Properties

Label 6-2166e3-1.1-c3e3-0-1
Degree $6$
Conductor $10161910296$
Sign $1$
Analytic cond. $2.08724\times 10^{6}$
Root an. cond. $11.3047$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 6·2-s − 9·3-s + 24·4-s + 2·5-s + 54·6-s − 17·7-s − 80·8-s + 54·9-s − 12·10-s − 52·11-s − 216·12-s + 75·13-s + 102·14-s − 18·15-s + 240·16-s − 48·17-s − 324·18-s + 48·20-s + 153·21-s + 312·22-s − 238·23-s + 720·24-s − 71·25-s − 450·26-s − 270·27-s − 408·28-s − 8·29-s + ⋯
L(s)  = 1  − 2.12·2-s − 1.73·3-s + 3·4-s + 0.178·5-s + 3.67·6-s − 0.917·7-s − 3.53·8-s + 2·9-s − 0.379·10-s − 1.42·11-s − 5.19·12-s + 1.60·13-s + 1.94·14-s − 0.309·15-s + 15/4·16-s − 0.684·17-s − 4.24·18-s + 0.536·20-s + 1.58·21-s + 3.02·22-s − 2.15·23-s + 6.12·24-s − 0.567·25-s − 3.39·26-s − 1.92·27-s − 2.75·28-s − 0.0512·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(2^{3} \cdot 3^{3} \cdot 19^{6}\)
Sign: $1$
Analytic conductor: \(2.08724\times 10^{6}\)
Root analytic conductor: \(11.3047\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2166} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 2^{3} \cdot 3^{3} \cdot 19^{6} ,\ ( \ : 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(0.3135547893\)
\(L(\frac12)\) \(\approx\) \(0.3135547893\)
\(L(\frac{5}{2})\) 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$C_1$ \( ( 1 + p T )^{3} \)
3$C_1$ \( ( 1 + p T )^{3} \)
19 \( 1 \)
good5$S_4\times C_2$ \( 1 - 2 T + 3 p^{2} T^{2} - 1868 T^{3} + 3 p^{5} T^{4} - 2 p^{6} T^{5} + p^{9} T^{6} \)
7$S_4\times C_2$ \( 1 + 17 T + 88 T^{2} - 6791 T^{3} + 88 p^{3} T^{4} + 17 p^{6} T^{5} + p^{9} T^{6} \)
11$S_4\times C_2$ \( 1 + 52 T + 3105 T^{2} + 105736 T^{3} + 3105 p^{3} T^{4} + 52 p^{6} T^{5} + p^{9} T^{6} \)
13$S_4\times C_2$ \( 1 - 75 T + 570 p T^{2} - 316343 T^{3} + 570 p^{4} T^{4} - 75 p^{6} T^{5} + p^{9} T^{6} \)
17$S_4\times C_2$ \( 1 + 48 T + 13011 T^{2} + 482016 T^{3} + 13011 p^{3} T^{4} + 48 p^{6} T^{5} + p^{9} T^{6} \)
23$S_4\times C_2$ \( 1 + 238 T + 13449 T^{2} - 313172 T^{3} + 13449 p^{3} T^{4} + 238 p^{6} T^{5} + p^{9} T^{6} \)
29$S_4\times C_2$ \( 1 + 8 T + 30351 T^{2} + 83792 T^{3} + 30351 p^{3} T^{4} + 8 p^{6} T^{5} + p^{9} T^{6} \)
31$S_4\times C_2$ \( 1 - 107 T + 75368 T^{2} - 4940027 T^{3} + 75368 p^{3} T^{4} - 107 p^{6} T^{5} + p^{9} T^{6} \)
37$S_4\times C_2$ \( 1 - 305 T + 26786 T^{2} - 1997981 T^{3} + 26786 p^{3} T^{4} - 305 p^{6} T^{5} + p^{9} T^{6} \)
41$S_4\times C_2$ \( 1 - 16 T + 40203 T^{2} - 9213088 T^{3} + 40203 p^{3} T^{4} - 16 p^{6} T^{5} + p^{9} T^{6} \)
43$S_4\times C_2$ \( 1 + 331 T + 154580 T^{2} + 55755235 T^{3} + 154580 p^{3} T^{4} + 331 p^{6} T^{5} + p^{9} T^{6} \)
47$S_4\times C_2$ \( 1 + 766 T + 402777 T^{2} + 138860332 T^{3} + 402777 p^{3} T^{4} + 766 p^{6} T^{5} + p^{9} T^{6} \)
53$S_4\times C_2$ \( 1 + 118 T + 228651 T^{2} - 5659004 T^{3} + 228651 p^{3} T^{4} + 118 p^{6} T^{5} + p^{9} T^{6} \)
59$S_4\times C_2$ \( 1 - 936 T + 766257 T^{2} - 352800000 T^{3} + 766257 p^{3} T^{4} - 936 p^{6} T^{5} + p^{9} T^{6} \)
61$S_4\times C_2$ \( 1 + 399 T + 471354 T^{2} + 102958675 T^{3} + 471354 p^{3} T^{4} + 399 p^{6} T^{5} + p^{9} T^{6} \)
67$S_4\times C_2$ \( 1 - 61 T + 714028 T^{2} - 959687 p T^{3} + 714028 p^{3} T^{4} - 61 p^{6} T^{5} + p^{9} T^{6} \)
71$S_4\times C_2$ \( 1 - 974 T + 1095153 T^{2} - 680220716 T^{3} + 1095153 p^{3} T^{4} - 974 p^{6} T^{5} + p^{9} T^{6} \)
73$S_4\times C_2$ \( 1 - 91 T + 598750 T^{2} + 96409345 T^{3} + 598750 p^{3} T^{4} - 91 p^{6} T^{5} + p^{9} T^{6} \)
79$S_4\times C_2$ \( 1 + 321 T + 1051536 T^{2} + 380263561 T^{3} + 1051536 p^{3} T^{4} + 321 p^{6} T^{5} + p^{9} T^{6} \)
83$S_4\times C_2$ \( 1 + 2148 T + 2987025 T^{2} + 2667812568 T^{3} + 2987025 p^{3} T^{4} + 2148 p^{6} T^{5} + p^{9} T^{6} \)
89$S_4\times C_2$ \( 1 - 1116 T + 2360067 T^{2} - 1584528840 T^{3} + 2360067 p^{3} T^{4} - 1116 p^{6} T^{5} + p^{9} T^{6} \)
97$S_4\times C_2$ \( 1 - 1382 T + 3339119 T^{2} - 2601906260 T^{3} + 3339119 p^{3} T^{4} - 1382 p^{6} T^{5} + p^{9} T^{6} \)
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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

−7.940061135886933130302910230808, −7.60537801940669381003094039123, −6.99225522182581726642501902608, −6.83880998781424743445759696516, −6.72820556512022503093270612843, −6.46609059890285077924227828719, −6.22773854425556557886805027433, −5.78825269029713768770666533443, −5.76218083376804267307618514750, −5.67803190946944679896940016732, −5.17021378162092595996187648728, −4.65114006168017633866386808727, −4.61434593765752845778438261117, −4.05113348020977982625411338437, −3.62156097032207006420325212441, −3.60796648826634215972802042430, −3.03642649313383325523222309059, −2.55954501782675289260351456091, −2.42869276302355518498675072260, −1.89622152809635654764390417230, −1.51416382320024480348750390503, −1.44285264453838696995812956534, −0.68727987613307161852698614907, −0.44095997027413699451940590790, −0.27530251122629034118367200354, 0.27530251122629034118367200354, 0.44095997027413699451940590790, 0.68727987613307161852698614907, 1.44285264453838696995812956534, 1.51416382320024480348750390503, 1.89622152809635654764390417230, 2.42869276302355518498675072260, 2.55954501782675289260351456091, 3.03642649313383325523222309059, 3.60796648826634215972802042430, 3.62156097032207006420325212441, 4.05113348020977982625411338437, 4.61434593765752845778438261117, 4.65114006168017633866386808727, 5.17021378162092595996187648728, 5.67803190946944679896940016732, 5.76218083376804267307618514750, 5.78825269029713768770666533443, 6.22773854425556557886805027433, 6.46609059890285077924227828719, 6.72820556512022503093270612843, 6.83880998781424743445759696516, 6.99225522182581726642501902608, 7.60537801940669381003094039123, 7.940061135886933130302910230808

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