Properties

Label 6-1904e3-1.1-c1e3-0-2
Degree $6$
Conductor $6902411264$
Sign $-1$
Analytic cond. $3514.24$
Root an. cond. $3.89916$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $3$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 3·5-s − 3·7-s − 3·9-s − 6·11-s − 3·15-s − 3·17-s − 4·19-s + 3·21-s − 4·23-s − 5·25-s − 27-s − 8·29-s − 3·31-s + 6·33-s − 9·35-s − 4·37-s + 9·41-s + 43-s − 9·45-s − 8·47-s + 6·49-s + 3·51-s + 19·53-s − 18·55-s + 4·57-s − 14·59-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.34·5-s − 1.13·7-s − 9-s − 1.80·11-s − 0.774·15-s − 0.727·17-s − 0.917·19-s + 0.654·21-s − 0.834·23-s − 25-s − 0.192·27-s − 1.48·29-s − 0.538·31-s + 1.04·33-s − 1.52·35-s − 0.657·37-s + 1.40·41-s + 0.152·43-s − 1.34·45-s − 1.16·47-s + 6/7·49-s + 0.420·51-s + 2.60·53-s − 2.42·55-s + 0.529·57-s − 1.82·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 7^{3} \cdot 17^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 7^{3} \cdot 17^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(2^{12} \cdot 7^{3} \cdot 17^{3}\)
Sign: $-1$
Analytic conductor: \(3514.24\)
Root analytic conductor: \(3.89916\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(3\)
Selberg data: \((6,\ 2^{12} \cdot 7^{3} \cdot 17^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ -1 )\)

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 \( 1 \)
7$C_1$ \( ( 1 + T )^{3} \)
17$C_1$ \( ( 1 + T )^{3} \)
good3$S_4\times C_2$ \( 1 + T + 4 T^{2} + 8 T^{3} + 4 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \)
5$S_4\times C_2$ \( 1 - 3 T + 14 T^{2} - 26 T^{3} + 14 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \)
11$S_4\times C_2$ \( 1 + 6 T + 29 T^{2} + 100 T^{3} + 29 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
13$C_2$ \( ( 1 + p T^{2} )^{3} \)
19$S_4\times C_2$ \( 1 + 4 T + 41 T^{2} + 96 T^{3} + 41 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
23$S_4\times C_2$ \( 1 + 4 T + 25 T^{2} + 216 T^{3} + 25 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
29$S_4\times C_2$ \( 1 + 8 T + 3 p T^{2} + 456 T^{3} + 3 p^{2} T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 + 3 T + 92 T^{2} + 182 T^{3} + 92 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 + 4 T + 95 T^{2} + 240 T^{3} + 95 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 - 9 T + 12 T^{2} + 268 T^{3} + 12 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 - T - 2 T^{2} + 98 T^{3} - 2 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 + 8 T + 77 T^{2} + 304 T^{3} + 77 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 - 19 T + 260 T^{2} - 40 p T^{3} + 260 p T^{4} - 19 p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 + 14 T + 113 T^{2} + 724 T^{3} + 113 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 - T + 46 T^{2} + 2 T^{3} + 46 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 + 7 T + 20 T^{2} - 570 T^{3} + 20 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 + 6 T + 81 T^{2} + 788 T^{3} + 81 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 + 5 T + 212 T^{2} + 732 T^{3} + 212 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \)
79$C_2$ \( ( 1 + 8 T + p T^{2} )^{3} \)
83$S_4\times C_2$ \( 1 + 4 T + 25 T^{2} + 1056 T^{3} + 25 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 - 10 T + 251 T^{2} - 1748 T^{3} + 251 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 - 13 T + 316 T^{2} - 2476 T^{3} + 316 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \)
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

−8.573186758291310578008271090905, −8.420861538605539450280535626324, −8.050094262371263042663552435426, −7.67612604488545837635398672755, −7.51542890178893285120226536209, −7.17938206566582065798415108528, −7.15670972614571563604606766310, −6.37955443424674527728166143691, −6.36164833087686256864331119335, −6.10351554236201601384242826197, −5.83623317887716347547937735355, −5.72990585160382804511369732012, −5.42945206776228320631667181750, −5.15576716399970425719434571391, −5.02525681740044008423686220107, −4.30450499563912716767404608256, −4.07305135997770552617002934518, −3.87652047629978531340788227141, −3.56891463428426713562520825859, −2.81276333203957859199325230635, −2.70552934607491041694729981692, −2.61711504428476638245585468727, −1.98043581593644227665291838762, −1.87599030954008853504029743434, −1.32108607976346260126523355971, 0, 0, 0, 1.32108607976346260126523355971, 1.87599030954008853504029743434, 1.98043581593644227665291838762, 2.61711504428476638245585468727, 2.70552934607491041694729981692, 2.81276333203957859199325230635, 3.56891463428426713562520825859, 3.87652047629978531340788227141, 4.07305135997770552617002934518, 4.30450499563912716767404608256, 5.02525681740044008423686220107, 5.15576716399970425719434571391, 5.42945206776228320631667181750, 5.72990585160382804511369732012, 5.83623317887716347547937735355, 6.10351554236201601384242826197, 6.36164833087686256864331119335, 6.37955443424674527728166143691, 7.15670972614571563604606766310, 7.17938206566582065798415108528, 7.51542890178893285120226536209, 7.67612604488545837635398672755, 8.050094262371263042663552435426, 8.420861538605539450280535626324, 8.573186758291310578008271090905

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