Properties

Label 6-3200e3-1.1-c1e3-0-4
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 $3$

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: \(3\)
Selberg data: \((6,\ 2^{21} \cdot 5^{6} ,\ ( \ : 1/2, 1/2, 1/2 ),\ -1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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

−8.061916963971535515451910583893, −7.74329040001408118332557337193, −7.53850948003489159010684628470, −7.28848271779358453382459274823, −6.84144942990417879782175032101, −6.75599148957352270341634295761, −6.43488057214508888743285331245, −6.25825971210638845216192345019, −6.23508442497104636865024695042, −5.78490112392852066806254395019, −5.47869448250385351962819572016, −5.14328100056141709087713094219, −5.14276002155579788487694588854, −4.71366110708683662573541108506, −4.58414961466488286491578627007, −4.19501424917633632855944506564, −3.57711524724079370275944885397, −3.53213132492645684856537405150, −3.38100836412571057686855863172, −2.71083007965054194087611686148, −2.66639596194841814838482813479, −2.48389997533222473437659389103, −1.71860582593336625004075770198, −1.45879773908393767285652525300, −1.00155140567354812934813386066, 0, 0, 0, 1.00155140567354812934813386066, 1.45879773908393767285652525300, 1.71860582593336625004075770198, 2.48389997533222473437659389103, 2.66639596194841814838482813479, 2.71083007965054194087611686148, 3.38100836412571057686855863172, 3.53213132492645684856537405150, 3.57711524724079370275944885397, 4.19501424917633632855944506564, 4.58414961466488286491578627007, 4.71366110708683662573541108506, 5.14276002155579788487694588854, 5.14328100056141709087713094219, 5.47869448250385351962819572016, 5.78490112392852066806254395019, 6.23508442497104636865024695042, 6.25825971210638845216192345019, 6.43488057214508888743285331245, 6.75599148957352270341634295761, 6.84144942990417879782175032101, 7.28848271779358453382459274823, 7.53850948003489159010684628470, 7.74329040001408118332557337193, 8.061916963971535515451910583893

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