Properties

Label 6-3200e3-1.1-c1e3-0-0
Degree $6$
Conductor $32768000000$
Sign $1$
Analytic cond. $16683.2$
Root an. cond. $5.05491$
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  + 2·3-s − 4·7-s − 9-s − 2·11-s + 8·13-s + 4·17-s − 2·19-s − 8·21-s − 8·23-s − 4·27-s − 6·29-s + 8·31-s − 4·33-s + 8·37-s + 16·39-s + 2·41-s + 14·43-s − 4·47-s − 49-s + 8·51-s + 16·53-s − 4·57-s + 2·59-s − 10·61-s + 4·63-s + 10·67-s − 16·69-s + ⋯
L(s)  = 1  + 1.15·3-s − 1.51·7-s − 1/3·9-s − 0.603·11-s + 2.21·13-s + 0.970·17-s − 0.458·19-s − 1.74·21-s − 1.66·23-s − 0.769·27-s − 1.11·29-s + 1.43·31-s − 0.696·33-s + 1.31·37-s + 2.56·39-s + 0.312·41-s + 2.13·43-s − 0.583·47-s − 1/7·49-s + 1.12·51-s + 2.19·53-s − 0.529·57-s + 0.260·59-s − 1.28·61-s + 0.503·63-s + 1.22·67-s − 1.92·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{21} \cdot 5^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{21} \cdot 5^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(2^{21} \cdot 5^{6}\)
Sign: $1$
Analytic conductor: \(16683.2\)
Root analytic conductor: \(5.05491\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{3200} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 2^{21} \cdot 5^{6} ,\ ( \ : 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(3.950106580\)
\(L(\frac12)\) \(\approx\) \(3.950106580\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
good3$S_4\times C_2$ \( 1 - 2 T + 5 T^{2} - 8 T^{3} + 5 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
7$S_4\times C_2$ \( 1 + 4 T + 17 T^{2} + 36 T^{3} + 17 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
11$S_4\times C_2$ \( 1 + 2 T + 13 T^{2} + 36 T^{3} + 13 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
13$S_4\times C_2$ \( 1 - 8 T + 47 T^{2} - 192 T^{3} + 47 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
17$S_4\times C_2$ \( 1 - 4 T + 35 T^{2} - 104 T^{3} + 35 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
19$S_4\times C_2$ \( 1 + 2 T + 21 T^{2} - 28 T^{3} + 21 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
23$S_4\times C_2$ \( 1 + 8 T + 81 T^{2} + 372 T^{3} + 81 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \)
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{3} \)
31$S_4\times C_2$ \( 1 - 8 T + 61 T^{2} - 368 T^{3} + 61 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 - 8 T + 79 T^{2} - 320 T^{3} + 79 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 - 2 T + 71 T^{2} + 20 T^{3} + 71 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 - 14 T + 149 T^{2} - 1104 T^{3} + 149 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 + 4 T + 113 T^{2} + 260 T^{3} + 113 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 - 16 T + 231 T^{2} - 1776 T^{3} + 231 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 - 2 T + 141 T^{2} - 132 T^{3} + 141 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 + 10 T + 155 T^{2} + 1212 T^{3} + 155 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 - 10 T + 141 T^{2} - 736 T^{3} + 141 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 + 12 T + 197 T^{2} + 1384 T^{3} + 197 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 - 20 T + 3 p T^{2} - 1736 T^{3} + 3 p^{2} T^{4} - 20 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 - 16 T + 269 T^{2} - 2400 T^{3} + 269 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 - 30 T + 509 T^{2} - 5504 T^{3} + 509 p T^{4} - 30 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 + 10 T + 151 T^{2} + 684 T^{3} + 151 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 - 28 T + 531 T^{2} - 6040 T^{3} + 531 p T^{4} - 28 p^{2} T^{5} + p^{3} T^{6} \)
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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

−7.80959392968645876183728114889, −7.60550200502344598342553327369, −7.35757565867414757516501283713, −6.82075854402774242069948601881, −6.43118157873599571573687003388, −6.41657002393830931539206937117, −6.28264717680533170321999305362, −5.94878579842365968820477616613, −5.63590454449603061374276801229, −5.62828377538297839016106892659, −5.10226577404750916164069958989, −4.83451532489689195433874734414, −4.35854511843336284965666949344, −3.93492390988198857089776231771, −3.93371884234546285162477885493, −3.70162419160270926244400326940, −3.18477104255624701687650277508, −3.14354575400268931518202721820, −2.94431944730512403354088444844, −2.27229583425703584225485956923, −2.20001635592051120872546725096, −2.02729712150983965517664289534, −1.10224269720422118375847820065, −0.910953786201455827732466803708, −0.42084165455400497960245270812, 0.42084165455400497960245270812, 0.910953786201455827732466803708, 1.10224269720422118375847820065, 2.02729712150983965517664289534, 2.20001635592051120872546725096, 2.27229583425703584225485956923, 2.94431944730512403354088444844, 3.14354575400268931518202721820, 3.18477104255624701687650277508, 3.70162419160270926244400326940, 3.93371884234546285162477885493, 3.93492390988198857089776231771, 4.35854511843336284965666949344, 4.83451532489689195433874734414, 5.10226577404750916164069958989, 5.62828377538297839016106892659, 5.63590454449603061374276801229, 5.94878579842365968820477616613, 6.28264717680533170321999305362, 6.41657002393830931539206937117, 6.43118157873599571573687003388, 6.82075854402774242069948601881, 7.35757565867414757516501283713, 7.60550200502344598342553327369, 7.80959392968645876183728114889

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