \( 1 - 28768 T + 4017714654 T^{2} - 28768 p^{5} T^{3} + p^{10} T^{4} \)
83$D_{4}$ \( 1 - 15066 T - 2258995409 T^{2} - 15066 p^{5} T^{3} + p^{10} T^{4} \)
89$D_{4}$ \( 1 - 178848 T + 18739138030 T^{2} - 178848 p^{5} T^{3} + p^{10} T^{4} \)
97$D_{4}$ \( 1 - 88942 T + 13779025491 T^{2} - 88942 p^{5} T^{3} + p^{10} T^{4} \)
show more
show less
\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−12.66321745753485678525558099401, −11.73221934650396198378854446496, −11.63273864058172205143549725506, −10.89412472095523456153149960272, −10.20969250921568759555062763524, −10.07402872452429506893550481858, −9.199154689048109096497305727703, −8.719751327649295135810788781785, −7.88475169166221899770692311595, −7.79586544670057594365706128324, −6.75818820268687459738952615895, −6.25936863139840133919971146252, −5.59754178581492614688608002415, −4.94224891656756950196435959752, −3.98570935811126743670208001395, −3.56943105237776466554535272994, −2.15948891911118990100055856966, −1.99809270355761564298561986659, 0, 0, 1.99809270355761564298561986659, 2.15948891911118990100055856966, 3.56943105237776466554535272994, 3.98570935811126743670208001395, 4.94224891656756950196435959752, 5.59754178581492614688608002415, 6.25936863139840133919971146252, 6.75818820268687459738952615895, 7.79586544670057594365706128324, 7.88475169166221899770692311595, 8.719751327649295135810788781785, 9.199154689048109096497305727703, 10.07402872452429506893550481858, 10.20969250921568759555062763524, 10.89412472095523456153149960272, 11.63273864058172205143549725506, 11.73221934650396198378854446496, 12.66321745753485678525558099401

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