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Properties

Label 12-143e6-1.1-c1e6-0-0
Degree $12$
Conductor $8.551\times 10^{12}$
Sign $1$
Analytic cond. $2.21655$
Root an. cond. $1.06857$
Motivic weight $1$
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  + 2-s − 3-s − 2·5-s − 6-s − 5·7-s + 5·8-s + 3·9-s − 2·10-s − 3·11-s − 15·13-s − 5·14-s + 2·15-s + 8·16-s + 7·17-s + 3·18-s − 2·19-s + 5·21-s − 3·22-s − 3·23-s − 5·24-s − 15·25-s − 15·26-s − 10·27-s + 4·29-s + 2·30-s + 2·31-s + 4·32-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s − 0.894·5-s − 0.408·6-s − 1.88·7-s + 1.76·8-s + 9-s − 0.632·10-s − 0.904·11-s − 4.16·13-s − 1.33·14-s + 0.516·15-s + 2·16-s + 1.69·17-s + 0.707·18-s − 0.458·19-s + 1.09·21-s − 0.639·22-s − 0.625·23-s − 1.02·24-s − 3·25-s − 2.94·26-s − 1.92·27-s + 0.742·29-s + 0.365·30-s + 0.359·31-s + 0.707·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(11^{6} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(11^{6} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(11^{6} \cdot 13^{6}\)
Sign: $1$
Analytic conductor: \(2.21655\)
Root analytic conductor: \(1.06857\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 11^{6} \cdot 13^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.2866594931\)
\(L(\frac12)\) \(\approx\) \(0.2866594931\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( ( 1 + T + T^{2} )^{3} \)
13 \( ( 1 + 5 T + p T^{2} )^{3} \)
good2 \( 1 - T + T^{2} - 3 p T^{3} + 3 T^{4} - p^{2} T^{5} + 21 T^{6} - p^{3} T^{7} + 3 p^{2} T^{8} - 3 p^{4} T^{9} + p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \)
3 \( 1 + T - 2 T^{2} + 5 T^{3} + T^{4} - 4 p T^{5} + 19 T^{6} - 4 p^{2} T^{7} + p^{2} T^{8} + 5 p^{3} T^{9} - 2 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} \)
5 \( ( 1 + T + 9 T^{2} + 3 T^{3} + 9 p T^{4} + p^{2} T^{5} + p^{3} T^{6} )^{2} \)
7 \( 1 + 5 T + 2 T^{2} - 23 T^{3} + 23 T^{4} + 222 T^{5} + 519 T^{6} + 222 p T^{7} + 23 p^{2} T^{8} - 23 p^{3} T^{9} + 2 p^{4} T^{10} + 5 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 - 7 T - 12 T^{2} + 35 T^{3} + 1067 T^{4} - 1596 T^{5} - 14231 T^{6} - 1596 p T^{7} + 1067 p^{2} T^{8} + 35 p^{3} T^{9} - 12 p^{4} T^{10} - 7 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 + 2 T - 10 T^{2} - 278 T^{3} - 376 T^{4} + 1800 T^{5} + 31959 T^{6} + 1800 p T^{7} - 376 p^{2} T^{8} - 278 p^{3} T^{9} - 10 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 + 3 T - 44 T^{2} - 119 T^{3} + 51 p T^{4} + 1640 T^{5} - 25713 T^{6} + 1640 p T^{7} + 51 p^{3} T^{8} - 119 p^{3} T^{9} - 44 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 - 4 T + 6 T^{2} + 326 T^{3} - 922 T^{4} - 2874 T^{5} + 65911 T^{6} - 2874 p T^{7} - 922 p^{2} T^{8} + 326 p^{3} T^{9} + 6 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
31 \( ( 1 - T + 68 T^{2} - 3 p T^{3} + 68 p T^{4} - p^{2} T^{5} + p^{3} T^{6} )^{2} \)
37 \( 1 + 12 T + 4 T^{2} - 34 T^{3} + 4136 T^{4} + 12088 T^{5} - 86581 T^{6} + 12088 p T^{7} + 4136 p^{2} T^{8} - 34 p^{3} T^{9} + 4 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 + T - 40 T^{2} - 185 T^{3} - 113 T^{4} + 72 p T^{5} + 80009 T^{6} + 72 p^{2} T^{7} - 113 p^{2} T^{8} - 185 p^{3} T^{9} - 40 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 - 4 T + 2 T^{2} - 530 T^{3} + 542 T^{4} + 738 T^{5} + 177171 T^{6} + 738 p T^{7} + 542 p^{2} T^{8} - 530 p^{3} T^{9} + 2 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
47 \( ( 1 + 8 T + 80 T^{2} + 493 T^{3} + 80 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
53 \( ( 1 + 9 T + 167 T^{2} + 905 T^{3} + 167 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
59 \( 1 + 30 T + 442 T^{2} + 5078 T^{3} + 52812 T^{4} + 466708 T^{5} + 3659343 T^{6} + 466708 p T^{7} + 52812 p^{2} T^{8} + 5078 p^{3} T^{9} + 442 p^{4} T^{10} + 30 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 - 18 T + 90 T^{2} + 194 T^{3} + 828 T^{4} - 68616 T^{5} + 761451 T^{6} - 68616 p T^{7} + 828 p^{2} T^{8} + 194 p^{3} T^{9} + 90 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 + 15 T - 32 T^{2} - 187 T^{3} + 18515 T^{4} + 70162 T^{5} - 749341 T^{6} + 70162 p T^{7} + 18515 p^{2} T^{8} - 187 p^{3} T^{9} - 32 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 - 11 T - 126 T^{2} + 469 T^{3} + 25949 T^{4} - 63108 T^{5} - 1688129 T^{6} - 63108 p T^{7} + 25949 p^{2} T^{8} + 469 p^{3} T^{9} - 126 p^{4} T^{10} - 11 p^{5} T^{11} + p^{6} T^{12} \)
73 \( ( 1 + 2 T + p T^{2} )^{6} \)
79 \( ( 1 - 12 T + 209 T^{2} - 1808 T^{3} + 209 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
83 \( ( 1 + 14 T + 232 T^{2} + 1983 T^{3} + 232 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
89 \( 1 - 17 T - 68 T^{2} + 285 T^{3} + 41667 T^{4} - 188564 T^{5} - 2128863 T^{6} - 188564 p T^{7} + 41667 p^{2} T^{8} + 285 p^{3} T^{9} - 68 p^{4} T^{10} - 17 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 + 11 T - 109 T^{2} - 1724 T^{3} + 7193 T^{4} + 101889 T^{5} + 79686 T^{6} + 101889 p T^{7} + 7193 p^{2} T^{8} - 1724 p^{3} T^{9} - 109 p^{4} T^{10} + 11 p^{5} T^{11} + p^{6} T^{12} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.45423016647127377876987861942, −7.32629102233358142052643440996, −7.06993337827377262606886267760, −6.74317485129549731901713628985, −6.49918513508715073043005549322, −6.36935453401415002536656161307, −6.03268270470242332116633811549, −5.79953328225383607209502418739, −5.72732559575342825455081354380, −5.32814487006313540232208881472, −5.18891349754117459711667445904, −5.03267913168695364035468188565, −4.59405422214158316108982665357, −4.51565741648276660295563571475, −4.43940646089086115585384385088, −4.28452767100056578507054895757, −3.65929259269781138913792962529, −3.61677235568673405868551834998, −3.19542837096067913875252852059, −3.18979186108957336633432038853, −2.78940096442065197352792970533, −2.06327866109594052857626405340, −1.91346846277548245704252819049, −1.84600934585149180624243899777, −0.31571882517742124963232595607, 0.31571882517742124963232595607, 1.84600934585149180624243899777, 1.91346846277548245704252819049, 2.06327866109594052857626405340, 2.78940096442065197352792970533, 3.18979186108957336633432038853, 3.19542837096067913875252852059, 3.61677235568673405868551834998, 3.65929259269781138913792962529, 4.28452767100056578507054895757, 4.43940646089086115585384385088, 4.51565741648276660295563571475, 4.59405422214158316108982665357, 5.03267913168695364035468188565, 5.18891349754117459711667445904, 5.32814487006313540232208881472, 5.72732559575342825455081354380, 5.79953328225383607209502418739, 6.03268270470242332116633811549, 6.36935453401415002536656161307, 6.49918513508715073043005549322, 6.74317485129549731901713628985, 7.06993337827377262606886267760, 7.32629102233358142052643440996, 7.45423016647127377876987861942

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

Plot not available for L-functions of degree greater than 10.