Properties

Label 6-825e3-1.1-c3e3-0-2
Degree $6$
Conductor $561515625$
Sign $-1$
Analytic cond. $115334.$
Root an. cond. $6.97686$
Motivic weight $3$
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  + 2·2-s − 9·3-s + 5·4-s − 18·6-s − 10·7-s − 4·8-s + 54·9-s + 33·11-s − 45·12-s − 114·13-s − 20·14-s − 27·16-s + 104·17-s + 108·18-s − 58·19-s + 90·21-s + 66·22-s − 120·23-s + 36·24-s − 228·26-s − 270·27-s − 50·28-s − 220·29-s + 248·31-s + 6·32-s − 297·33-s + 208·34-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.73·3-s + 5/8·4-s − 1.22·6-s − 0.539·7-s − 0.176·8-s + 2·9-s + 0.904·11-s − 1.08·12-s − 2.43·13-s − 0.381·14-s − 0.421·16-s + 1.48·17-s + 1.41·18-s − 0.700·19-s + 0.935·21-s + 0.639·22-s − 1.08·23-s + 0.306·24-s − 1.71·26-s − 1.92·27-s − 0.337·28-s − 1.40·29-s + 1.43·31-s + 0.0331·32-s − 1.56·33-s + 1.04·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 5^{6} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 5^{6} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(3^{3} \cdot 5^{6} \cdot 11^{3}\)
Sign: $-1$
Analytic conductor: \(115334.\)
Root analytic conductor: \(6.97686\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{825} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(3\)
Selberg data: \((6,\ 3^{3} \cdot 5^{6} \cdot 11^{3} ,\ ( \ : 3/2, 3/2, 3/2 ),\ -1 )\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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$C_1$ \( ( 1 + p T )^{3} \)
5 \( 1 \)
11$C_1$ \( ( 1 - p T )^{3} \)
good2$S_4\times C_2$ \( 1 - p T - T^{2} + p^{4} T^{3} - p^{3} T^{4} - p^{7} T^{5} + p^{9} T^{6} \)
7$S_4\times C_2$ \( 1 + 10 T + 425 T^{2} + 3412 T^{3} + 425 p^{3} T^{4} + 10 p^{6} T^{5} + p^{9} T^{6} \)
13$S_4\times C_2$ \( 1 + 114 T + 10303 T^{2} + 538132 T^{3} + 10303 p^{3} T^{4} + 114 p^{6} T^{5} + p^{9} T^{6} \)
17$S_4\times C_2$ \( 1 - 104 T + 17531 T^{2} - 1030352 T^{3} + 17531 p^{3} T^{4} - 104 p^{6} T^{5} + p^{9} T^{6} \)
19$S_4\times C_2$ \( 1 + 58 T + 9081 T^{2} + 861164 T^{3} + 9081 p^{3} T^{4} + 58 p^{6} T^{5} + p^{9} T^{6} \)
23$S_4\times C_2$ \( 1 + 120 T + 25765 T^{2} + 2771856 T^{3} + 25765 p^{3} T^{4} + 120 p^{6} T^{5} + p^{9} T^{6} \)
29$S_4\times C_2$ \( 1 + 220 T + 58859 T^{2} + 10101400 T^{3} + 58859 p^{3} T^{4} + 220 p^{6} T^{5} + p^{9} T^{6} \)
31$S_4\times C_2$ \( 1 - 8 p T + 50781 T^{2} - 5187088 T^{3} + 50781 p^{3} T^{4} - 8 p^{7} T^{5} + p^{9} T^{6} \)
37$S_4\times C_2$ \( 1 + 838 T + 375507 T^{2} + 103501764 T^{3} + 375507 p^{3} T^{4} + 838 p^{6} T^{5} + p^{9} T^{6} \)
41$S_4\times C_2$ \( 1 - 156 T + 100167 T^{2} - 24516984 T^{3} + 100167 p^{3} T^{4} - 156 p^{6} T^{5} + p^{9} T^{6} \)
43$S_4\times C_2$ \( 1 + 122 T + 193581 T^{2} + 17954308 T^{3} + 193581 p^{3} T^{4} + 122 p^{6} T^{5} + p^{9} T^{6} \)
47$S_4\times C_2$ \( 1 + 504 T + 393661 T^{2} + 109025808 T^{3} + 393661 p^{3} T^{4} + 504 p^{6} T^{5} + p^{9} T^{6} \)
53$S_4\times C_2$ \( 1 + 282 T + 224563 T^{2} + 87620892 T^{3} + 224563 p^{3} T^{4} + 282 p^{6} T^{5} + p^{9} T^{6} \)
59$S_4\times C_2$ \( 1 - 548 T + 11419 p T^{2} - 226302104 T^{3} + 11419 p^{4} T^{4} - 548 p^{6} T^{5} + p^{9} T^{6} \)
61$S_4\times C_2$ \( 1 - 414 T - 389 T^{2} + 154404524 T^{3} - 389 p^{3} T^{4} - 414 p^{6} T^{5} + p^{9} T^{6} \)
67$S_4\times C_2$ \( 1 - 428 T + 695537 T^{2} - 249317576 T^{3} + 695537 p^{3} T^{4} - 428 p^{6} T^{5} + p^{9} T^{6} \)
71$S_4\times C_2$ \( 1 + 912 T + 1171621 T^{2} + 649961952 T^{3} + 1171621 p^{3} T^{4} + 912 p^{6} T^{5} + p^{9} T^{6} \)
73$S_4\times C_2$ \( 1 + 618 T + 1066507 T^{2} + 454366420 T^{3} + 1066507 p^{3} T^{4} + 618 p^{6} T^{5} + p^{9} T^{6} \)
79$S_4\times C_2$ \( 1 + 542 T + 1317893 T^{2} + 445950836 T^{3} + 1317893 p^{3} T^{4} + 542 p^{6} T^{5} + p^{9} T^{6} \)
83$S_4\times C_2$ \( 1 + 624021 T^{2} + 434328048 T^{3} + 624021 p^{3} T^{4} + p^{9} T^{6} \)
89$S_4\times C_2$ \( 1 - 790 T - 3625 T^{2} + 827778380 T^{3} - 3625 p^{3} T^{4} - 790 p^{6} T^{5} + p^{9} T^{6} \)
97$S_4\times C_2$ \( 1 + 2074 T + 3705231 T^{2} + 3687692268 T^{3} + 3705231 p^{3} T^{4} + 2074 p^{6} T^{5} + p^{9} T^{6} \)
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

−9.455059253445627846431972477339, −8.789070329456716328700463922895, −8.507383788613095679588096596369, −8.400916356948033273541867174937, −7.66971650226535703084286455320, −7.59864198458999920060838161953, −7.22476365326484019209834724176, −7.04021622172011983038074682840, −6.67428184122129793160850072834, −6.42590762522689710887739135138, −6.19632369292971837073748416110, −5.94190570289524022226308542413, −5.48066244628059150307027467943, −5.25841283615790482793916750187, −4.95270927244098311351284433210, −4.86201891524306016678154184888, −4.27893732612711307807493969845, −3.98717575804290234756498577318, −3.65529333290895452112174093378, −3.17526088588212173964864198726, −2.96090904958045158740593924008, −2.24488153647382894389430809733, −2.01897951929802681699068491839, −1.46064607255351147116592512346, −1.09068984393183498145687599081, 0, 0, 0, 1.09068984393183498145687599081, 1.46064607255351147116592512346, 2.01897951929802681699068491839, 2.24488153647382894389430809733, 2.96090904958045158740593924008, 3.17526088588212173964864198726, 3.65529333290895452112174093378, 3.98717575804290234756498577318, 4.27893732612711307807493969845, 4.86201891524306016678154184888, 4.95270927244098311351284433210, 5.25841283615790482793916750187, 5.48066244628059150307027467943, 5.94190570289524022226308542413, 6.19632369292971837073748416110, 6.42590762522689710887739135138, 6.67428184122129793160850072834, 7.04021622172011983038074682840, 7.22476365326484019209834724176, 7.59864198458999920060838161953, 7.66971650226535703084286455320, 8.400916356948033273541867174937, 8.507383788613095679588096596369, 8.789070329456716328700463922895, 9.455059253445627846431972477339

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