Properties

Label 6-8664e3-1.1-c1e3-0-2
Degree $6$
Conductor $650362258944$
Sign $1$
Analytic cond. $331120.$
Root an. cond. $8.31759$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s + 3·7-s + 6·9-s − 6·11-s + 3·13-s − 9·21-s − 3·25-s − 10·27-s + 12·29-s + 15·31-s + 18·33-s − 3·37-s − 9·39-s + 12·41-s + 9·43-s − 18·47-s − 3·49-s − 6·59-s − 3·61-s + 18·63-s − 3·67-s + 6·71-s − 9·73-s + 9·75-s − 18·77-s + 27·79-s + 15·81-s + ⋯
L(s)  = 1  − 1.73·3-s + 1.13·7-s + 2·9-s − 1.80·11-s + 0.832·13-s − 1.96·21-s − 3/5·25-s − 1.92·27-s + 2.22·29-s + 2.69·31-s + 3.13·33-s − 0.493·37-s − 1.44·39-s + 1.87·41-s + 1.37·43-s − 2.62·47-s − 3/7·49-s − 0.781·59-s − 0.384·61-s + 2.26·63-s − 0.366·67-s + 0.712·71-s − 1.05·73-s + 1.03·75-s − 2.05·77-s + 3.03·79-s + 5/3·81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{3} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{3} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(2^{9} \cdot 3^{3} \cdot 19^{6}\)
Sign: $1$
Analytic conductor: \(331120.\)
Root analytic conductor: \(8.31759\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 2^{9} \cdot 3^{3} \cdot 19^{6} ,\ ( \ : 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(3.756344149\)
\(L(\frac12)\) \(\approx\) \(3.756344149\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
3$C_1$ \( ( 1 + T )^{3} \)
19 \( 1 \)
good5$A_4\times C_2$ \( 1 + 3 T^{2} + 8 T^{3} + 3 p T^{4} + p^{3} T^{6} \)
7$A_4\times C_2$ \( 1 - 3 T + 12 T^{2} - 39 T^{3} + 12 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \)
11$A_4\times C_2$ \( 1 + 6 T + 9 T^{2} - 4 T^{3} + 9 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
13$C_2$ \( ( 1 - T + p T^{2} )^{3} \)
17$C_6$ \( 1 + 3 T^{2} + 64 T^{3} + 3 p T^{4} + p^{3} T^{6} \)
23$A_4\times C_2$ \( 1 + 57 T^{2} + 8 T^{3} + 57 p T^{4} + p^{3} T^{6} \)
29$A_4\times C_2$ \( 1 - 12 T + 3 p T^{2} - 504 T^{3} + 3 p^{2} T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \)
31$A_4\times C_2$ \( 1 - 15 T + 132 T^{2} - 803 T^{3} + 132 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \)
37$A_4\times C_2$ \( 1 + 3 T + 66 T^{2} + 3 p T^{3} + 66 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
41$A_4\times C_2$ \( 1 - 12 T + 3 p T^{2} - 792 T^{3} + 3 p^{2} T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \)
43$A_4\times C_2$ \( 1 - 9 T + 144 T^{2} - 757 T^{3} + 144 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \)
47$C_2$ \( ( 1 + 6 T + p T^{2} )^{3} \)
53$A_4\times C_2$ \( 1 + 75 T^{2} + 296 T^{3} + 75 p T^{4} + p^{3} T^{6} \)
59$A_4\times C_2$ \( 1 + 6 T + 81 T^{2} + 284 T^{3} + 81 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
61$A_4\times C_2$ \( 1 + 3 T + 138 T^{2} + 383 T^{3} + 138 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
67$A_4\times C_2$ \( 1 + 3 T + 168 T^{2} + 439 T^{3} + 168 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
71$A_4\times C_2$ \( 1 - 6 T + 81 T^{2} + 4 T^{3} + 81 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
73$A_4\times C_2$ \( 1 + 9 T + 198 T^{2} + 1133 T^{3} + 198 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \)
79$A_4\times C_2$ \( 1 - 27 T + 396 T^{2} - 3943 T^{3} + 396 p T^{4} - 27 p^{2} T^{5} + p^{3} T^{6} \)
83$A_4\times C_2$ \( 1 + 12 T + 153 T^{2} + 2056 T^{3} + 153 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \)
89$A_4\times C_2$ \( 1 - 18 T + 339 T^{2} - 3132 T^{3} + 339 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \)
97$A_4\times C_2$ \( 1 - 6 T + 159 T^{2} - 308 T^{3} + 159 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−6.81507141414860135014613983944, −6.46162484748277189807391197975, −6.36636681306324693901955959042, −6.03462825957970128834091706652, −5.96718746908024929638378101331, −5.94826416969096767814560529695, −5.33690174097743331251494902255, −5.11224477473601325332915738646, −5.05824103254662511756596457328, −4.82663913882493658419674650390, −4.54398268716957126516695490866, −4.39234780928039679050893997357, −4.36648975279828454686234965199, −3.64818927356256131670735376497, −3.64575463333201338019714295700, −3.19187081965832506254021215981, −2.72362863093109388166015023493, −2.68562951122999337768648475962, −2.50556313676949014595363519583, −1.77930265050909609774735114436, −1.66819619698039960608345897634, −1.51100052800315348506608389244, −0.71569358183342418998040947521, −0.61307013884533657001285808745, −0.60310524256666930289876194525, 0.60310524256666930289876194525, 0.61307013884533657001285808745, 0.71569358183342418998040947521, 1.51100052800315348506608389244, 1.66819619698039960608345897634, 1.77930265050909609774735114436, 2.50556313676949014595363519583, 2.68562951122999337768648475962, 2.72362863093109388166015023493, 3.19187081965832506254021215981, 3.64575463333201338019714295700, 3.64818927356256131670735376497, 4.36648975279828454686234965199, 4.39234780928039679050893997357, 4.54398268716957126516695490866, 4.82663913882493658419674650390, 5.05824103254662511756596457328, 5.11224477473601325332915738646, 5.33690174097743331251494902255, 5.94826416969096767814560529695, 5.96718746908024929638378101331, 6.03462825957970128834091706652, 6.36636681306324693901955959042, 6.46162484748277189807391197975, 6.81507141414860135014613983944

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