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Properties

Label 12-630e6-1.1-c1e6-0-2
Degree $12$
Conductor $6.252\times 10^{16}$
Sign $1$
Analytic cond. $16207.1$
Root an. cond. $2.24289$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·2-s − 3-s + 3·4-s + 3·5-s + 3·6-s − 3·7-s + 2·8-s − 3·9-s − 9·10-s + 3·11-s − 3·12-s + 9·14-s − 3·15-s − 9·16-s − 18·17-s + 9·18-s − 6·19-s + 9·20-s + 3·21-s − 9·22-s + 6·23-s − 2·24-s + 3·25-s + 4·27-s − 9·28-s + 12·29-s + 9·30-s + ⋯
L(s)  = 1  − 2.12·2-s − 0.577·3-s + 3/2·4-s + 1.34·5-s + 1.22·6-s − 1.13·7-s + 0.707·8-s − 9-s − 2.84·10-s + 0.904·11-s − 0.866·12-s + 2.40·14-s − 0.774·15-s − 9/4·16-s − 4.36·17-s + 2.12·18-s − 1.37·19-s + 2.01·20-s + 0.654·21-s − 1.91·22-s + 1.25·23-s − 0.408·24-s + 3/5·25-s + 0.769·27-s − 1.70·28-s + 2.22·29-s + 1.64·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{12} \cdot 5^{6} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{12} \cdot 5^{6} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{6} \cdot 3^{12} \cdot 5^{6} \cdot 7^{6}\)
Sign: $1$
Analytic conductor: \(16207.1\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{6} \cdot 3^{12} \cdot 5^{6} \cdot 7^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.6175135025\)
\(L(\frac12)\) \(\approx\) \(0.6175135025\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 + T + T^{2} )^{3} \)
3 \( 1 + T + 4 T^{2} + p T^{3} + 4 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \)
5 \( ( 1 - T + T^{2} )^{3} \)
7 \( ( 1 + T + T^{2} )^{3} \)
good11 \( 1 - 3 T - 9 T^{2} + 96 T^{3} - 9 p T^{4} - 525 T^{5} + 3634 T^{6} - 525 p T^{7} - 9 p^{3} T^{8} + 96 p^{3} T^{9} - 9 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 - 21 T^{2} - 4 p T^{3} + 168 T^{4} + 42 p T^{5} - 51 p T^{6} + 42 p^{2} T^{7} + 168 p^{2} T^{8} - 4 p^{4} T^{9} - 21 p^{4} T^{10} + p^{6} T^{12} \)
17 \( ( 1 + 9 T + 54 T^{2} + 225 T^{3} + 54 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
19 \( ( 1 + 3 T + 36 T^{2} + 127 T^{3} + 36 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
23 \( 1 - 6 T - 27 T^{2} + 66 T^{3} + 1188 T^{4} + 228 T^{5} - 36713 T^{6} + 228 p T^{7} + 1188 p^{2} T^{8} + 66 p^{3} T^{9} - 27 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 - 12 T + 27 T^{2} + 24 T^{3} + 1728 T^{4} - 12198 T^{5} + 40489 T^{6} - 12198 p T^{7} + 1728 p^{2} T^{8} + 24 p^{3} T^{9} + 27 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
31 \( ( 1 - 11 T + p T^{2} )^{3}( 1 + 7 T + p T^{2} )^{3} \)
37 \( ( 1 + 63 T^{2} + 34 T^{3} + 63 p T^{4} + p^{3} T^{6} )^{2} \)
41 \( 1 - 15 T + 45 T^{2} - 222 T^{3} + 6435 T^{4} - 26835 T^{5} - 33662 T^{6} - 26835 p T^{7} + 6435 p^{2} T^{8} - 222 p^{3} T^{9} + 45 p^{4} T^{10} - 15 p^{5} T^{11} + p^{6} T^{12} \)
43 \( ( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} )^{3} \)
47 \( 1 - 6 T + 27 T^{2} - 798 T^{3} + 1890 T^{4} - 4830 T^{5} + 262699 T^{6} - 4830 p T^{7} + 1890 p^{2} T^{8} - 798 p^{3} T^{9} + 27 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
53 \( ( 1 + 12 T + 3 p T^{2} + 1110 T^{3} + 3 p^{2} T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
59 \( 1 - 3 T - 87 T^{2} + 582 T^{3} + 2247 T^{4} - 18219 T^{5} + 24910 T^{6} - 18219 p T^{7} + 2247 p^{2} T^{8} + 582 p^{3} T^{9} - 87 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 - 147 T^{2} + 164 T^{3} + 12642 T^{4} - 12054 T^{5} - 857463 T^{6} - 12054 p T^{7} + 12642 p^{2} T^{8} + 164 p^{3} T^{9} - 147 p^{4} T^{10} + p^{6} T^{12} \)
67 \( 1 - 9 T - 39 T^{2} + 1226 T^{3} - 4017 T^{4} - 34041 T^{5} + 602526 T^{6} - 34041 p T^{7} - 4017 p^{2} T^{8} + 1226 p^{3} T^{9} - 39 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
71 \( ( 1 - 12 T + 3 p T^{2} - 1542 T^{3} + 3 p^{2} T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
73 \( ( 1 + 3 T + 126 T^{2} + 55 T^{3} + 126 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
79 \( 1 + 6 T - 117 T^{2} - 1186 T^{3} + 5010 T^{4} + 53358 T^{5} + 18795 T^{6} + 53358 p T^{7} + 5010 p^{2} T^{8} - 1186 p^{3} T^{9} - 117 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 - 24 T + 279 T^{2} - 1752 T^{3} - 954 T^{4} + 219576 T^{5} - 2920493 T^{6} + 219576 p T^{7} - 954 p^{2} T^{8} - 1752 p^{3} T^{9} + 279 p^{4} T^{10} - 24 p^{5} T^{11} + p^{6} T^{12} \)
89 \( ( 1 + 18 T + 279 T^{2} + 2556 T^{3} + 279 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
97 \( 1 - 9 T - 165 T^{2} + 1316 T^{3} + 20769 T^{4} - 89355 T^{5} - 1691610 T^{6} - 89355 p T^{7} + 20769 p^{2} T^{8} + 1316 p^{3} T^{9} - 165 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
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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

−5.72951491323466908361734039616, −5.57945504550285107758023864472, −5.56128040924647329352457343974, −5.31448930947522462486530065926, −4.83464531789854644802397747510, −4.62946107162488130544512347465, −4.59611858871469881722649348442, −4.53495970709620343018153185570, −4.32221874603505322960934628457, −4.27787571569076114443279907230, −4.16117980172340582968552107102, −3.64902212601791754392334373003, −3.39199033403879473966815180087, −3.20213029311347168632419437690, −2.87708131283625791577493196856, −2.81039751340013242622911338009, −2.55201183197213906563888028149, −2.30470178997068449334091354006, −2.10035769400138954647308960165, −2.08002687716388397158075342787, −1.57956346084266550030553452824, −1.38479270518585374596424950258, −0.75693000417808372245492548393, −0.53530645338903379925161210169, −0.51229161615569050963759293030, 0.51229161615569050963759293030, 0.53530645338903379925161210169, 0.75693000417808372245492548393, 1.38479270518585374596424950258, 1.57956346084266550030553452824, 2.08002687716388397158075342787, 2.10035769400138954647308960165, 2.30470178997068449334091354006, 2.55201183197213906563888028149, 2.81039751340013242622911338009, 2.87708131283625791577493196856, 3.20213029311347168632419437690, 3.39199033403879473966815180087, 3.64902212601791754392334373003, 4.16117980172340582968552107102, 4.27787571569076114443279907230, 4.32221874603505322960934628457, 4.53495970709620343018153185570, 4.59611858871469881722649348442, 4.62946107162488130544512347465, 4.83464531789854644802397747510, 5.31448930947522462486530065926, 5.56128040924647329352457343974, 5.57945504550285107758023864472, 5.72951491323466908361734039616

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

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