Properties

Label 6-69e3-1.1-c5e3-0-0
Degree $6$
Conductor $328509$
Sign $-1$
Analytic cond. $1355.27$
Root an. cond. $3.32663$
Motivic weight $5$
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  − 8·2-s + 27·3-s − 5·4-s − 56·5-s − 216·6-s − 114·7-s + 126·8-s + 486·9-s + 448·10-s − 376·11-s − 135·12-s − 858·13-s + 912·14-s − 1.51e3·15-s + 857·16-s − 2.54e3·17-s − 3.88e3·18-s − 2.84e3·19-s + 280·20-s − 3.07e3·21-s + 3.00e3·22-s + 1.58e3·23-s + 3.40e3·24-s − 2.74e3·25-s + 6.86e3·26-s + 7.29e3·27-s + 570·28-s + ⋯
L(s)  = 1  − 1.41·2-s + 1.73·3-s − 0.156·4-s − 1.00·5-s − 2.44·6-s − 0.879·7-s + 0.696·8-s + 2·9-s + 1.41·10-s − 0.936·11-s − 0.270·12-s − 1.40·13-s + 1.24·14-s − 1.73·15-s + 0.836·16-s − 2.13·17-s − 2.82·18-s − 1.80·19-s + 0.156·20-s − 1.52·21-s + 1.32·22-s + 0.625·23-s + 1.20·24-s − 0.877·25-s + 1.99·26-s + 1.92·27-s + 0.137·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 328509 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 328509 ^{s/2} \, \Gamma_{\C}(s+5/2)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(328509\)    =    \(3^{3} \cdot 23^{3}\)
Sign: $-1$
Analytic conductor: \(1355.27\)
Root analytic conductor: \(3.32663\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{69} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(3\)
Selberg data: \((6,\ 328509,\ (\ :5/2, 5/2, 5/2),\ -1)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) 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)$
bad3$C_1$ \( ( 1 - p^{2} T )^{3} \)
23$C_1$ \( ( 1 - p^{2} T )^{3} \)
good2$S_4\times C_2$ \( 1 + p^{3} T + 69 T^{2} + 233 p T^{3} + 69 p^{5} T^{4} + p^{13} T^{5} + p^{15} T^{6} \)
5$S_4\times C_2$ \( 1 + 56 T + 5879 T^{2} + 154424 T^{3} + 5879 p^{5} T^{4} + 56 p^{10} T^{5} + p^{15} T^{6} \)
7$S_4\times C_2$ \( 1 + 114 T + 29473 T^{2} + 3883732 T^{3} + 29473 p^{5} T^{4} + 114 p^{10} T^{5} + p^{15} T^{6} \)
11$S_4\times C_2$ \( 1 + 376 T + 252481 T^{2} + 142251704 T^{3} + 252481 p^{5} T^{4} + 376 p^{10} T^{5} + p^{15} T^{6} \)
13$S_4\times C_2$ \( 1 + 66 p T + 673923 T^{2} + 268856412 T^{3} + 673923 p^{5} T^{4} + 66 p^{11} T^{5} + p^{15} T^{6} \)
17$S_4\times C_2$ \( 1 + 2548 T + 5764387 T^{2} + 7106478632 T^{3} + 5764387 p^{5} T^{4} + 2548 p^{10} T^{5} + p^{15} T^{6} \)
19$S_4\times C_2$ \( 1 + 2846 T + 487079 p T^{2} + 14089642340 T^{3} + 487079 p^{6} T^{4} + 2846 p^{10} T^{5} + p^{15} T^{6} \)
29$S_4\times C_2$ \( 1 + 16370 T + 143935155 T^{2} + 789370759340 T^{3} + 143935155 p^{5} T^{4} + 16370 p^{10} T^{5} + p^{15} T^{6} \)
31$S_4\times C_2$ \( 1 + 476 p T + 138580077 T^{2} + 861567646264 T^{3} + 138580077 p^{5} T^{4} + 476 p^{11} T^{5} + p^{15} T^{6} \)
37$S_4\times C_2$ \( 1 - 15874 T + 283329963 T^{2} - 2305001420148 T^{3} + 283329963 p^{5} T^{4} - 15874 p^{10} T^{5} + p^{15} T^{6} \)
41$S_4\times C_2$ \( 1 - 12606 T + 331997335 T^{2} - 2707384025828 T^{3} + 331997335 p^{5} T^{4} - 12606 p^{10} T^{5} + p^{15} T^{6} \)
43$S_4\times C_2$ \( 1 - 3154 T + 339001869 T^{2} - 517383556556 T^{3} + 339001869 p^{5} T^{4} - 3154 p^{10} T^{5} + p^{15} T^{6} \)
47$S_4\times C_2$ \( 1 - 29928 T + 824463709 T^{2} - 13203001936176 T^{3} + 824463709 p^{5} T^{4} - 29928 p^{10} T^{5} + p^{15} T^{6} \)
53$S_4\times C_2$ \( 1 + 44084 T + 710047655 T^{2} + 6683794054544 T^{3} + 710047655 p^{5} T^{4} + 44084 p^{10} T^{5} + p^{15} T^{6} \)
59$S_4\times C_2$ \( 1 + 29300 T + 1744914257 T^{2} + 37824050997176 T^{3} + 1744914257 p^{5} T^{4} + 29300 p^{10} T^{5} + p^{15} T^{6} \)
61$S_4\times C_2$ \( 1 - 54010 T + 3288997283 T^{2} - 91927672344788 T^{3} + 3288997283 p^{5} T^{4} - 54010 p^{10} T^{5} + p^{15} T^{6} \)
67$S_4\times C_2$ \( 1 - 43390 T + 1100710197 T^{2} + 11754561587628 T^{3} + 1100710197 p^{5} T^{4} - 43390 p^{10} T^{5} + p^{15} T^{6} \)
71$S_4\times C_2$ \( 1 - 23424 T + 4239683749 T^{2} - 58278976303616 T^{3} + 4239683749 p^{5} T^{4} - 23424 p^{10} T^{5} + p^{15} T^{6} \)
73$S_4\times C_2$ \( 1 + 91402 T + 5485513767 T^{2} + 259506540394092 T^{3} + 5485513767 p^{5} T^{4} + 91402 p^{10} T^{5} + p^{15} T^{6} \)
79$S_4\times C_2$ \( 1 + 49398 T + 5918352025 T^{2} + 210022241166364 T^{3} + 5918352025 p^{5} T^{4} + 49398 p^{10} T^{5} + p^{15} T^{6} \)
83$S_4\times C_2$ \( 1 + 103936 T + 5338515353 T^{2} + 124565773223896 T^{3} + 5338515353 p^{5} T^{4} + 103936 p^{10} T^{5} + p^{15} T^{6} \)
89$S_4\times C_2$ \( 1 - 96112 T + 15326136523 T^{2} - 970745006227616 T^{3} + 15326136523 p^{5} T^{4} - 96112 p^{10} T^{5} + p^{15} T^{6} \)
97$S_4\times C_2$ \( 1 + 135318 T + 21864400719 T^{2} + 2241137600844532 T^{3} + 21864400719 p^{5} T^{4} + 135318 p^{10} T^{5} + p^{15} T^{6} \)
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

−13.02824546645033867602483966886, −12.72646068368018810513857599117, −12.39929156605579735569153989201, −11.45777910379020830157247897288, −11.12279833255946615410651079087, −10.89030682676598167931342723265, −10.40776779568168841346057068582, −9.683486347078546532490312006770, −9.429406645727139011833165983230, −9.357905279675207803957630838114, −9.100166308007696524322754992864, −8.624749963165132819704580509927, −8.219743289068679694242388266965, −7.68341097708939397240904285760, −7.60390341143152664273956425077, −7.22676723287847310542670137545, −6.58143900012162579823380391934, −5.84392042839921085134309680316, −5.27476984152840704018611085494, −4.29743650348217054499033465717, −4.10516298989039327972984346540, −3.73197627905230858390345858811, −2.79437311095028638023727714666, −2.37827262495584277283687295674, −1.78100448633095974814191542570, 0, 0, 0, 1.78100448633095974814191542570, 2.37827262495584277283687295674, 2.79437311095028638023727714666, 3.73197627905230858390345858811, 4.10516298989039327972984346540, 4.29743650348217054499033465717, 5.27476984152840704018611085494, 5.84392042839921085134309680316, 6.58143900012162579823380391934, 7.22676723287847310542670137545, 7.60390341143152664273956425077, 7.68341097708939397240904285760, 8.219743289068679694242388266965, 8.624749963165132819704580509927, 9.100166308007696524322754992864, 9.357905279675207803957630838114, 9.429406645727139011833165983230, 9.683486347078546532490312006770, 10.40776779568168841346057068582, 10.89030682676598167931342723265, 11.12279833255946615410651079087, 11.45777910379020830157247897288, 12.39929156605579735569153989201, 12.72646068368018810513857599117, 13.02824546645033867602483966886

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