Properties

Label 6-1900e3-1.1-c1e3-0-0
Degree $6$
Conductor $6859000000$
Sign $1$
Analytic cond. $3492.14$
Root an. cond. $3.89507$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s + 2·7-s − 2·9-s − 11-s + 13-s + 2·17-s − 3·19-s + 4·21-s + 8·23-s − 5·27-s − 7·29-s + 11·31-s − 2·33-s − 5·37-s + 2·39-s + 13·41-s + 11·43-s + 19·47-s − 14·49-s + 4·51-s + 11·53-s − 6·57-s − 6·59-s + 7·61-s − 4·63-s + 3·67-s + 16·69-s + ⋯
L(s)  = 1  + 1.15·3-s + 0.755·7-s − 2/3·9-s − 0.301·11-s + 0.277·13-s + 0.485·17-s − 0.688·19-s + 0.872·21-s + 1.66·23-s − 0.962·27-s − 1.29·29-s + 1.97·31-s − 0.348·33-s − 0.821·37-s + 0.320·39-s + 2.03·41-s + 1.67·43-s + 2.77·47-s − 2·49-s + 0.560·51-s + 1.51·53-s − 0.794·57-s − 0.781·59-s + 0.896·61-s − 0.503·63-s + 0.366·67-s + 1.92·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 5^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 5^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(2^{6} \cdot 5^{6} \cdot 19^{3}\)
Sign: $1$
Analytic conductor: \(3492.14\)
Root analytic conductor: \(3.89507\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 2^{6} \cdot 5^{6} \cdot 19^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(5.731591719\)
\(L(\frac12)\) \(\approx\) \(5.731591719\)
\(L(\frac{3}{2})\) 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 \( 1 \)
19$C_1$ \( ( 1 + T )^{3} \)
good3$S_4\times C_2$ \( 1 - 2 T + 2 p T^{2} - 11 T^{3} + 2 p^{2} T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
7$S_4\times C_2$ \( 1 - 2 T + 18 T^{2} - 27 T^{3} + 18 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
11$S_4\times C_2$ \( 1 + T + 7 T^{2} + 37 T^{3} + 7 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \)
13$S_4\times C_2$ \( 1 - T + 31 T^{2} - 23 T^{3} + 31 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \)
17$S_4\times C_2$ \( 1 - 2 T + 48 T^{2} - 67 T^{3} + 48 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
23$S_4\times C_2$ \( 1 - 8 T + 66 T^{2} - 359 T^{3} + 66 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
29$S_4\times C_2$ \( 1 + 7 T + 77 T^{2} + 381 T^{3} + 77 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 - 11 T + 87 T^{2} - 441 T^{3} + 87 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 + 5 T + 39 T^{2} - 35 T^{3} + 39 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 - 13 T + 159 T^{2} - 1091 T^{3} + 159 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 - 11 T + 152 T^{2} - 919 T^{3} + 152 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 - 19 T + 209 T^{2} - 1723 T^{3} + 209 p T^{4} - 19 p^{2} T^{5} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 - 11 T + 182 T^{2} - 1139 T^{3} + 182 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 + 6 T + T^{2} - 148 T^{3} + p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 - 7 T + 153 T^{2} - 879 T^{3} + 153 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 - 3 T + 73 T^{2} + 197 T^{3} + 73 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 + 5 T + 140 T^{2} + 833 T^{3} + 140 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 - 9 T + 201 T^{2} - 1287 T^{3} + 201 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 + 18 T + 238 T^{2} + 2237 T^{3} + 238 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 - 3 T + T^{2} + 251 T^{3} + p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 + 84 T^{2} - 857 T^{3} + 84 p T^{4} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 + 3 T + 219 T^{2} + 501 T^{3} + 219 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.481972547614559359798996326538, −7.80931231480898487064626819310, −7.66332254029826154735186500848, −7.52575933911007915414622087769, −7.37840270108168496588298059051, −6.86637242952064693874560778884, −6.67391065364349984203915423216, −6.14042138886561357707251506789, −6.06926343718301189502592623341, −5.78432413480489783661529548490, −5.39461165107585422156210048695, −5.24356054433342705557098206047, −4.86441556924652883129086445900, −4.46734873583733420544104633060, −4.19941150245915004172982549530, −4.10668227728553220784954911106, −3.40982862762143731522495822904, −3.16893294686290915309296093749, −3.15178553107687765791951189834, −2.41156306355030215675629787629, −2.32866663840395160526288463377, −2.25951248532388438784273882041, −1.40318524313527581858812944527, −0.972027755179263679599206240816, −0.59576588304875937712499062945, 0.59576588304875937712499062945, 0.972027755179263679599206240816, 1.40318524313527581858812944527, 2.25951248532388438784273882041, 2.32866663840395160526288463377, 2.41156306355030215675629787629, 3.15178553107687765791951189834, 3.16893294686290915309296093749, 3.40982862762143731522495822904, 4.10668227728553220784954911106, 4.19941150245915004172982549530, 4.46734873583733420544104633060, 4.86441556924652883129086445900, 5.24356054433342705557098206047, 5.39461165107585422156210048695, 5.78432413480489783661529548490, 6.06926343718301189502592623341, 6.14042138886561357707251506789, 6.67391065364349984203915423216, 6.86637242952064693874560778884, 7.37840270108168496588298059051, 7.52575933911007915414622087769, 7.66332254029826154735186500848, 7.80931231480898487064626819310, 8.481972547614559359798996326538

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