Properties

Label 12-7406e6-1.1-c1e6-0-2
Degree $12$
Conductor $1.650\times 10^{23}$
Sign $1$
Analytic cond. $4.27725\times 10^{10}$
Root an. cond. $7.69007$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $6$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 6·2-s + 2·3-s + 21·4-s − 4·5-s + 12·6-s − 6·7-s + 56·8-s − 4·9-s − 24·10-s − 16·11-s + 42·12-s − 36·14-s − 8·15-s + 126·16-s − 2·17-s − 24·18-s − 18·19-s − 84·20-s − 12·21-s − 96·22-s + 112·24-s + 25-s − 18·27-s − 126·28-s + 4·29-s − 48·30-s − 2·31-s + ⋯
L(s)  = 1  + 4.24·2-s + 1.15·3-s + 21/2·4-s − 1.78·5-s + 4.89·6-s − 2.26·7-s + 19.7·8-s − 4/3·9-s − 7.58·10-s − 4.82·11-s + 12.1·12-s − 9.62·14-s − 2.06·15-s + 63/2·16-s − 0.485·17-s − 5.65·18-s − 4.12·19-s − 18.7·20-s − 2.61·21-s − 20.4·22-s + 22.8·24-s + 1/5·25-s − 3.46·27-s − 23.8·28-s + 0.742·29-s − 8.76·30-s − 0.359·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 7^{6} \cdot 23^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 7^{6} \cdot 23^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{6} \cdot 7^{6} \cdot 23^{12}\)
Sign: $1$
Analytic conductor: \(4.27725\times 10^{10}\)
Root analytic conductor: \(7.69007\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(6\)
Selberg data: \((12,\ 2^{6} \cdot 7^{6} \cdot 23^{12} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

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)$
bad2 \( ( 1 - T )^{6} \)
7 \( ( 1 + T )^{6} \)
23 \( 1 \)
good3 \( 1 - 2 T + 8 T^{2} - 2 p T^{3} + 7 p T^{4} + 8 T^{5} + 37 T^{6} + 8 p T^{7} + 7 p^{3} T^{8} - 2 p^{4} T^{9} + 8 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
5 \( 1 + 4 T + 3 p T^{2} + 46 T^{3} + 152 T^{4} + 366 T^{5} + 892 T^{6} + 366 p T^{7} + 152 p^{2} T^{8} + 46 p^{3} T^{9} + 3 p^{5} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 + 16 T + 161 T^{2} + 104 p T^{3} + 6352 T^{4} + 28296 T^{5} + 103496 T^{6} + 28296 p T^{7} + 6352 p^{2} T^{8} + 104 p^{4} T^{9} + 161 p^{4} T^{10} + 16 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 + 27 T^{2} + 18 T^{3} + 396 T^{4} + 594 T^{5} + 320 p T^{6} + 594 p T^{7} + 396 p^{2} T^{8} + 18 p^{3} T^{9} + 27 p^{4} T^{10} + p^{6} T^{12} \)
17 \( 1 + 2 T + 49 T^{2} - 6 T^{3} + 1182 T^{4} - 1102 T^{5} + 23109 T^{6} - 1102 p T^{7} + 1182 p^{2} T^{8} - 6 p^{3} T^{9} + 49 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 + 18 T + 205 T^{2} + 1718 T^{3} + 11558 T^{4} + 64306 T^{5} + 304481 T^{6} + 64306 p T^{7} + 11558 p^{2} T^{8} + 1718 p^{3} T^{9} + 205 p^{4} T^{10} + 18 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 - 4 T + 118 T^{2} - 276 T^{3} + 5943 T^{4} - 7576 T^{5} + 195252 T^{6} - 7576 p T^{7} + 5943 p^{2} T^{8} - 276 p^{3} T^{9} + 118 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 + 2 T + 52 T^{2} - 112 T^{3} + 1724 T^{4} - 8230 T^{5} + 47810 T^{6} - 8230 p T^{7} + 1724 p^{2} T^{8} - 112 p^{3} T^{9} + 52 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 + 4 T + 116 T^{2} + 8 p T^{3} + 7076 T^{4} + 17124 T^{5} + 318138 T^{6} + 17124 p T^{7} + 7076 p^{2} T^{8} + 8 p^{4} T^{9} + 116 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 + 139 T^{2} - 146 T^{3} + 10356 T^{4} - 11266 T^{5} + 523800 T^{6} - 11266 p T^{7} + 10356 p^{2} T^{8} - 146 p^{3} T^{9} + 139 p^{4} T^{10} + p^{6} T^{12} \)
43 \( 1 + 8 T + 217 T^{2} + 1544 T^{3} + 21110 T^{4} + 126056 T^{5} + 1170941 T^{6} + 126056 p T^{7} + 21110 p^{2} T^{8} + 1544 p^{3} T^{9} + 217 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 - 8 T + 59 T^{2} - 110 T^{3} - 692 T^{4} + 22722 T^{5} - 107248 T^{6} + 22722 p T^{7} - 692 p^{2} T^{8} - 110 p^{3} T^{9} + 59 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 - 8 T + 142 T^{2} - 744 T^{3} + 9687 T^{4} - 35120 T^{5} + 463908 T^{6} - 35120 p T^{7} + 9687 p^{2} T^{8} - 744 p^{3} T^{9} + 142 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 - 2 T + 227 T^{2} - 566 T^{3} + 26431 T^{4} - 59100 T^{5} + 1948874 T^{6} - 59100 p T^{7} + 26431 p^{2} T^{8} - 566 p^{3} T^{9} + 227 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 + 28 T + 570 T^{2} + 8348 T^{3} + 101159 T^{4} + 1005528 T^{5} + 8564236 T^{6} + 1005528 p T^{7} + 101159 p^{2} T^{8} + 8348 p^{3} T^{9} + 570 p^{4} T^{10} + 28 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 + 16 T + 419 T^{2} + 4976 T^{3} + 71015 T^{4} + 642096 T^{5} + 6358458 T^{6} + 642096 p T^{7} + 71015 p^{2} T^{8} + 4976 p^{3} T^{9} + 419 p^{4} T^{10} + 16 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 + 10 T + 141 T^{2} + 820 T^{3} + 12788 T^{4} + 108246 T^{5} + 1302304 T^{6} + 108246 p T^{7} + 12788 p^{2} T^{8} + 820 p^{3} T^{9} + 141 p^{4} T^{10} + 10 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 + 28 T + 681 T^{2} + 10808 T^{3} + 149282 T^{4} + 1612092 T^{5} + 15305425 T^{6} + 1612092 p T^{7} + 149282 p^{2} T^{8} + 10808 p^{3} T^{9} + 681 p^{4} T^{10} + 28 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 + 16 T + 388 T^{2} + 4192 T^{3} + 62912 T^{4} + 548272 T^{5} + 6263462 T^{6} + 548272 p T^{7} + 62912 p^{2} T^{8} + 4192 p^{3} T^{9} + 388 p^{4} T^{10} + 16 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 + 24 T + 544 T^{2} + 7654 T^{3} + 105333 T^{4} + 1097408 T^{5} + 11291661 T^{6} + 1097408 p T^{7} + 105333 p^{2} T^{8} + 7654 p^{3} T^{9} + 544 p^{4} T^{10} + 24 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 - 46 T + 1316 T^{2} - 26584 T^{3} + 419236 T^{4} - 5307690 T^{5} + 55200866 T^{6} - 5307690 p T^{7} + 419236 p^{2} T^{8} - 26584 p^{3} T^{9} + 1316 p^{4} T^{10} - 46 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 + 2 T + 196 T^{2} - 568 T^{3} + 25892 T^{4} - 15034 T^{5} + 3509522 T^{6} - 15034 p T^{7} + 25892 p^{2} T^{8} - 568 p^{3} T^{9} + 196 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
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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

−4.34620525561960837071960713759, −4.32301318433000456814369437037, −4.04477453025874407176641041294, −3.99402720087031829744866676459, −3.96264042057271052086279080958, −3.73145138868536623130582875716, −3.67669840065325997346886015170, −3.44469682083999095209406702251, −3.42896731591383347498394277067, −3.34361748455180359880207710617, −2.94515622503574439386007189191, −2.90201363143528754080179495928, −2.81813225401038648624155992058, −2.65246803576628413959556033637, −2.63541234565344483309912960483, −2.49848320315989384043428394536, −2.49623890399405090888620129157, −2.46851943933054731914734786381, −2.45908323389126442838414061225, −1.89553137394915353792822730411, −1.71147234282659639863037670415, −1.56741313943896551712875272925, −1.44395337706535969583030970733, −1.43190627487245093459295544916, −0.77560549393060987822880403083, 0, 0, 0, 0, 0, 0, 0.77560549393060987822880403083, 1.43190627487245093459295544916, 1.44395337706535969583030970733, 1.56741313943896551712875272925, 1.71147234282659639863037670415, 1.89553137394915353792822730411, 2.45908323389126442838414061225, 2.46851943933054731914734786381, 2.49623890399405090888620129157, 2.49848320315989384043428394536, 2.63541234565344483309912960483, 2.65246803576628413959556033637, 2.81813225401038648624155992058, 2.90201363143528754080179495928, 2.94515622503574439386007189191, 3.34361748455180359880207710617, 3.42896731591383347498394277067, 3.44469682083999095209406702251, 3.67669840065325997346886015170, 3.73145138868536623130582875716, 3.96264042057271052086279080958, 3.99402720087031829744866676459, 4.04477453025874407176641041294, 4.32301318433000456814369437037, 4.34620525561960837071960713759

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

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