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Properties

Label 12-4004e6-1.1-c1e6-0-0
Degree $12$
Conductor $4.121\times 10^{21}$
Sign $1$
Analytic cond. $1.06813\times 10^{9}$
Root an. cond. $5.65438$
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  − 2·3-s − 3·5-s + 6·7-s − 5·9-s − 6·11-s − 6·13-s + 6·15-s − 17-s − 12·19-s − 12·21-s + 5·23-s − 12·25-s + 12·27-s − 14·29-s − 4·31-s + 12·33-s − 18·35-s − 3·37-s + 12·39-s + 6·41-s − 14·43-s + 15·45-s + 2·47-s + 21·49-s + 2·51-s − 3·53-s + 18·55-s + ⋯
L(s)  = 1  − 1.15·3-s − 1.34·5-s + 2.26·7-s − 5/3·9-s − 1.80·11-s − 1.66·13-s + 1.54·15-s − 0.242·17-s − 2.75·19-s − 2.61·21-s + 1.04·23-s − 2.39·25-s + 2.30·27-s − 2.59·29-s − 0.718·31-s + 2.08·33-s − 3.04·35-s − 0.493·37-s + 1.92·39-s + 0.937·41-s − 2.13·43-s + 2.23·45-s + 0.291·47-s + 3·49-s + 0.280·51-s − 0.412·53-s + 2.42·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 7^{6} \cdot 11^{6} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 7^{6} \cdot 11^{6} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{12} \cdot 7^{6} \cdot 11^{6} \cdot 13^{6}\)
Sign: $1$
Analytic conductor: \(1.06813\times 10^{9}\)
Root analytic conductor: \(5.65438\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(6\)
Selberg data: \((12,\ 2^{12} \cdot 7^{6} \cdot 11^{6} \cdot 13^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

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)$
bad2 \( 1 \)
7 \( ( 1 - T )^{6} \)
11 \( ( 1 + T )^{6} \)
13 \( ( 1 + T )^{6} \)
good3 \( 1 + 2 T + p^{2} T^{2} + 16 T^{3} + 14 p T^{4} + 67 T^{5} + 134 T^{6} + 67 p T^{7} + 14 p^{3} T^{8} + 16 p^{3} T^{9} + p^{6} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
5 \( 1 + 3 T + 21 T^{2} + 2 p^{2} T^{3} + 212 T^{4} + 421 T^{5} + 1319 T^{6} + 421 p T^{7} + 212 p^{2} T^{8} + 2 p^{5} T^{9} + 21 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 + T + 70 T^{2} + 49 T^{3} + 2404 T^{4} + 1316 T^{5} + 50846 T^{6} + 1316 p T^{7} + 2404 p^{2} T^{8} + 49 p^{3} T^{9} + 70 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 + 12 T + 131 T^{2} + 1004 T^{3} + 6562 T^{4} + 35859 T^{5} + 168496 T^{6} + 35859 p T^{7} + 6562 p^{2} T^{8} + 1004 p^{3} T^{9} + 131 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 - 5 T + 101 T^{2} - 382 T^{3} + 4460 T^{4} - 13183 T^{5} + 122637 T^{6} - 13183 p T^{7} + 4460 p^{2} T^{8} - 382 p^{3} T^{9} + 101 p^{4} T^{10} - 5 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 + 14 T + 215 T^{2} + 1916 T^{3} + 584 p T^{4} + 107937 T^{5} + 669980 T^{6} + 107937 p T^{7} + 584 p^{3} T^{8} + 1916 p^{3} T^{9} + 215 p^{4} T^{10} + 14 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 + 4 T + 91 T^{2} + 484 T^{3} + 3958 T^{4} + 28516 T^{5} + 130351 T^{6} + 28516 p T^{7} + 3958 p^{2} T^{8} + 484 p^{3} T^{9} + 91 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 + 3 T + 138 T^{2} + 314 T^{3} + 9027 T^{4} + 17766 T^{5} + 393998 T^{6} + 17766 p T^{7} + 9027 p^{2} T^{8} + 314 p^{3} T^{9} + 138 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 - 6 T + 166 T^{2} - 579 T^{3} + 11948 T^{4} - 28229 T^{5} + 572510 T^{6} - 28229 p T^{7} + 11948 p^{2} T^{8} - 579 p^{3} T^{9} + 166 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 + 14 T + 290 T^{2} + 2934 T^{3} + 33137 T^{4} + 249245 T^{5} + 1936322 T^{6} + 249245 p T^{7} + 33137 p^{2} T^{8} + 2934 p^{3} T^{9} + 290 p^{4} T^{10} + 14 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 - 2 T + 159 T^{2} - 180 T^{3} + 14256 T^{4} - 13271 T^{5} + 818896 T^{6} - 13271 p T^{7} + 14256 p^{2} T^{8} - 180 p^{3} T^{9} + 159 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 + 3 T + 211 T^{2} + 738 T^{3} + 21098 T^{4} + 77572 T^{5} + 1348262 T^{6} + 77572 p T^{7} + 21098 p^{2} T^{8} + 738 p^{3} T^{9} + 211 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 - 2 T + 230 T^{2} - 504 T^{3} + 25733 T^{4} - 60007 T^{5} + 1847032 T^{6} - 60007 p T^{7} + 25733 p^{2} T^{8} - 504 p^{3} T^{9} + 230 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 + 26 T + 526 T^{2} + 7131 T^{3} + 82356 T^{4} + 773629 T^{5} + 6526846 T^{6} + 773629 p T^{7} + 82356 p^{2} T^{8} + 7131 p^{3} T^{9} + 526 p^{4} T^{10} + 26 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 - 9 T + 254 T^{2} - 1897 T^{3} + 32416 T^{4} - 199521 T^{5} + 2614228 T^{6} - 199521 p T^{7} + 32416 p^{2} T^{8} - 1897 p^{3} T^{9} + 254 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 - 3 T + 212 T^{2} - 801 T^{3} + 25047 T^{4} - 107526 T^{5} + 2087912 T^{6} - 107526 p T^{7} + 25047 p^{2} T^{8} - 801 p^{3} T^{9} + 212 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 + 7 T + 278 T^{2} + 1322 T^{3} + 36673 T^{4} + 136952 T^{5} + 3198634 T^{6} + 136952 p T^{7} + 36673 p^{2} T^{8} + 1322 p^{3} T^{9} + 278 p^{4} T^{10} + 7 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 + 334 T^{2} + 378 T^{3} + 53773 T^{4} + 68977 T^{5} + 5331516 T^{6} + 68977 p T^{7} + 53773 p^{2} T^{8} + 378 p^{3} T^{9} + 334 p^{4} T^{10} + p^{6} T^{12} \)
83 \( 1 + 15 T + 328 T^{2} + 3830 T^{3} + 52011 T^{4} + 495484 T^{5} + 5109378 T^{6} + 495484 p T^{7} + 52011 p^{2} T^{8} + 3830 p^{3} T^{9} + 328 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 + T + 217 T^{2} - 684 T^{3} + 11028 T^{4} - 204809 T^{5} - 70125 T^{6} - 204809 p T^{7} + 11028 p^{2} T^{8} - 684 p^{3} T^{9} + 217 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 + 16 T + 593 T^{2} + 7277 T^{3} + 145318 T^{4} + 1366256 T^{5} + 18857263 T^{6} + 1366256 p T^{7} + 145318 p^{2} T^{8} + 7277 p^{3} T^{9} + 593 p^{4} T^{10} + 16 p^{5} T^{11} + p^{6} T^{12} \)
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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

−4.68222531836835975397824556532, −4.51715004744661902432888768167, −4.48323627906219234657414348249, −4.42066714566734548532905157294, −4.30397145177512306976329245055, −4.17257598933267995112390425364, −4.16740084367364015199764531273, −3.76214167775054304225808087565, −3.44584926093316005366426671634, −3.42865592252877289986919373698, −3.40273770595323142340531896588, −3.32945874645382723163322438158, −3.26507989967306897056761375091, −2.67260380322195818599227912329, −2.53092810104642267733891672857, −2.48639522852739071455490833134, −2.32438818691717494668125752618, −2.31149165283672743334768990903, −2.26798388521255216513550457394, −1.83003258650736147626617074960, −1.79719428793017416100306428858, −1.57601418903372879387098370448, −1.35718394343569597956690082645, −1.04773499514804901745420932284, −0.996813965770702996713113825648, 0, 0, 0, 0, 0, 0, 0.996813965770702996713113825648, 1.04773499514804901745420932284, 1.35718394343569597956690082645, 1.57601418903372879387098370448, 1.79719428793017416100306428858, 1.83003258650736147626617074960, 2.26798388521255216513550457394, 2.31149165283672743334768990903, 2.32438818691717494668125752618, 2.48639522852739071455490833134, 2.53092810104642267733891672857, 2.67260380322195818599227912329, 3.26507989967306897056761375091, 3.32945874645382723163322438158, 3.40273770595323142340531896588, 3.42865592252877289986919373698, 3.44584926093316005366426671634, 3.76214167775054304225808087565, 4.16740084367364015199764531273, 4.17257598933267995112390425364, 4.30397145177512306976329245055, 4.42066714566734548532905157294, 4.48323627906219234657414348249, 4.51715004744661902432888768167, 4.68222531836835975397824556532

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

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