Properties

Label 6-1080e3-1.1-c3e3-0-5
Degree $6$
Conductor $1259712000$
Sign $1$
Analytic cond. $258743.$
Root an. cond. $7.98261$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 15·5-s + 9·7-s + 18·11-s − 21·13-s + 84·17-s + 21·19-s + 48·23-s + 150·25-s − 36·29-s + 324·31-s + 135·35-s + 33·37-s + 114·41-s + 282·43-s + 282·47-s − 360·49-s + 222·53-s + 270·55-s − 276·59-s + 303·61-s − 315·65-s + 1.03e3·67-s + 510·71-s + 447·73-s + 162·77-s + 777·79-s + 78·83-s + ⋯
L(s)  = 1  + 1.34·5-s + 0.485·7-s + 0.493·11-s − 0.448·13-s + 1.19·17-s + 0.253·19-s + 0.435·23-s + 6/5·25-s − 0.230·29-s + 1.87·31-s + 0.651·35-s + 0.146·37-s + 0.434·41-s + 1.00·43-s + 0.875·47-s − 1.04·49-s + 0.575·53-s + 0.661·55-s − 0.609·59-s + 0.635·61-s − 0.601·65-s + 1.88·67-s + 0.852·71-s + 0.716·73-s + 0.239·77-s + 1.10·79-s + 0.103·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{9} \cdot 5^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{9} \cdot 5^{3}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(2^{9} \cdot 3^{9} \cdot 5^{3}\)
Sign: $1$
Analytic conductor: \(258743.\)
Root analytic conductor: \(7.98261\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1080} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 2^{9} \cdot 3^{9} \cdot 5^{3} ,\ ( \ : 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(12.63606558\)
\(L(\frac12)\) \(\approx\) \(12.63606558\)
\(L(\frac{5}{2})\) 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)$
bad2 \( 1 \)
3 \( 1 \)
5$C_1$ \( ( 1 - p T )^{3} \)
good7$S_4\times C_2$ \( 1 - 9 T + 9 p^{2} T^{2} - 650 T^{3} + 9 p^{5} T^{4} - 9 p^{6} T^{5} + p^{9} T^{6} \)
11$S_4\times C_2$ \( 1 - 18 T + 912 T^{2} + 28510 T^{3} + 912 p^{3} T^{4} - 18 p^{6} T^{5} + p^{9} T^{6} \)
13$S_4\times C_2$ \( 1 + 21 T + 2910 T^{2} + 4721 p T^{3} + 2910 p^{3} T^{4} + 21 p^{6} T^{5} + p^{9} T^{6} \)
17$S_4\times C_2$ \( 1 - 84 T + 14706 T^{2} - 738652 T^{3} + 14706 p^{3} T^{4} - 84 p^{6} T^{5} + p^{9} T^{6} \)
19$S_4\times C_2$ \( 1 - 21 T + 15189 T^{2} - 149614 T^{3} + 15189 p^{3} T^{4} - 21 p^{6} T^{5} + p^{9} T^{6} \)
23$S_4\times C_2$ \( 1 - 48 T + 36654 T^{2} - 1165994 T^{3} + 36654 p^{3} T^{4} - 48 p^{6} T^{5} + p^{9} T^{6} \)
29$S_4\times C_2$ \( 1 + 36 T + 31512 T^{2} - 1278578 T^{3} + 31512 p^{3} T^{4} + 36 p^{6} T^{5} + p^{9} T^{6} \)
31$S_4\times C_2$ \( 1 - 324 T + 70560 T^{2} - 10091392 T^{3} + 70560 p^{3} T^{4} - 324 p^{6} T^{5} + p^{9} T^{6} \)
37$S_4\times C_2$ \( 1 - 33 T + 107211 T^{2} - 491142 T^{3} + 107211 p^{3} T^{4} - 33 p^{6} T^{5} + p^{9} T^{6} \)
41$S_4\times C_2$ \( 1 - 114 T + 125619 T^{2} - 13866756 T^{3} + 125619 p^{3} T^{4} - 114 p^{6} T^{5} + p^{9} T^{6} \)
43$S_4\times C_2$ \( 1 - 282 T + 220578 T^{2} - 43728712 T^{3} + 220578 p^{3} T^{4} - 282 p^{6} T^{5} + p^{9} T^{6} \)
47$S_4\times C_2$ \( 1 - 6 p T + 152826 T^{2} - 69794468 T^{3} + 152826 p^{3} T^{4} - 6 p^{7} T^{5} + p^{9} T^{6} \)
53$S_4\times C_2$ \( 1 - 222 T + 282615 T^{2} - 72010284 T^{3} + 282615 p^{3} T^{4} - 222 p^{6} T^{5} + p^{9} T^{6} \)
59$C_2$ \( ( 1 + 92 T + p^{3} T^{2} )^{3} \)
61$S_4\times C_2$ \( 1 - 303 T + 620067 T^{2} - 124220266 T^{3} + 620067 p^{3} T^{4} - 303 p^{6} T^{5} + p^{9} T^{6} \)
67$S_4\times C_2$ \( 1 - 1035 T + 63873 T^{2} + 227096926 T^{3} + 63873 p^{3} T^{4} - 1035 p^{6} T^{5} + p^{9} T^{6} \)
71$S_4\times C_2$ \( 1 - 510 T + 1098285 T^{2} - 353629860 T^{3} + 1098285 p^{3} T^{4} - 510 p^{6} T^{5} + p^{9} T^{6} \)
73$S_4\times C_2$ \( 1 - 447 T + 657111 T^{2} - 269206770 T^{3} + 657111 p^{3} T^{4} - 447 p^{6} T^{5} + p^{9} T^{6} \)
79$S_4\times C_2$ \( 1 - 777 T + 14268 p T^{2} - 780182781 T^{3} + 14268 p^{4} T^{4} - 777 p^{6} T^{5} + p^{9} T^{6} \)
83$S_4\times C_2$ \( 1 - 78 T + 882717 T^{2} - 176745340 T^{3} + 882717 p^{3} T^{4} - 78 p^{6} T^{5} + p^{9} T^{6} \)
89$S_4\times C_2$ \( 1 + 324 T + 1606299 T^{2} + 549274824 T^{3} + 1606299 p^{3} T^{4} + 324 p^{6} T^{5} + p^{9} T^{6} \)
97$S_4\times C_2$ \( 1 - 1191 T + 2555007 T^{2} - 2142250738 T^{3} + 2555007 p^{3} T^{4} - 1191 p^{6} T^{5} + p^{9} T^{6} \)
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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

−8.523493204494339272888488025072, −7.995159477919477882055771280876, −7.83268059193649544533309695889, −7.73513346585348863211355120918, −7.25992122889666419306807737989, −6.86236280160104781324037033621, −6.74915740501309006672081450449, −6.27689104271426081196683858938, −6.22321186652569168025407021730, −5.81575373315982042057902107333, −5.29278563125356040144039520509, −5.26261367783732657998997704075, −5.19677863524265578883424733608, −4.46072733724947046754785373364, −4.21839909669030073962912585538, −4.17011298447007846164859932477, −3.27860737584968292133941066063, −3.13280884238573437061904901019, −3.00737685667512803137908904504, −2.18279923682525301531473074184, −2.07716151785008547603311381455, −1.85855150732230445912457256292, −0.992124289239819836623910230322, −0.865669043753499182111190132256, −0.64878227737191441963608796588, 0.64878227737191441963608796588, 0.865669043753499182111190132256, 0.992124289239819836623910230322, 1.85855150732230445912457256292, 2.07716151785008547603311381455, 2.18279923682525301531473074184, 3.00737685667512803137908904504, 3.13280884238573437061904901019, 3.27860737584968292133941066063, 4.17011298447007846164859932477, 4.21839909669030073962912585538, 4.46072733724947046754785373364, 5.19677863524265578883424733608, 5.26261367783732657998997704075, 5.29278563125356040144039520509, 5.81575373315982042057902107333, 6.22321186652569168025407021730, 6.27689104271426081196683858938, 6.74915740501309006672081450449, 6.86236280160104781324037033621, 7.25992122889666419306807737989, 7.73513346585348863211355120918, 7.83268059193649544533309695889, 7.995159477919477882055771280876, 8.523493204494339272888488025072

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