Properties

Label 12-671e6-1.1-c1e6-0-0
Degree $12$
Conductor $9.127\times 10^{16}$
Sign $1$
Analytic cond. $23659.0$
Root an. cond. $2.31472$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $6$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 5·4-s − 5-s − 5·7-s − 2·8-s − 11·9-s − 6·11-s + 5·12-s − 4·13-s + 15-s + 9·16-s − 5·17-s − 3·19-s + 5·20-s + 5·21-s − 3·23-s + 2·24-s − 20·25-s + 11·27-s + 25·28-s − 29-s − 10·31-s + 13·32-s + 6·33-s + 5·35-s + 55·36-s − 19·37-s + ⋯
L(s)  = 1  − 0.577·3-s − 5/2·4-s − 0.447·5-s − 1.88·7-s − 0.707·8-s − 3.66·9-s − 1.80·11-s + 1.44·12-s − 1.10·13-s + 0.258·15-s + 9/4·16-s − 1.21·17-s − 0.688·19-s + 1.11·20-s + 1.09·21-s − 0.625·23-s + 0.408·24-s − 4·25-s + 2.11·27-s + 4.72·28-s − 0.185·29-s − 1.79·31-s + 2.29·32-s + 1.04·33-s + 0.845·35-s + 55/6·36-s − 3.12·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(11^{6} \cdot 61^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(11^{6} \cdot 61^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(11^{6} \cdot 61^{6}\)
Sign: $1$
Analytic conductor: \(23659.0\)
Root analytic conductor: \(2.31472\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(6\)
Selberg data: \((12,\ 11^{6} \cdot 61^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( ( 1 + T )^{6} \)
61 \( ( 1 + T )^{6} \)
good2 \( 1 + 5 T^{2} + p T^{3} + p^{4} T^{4} + 7 T^{5} + 19 p T^{6} + 7 p T^{7} + p^{6} T^{8} + p^{4} T^{9} + 5 p^{4} T^{10} + p^{6} T^{12} \)
3 \( 1 + T + 4 p T^{2} + 4 p T^{3} + 8 p^{2} T^{4} + 64 T^{5} + 269 T^{6} + 64 p T^{7} + 8 p^{4} T^{8} + 4 p^{4} T^{9} + 4 p^{5} T^{10} + p^{5} T^{11} + p^{6} T^{12} \)
5 \( 1 + T + 21 T^{2} + 22 T^{3} + 201 T^{4} + 206 T^{5} + 1209 T^{6} + 206 p T^{7} + 201 p^{2} T^{8} + 22 p^{3} T^{9} + 21 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} \)
7 \( 1 + 5 T + 46 T^{2} + 164 T^{3} + 834 T^{4} + 2222 T^{5} + 7858 T^{6} + 2222 p T^{7} + 834 p^{2} T^{8} + 164 p^{3} T^{9} + 46 p^{4} T^{10} + 5 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 + 4 T + 66 T^{2} + 197 T^{3} + 1875 T^{4} + 4358 T^{5} + 30838 T^{6} + 4358 p T^{7} + 1875 p^{2} T^{8} + 197 p^{3} T^{9} + 66 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 + 5 T + 63 T^{2} + 282 T^{3} + 2011 T^{4} + 7720 T^{5} + 40860 T^{6} + 7720 p T^{7} + 2011 p^{2} T^{8} + 282 p^{3} T^{9} + 63 p^{4} T^{10} + 5 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 + 3 T + 105 T^{2} + 245 T^{3} + 4688 T^{4} + 8533 T^{5} + 116050 T^{6} + 8533 p T^{7} + 4688 p^{2} T^{8} + 245 p^{3} T^{9} + 105 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 + 3 T + 95 T^{2} + 271 T^{3} + 4115 T^{4} + 11062 T^{5} + 113245 T^{6} + 11062 p T^{7} + 4115 p^{2} T^{8} + 271 p^{3} T^{9} + 95 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 + T + 165 T^{2} + 133 T^{3} + 11578 T^{4} + 7373 T^{5} + 442770 T^{6} + 7373 p T^{7} + 11578 p^{2} T^{8} + 133 p^{3} T^{9} + 165 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 + 10 T + 211 T^{2} + 1529 T^{3} + 17373 T^{4} + 3033 p T^{5} + 731173 T^{6} + 3033 p^{2} T^{7} + 17373 p^{2} T^{8} + 1529 p^{3} T^{9} + 211 p^{4} T^{10} + 10 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 + 19 T + 254 T^{2} + 2282 T^{3} + 17160 T^{4} + 108036 T^{5} + 672701 T^{6} + 108036 p T^{7} + 17160 p^{2} T^{8} + 2282 p^{3} T^{9} + 254 p^{4} T^{10} + 19 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 + 7 T + 157 T^{2} + 1231 T^{3} + 11116 T^{4} + 93909 T^{5} + 522244 T^{6} + 93909 p T^{7} + 11116 p^{2} T^{8} + 1231 p^{3} T^{9} + 157 p^{4} T^{10} + 7 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 + 2 T + 184 T^{2} + 585 T^{3} + 15115 T^{4} + 56676 T^{5} + 778470 T^{6} + 56676 p T^{7} + 15115 p^{2} T^{8} + 585 p^{3} T^{9} + 184 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 - 5 T + 172 T^{2} - 706 T^{3} + 15414 T^{4} - 52066 T^{5} + 869866 T^{6} - 52066 p T^{7} + 15414 p^{2} T^{8} - 706 p^{3} T^{9} + 172 p^{4} T^{10} - 5 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 + 9 T + 236 T^{2} + 1806 T^{3} + 26894 T^{4} + 167050 T^{5} + 1820956 T^{6} + 167050 p T^{7} + 26894 p^{2} T^{8} + 1806 p^{3} T^{9} + 236 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 + 5 T + 5 p T^{2} + 1301 T^{3} + 39467 T^{4} + 144034 T^{5} + 3009281 T^{6} + 144034 p T^{7} + 39467 p^{2} T^{8} + 1301 p^{3} T^{9} + 5 p^{5} T^{10} + 5 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 + 14 T + 345 T^{2} + 2734 T^{3} + 39586 T^{4} + 203614 T^{5} + 2781137 T^{6} + 203614 p T^{7} + 39586 p^{2} T^{8} + 2734 p^{3} T^{9} + 345 p^{4} T^{10} + 14 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 + 14 T + 370 T^{2} + 4244 T^{3} + 60622 T^{4} + 557781 T^{5} + 5582929 T^{6} + 557781 p T^{7} + 60622 p^{2} T^{8} + 4244 p^{3} T^{9} + 370 p^{4} T^{10} + 14 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 + 14 T + 238 T^{2} + 1833 T^{3} + 26783 T^{4} + 214906 T^{5} + 2618754 T^{6} + 214906 p T^{7} + 26783 p^{2} T^{8} + 1833 p^{3} T^{9} + 238 p^{4} T^{10} + 14 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 - 5 T + 311 T^{2} - 1249 T^{3} + 49564 T^{4} - 164781 T^{5} + 4854524 T^{6} - 164781 p T^{7} + 49564 p^{2} T^{8} - 1249 p^{3} T^{9} + 311 p^{4} T^{10} - 5 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 - 17 T + 327 T^{2} - 4281 T^{3} + 55034 T^{4} - 551657 T^{5} + 5713940 T^{6} - 551657 p T^{7} + 55034 p^{2} T^{8} - 4281 p^{3} T^{9} + 327 p^{4} T^{10} - 17 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 + 25 T + 649 T^{2} + 10981 T^{3} + 159015 T^{4} + 1943100 T^{5} + 19432603 T^{6} + 1943100 p T^{7} + 159015 p^{2} T^{8} + 10981 p^{3} T^{9} + 649 p^{4} T^{10} + 25 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 + 24 T + 678 T^{2} + 10373 T^{3} + 170592 T^{4} + 1896695 T^{5} + 22185859 T^{6} + 1896695 p T^{7} + 170592 p^{2} T^{8} + 10373 p^{3} T^{9} + 678 p^{4} T^{10} + 24 p^{5} T^{11} + p^{6} T^{12} \)
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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

−6.19305739402128597098919339412, −5.74284747054797868463202228358, −5.73028205015805746939969562793, −5.54528460244094820338802018239, −5.42708133505674306197001399466, −5.39958313256199352887360687871, −5.33484025054038845679638157171, −4.84912243412886928715782536764, −4.75639753735756710041031284030, −4.72823837303605945614262468954, −4.37257687706191603074494445033, −4.31721828151194991557550212682, −4.13090899758833830515683112485, −3.84736500072943321524358003967, −3.48277964736747044957505906713, −3.40615673315827058921822313712, −3.32227902790717779345396195603, −3.25366504677420812714213157981, −2.89755663255118339419698141548, −2.81246843783612844611606967097, −2.56562691549407031890805347774, −2.16492764116722575854130681317, −1.97485106343329027577602507079, −1.90960175100308518588471795197, −1.46372237860879810016738663848, 0, 0, 0, 0, 0, 0, 1.46372237860879810016738663848, 1.90960175100308518588471795197, 1.97485106343329027577602507079, 2.16492764116722575854130681317, 2.56562691549407031890805347774, 2.81246843783612844611606967097, 2.89755663255118339419698141548, 3.25366504677420812714213157981, 3.32227902790717779345396195603, 3.40615673315827058921822313712, 3.48277964736747044957505906713, 3.84736500072943321524358003967, 4.13090899758833830515683112485, 4.31721828151194991557550212682, 4.37257687706191603074494445033, 4.72823837303605945614262468954, 4.75639753735756710041031284030, 4.84912243412886928715782536764, 5.33484025054038845679638157171, 5.39958313256199352887360687871, 5.42708133505674306197001399466, 5.54528460244094820338802018239, 5.73028205015805746939969562793, 5.74284747054797868463202228358, 6.19305739402128597098919339412

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

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