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Properties

Label 12-8280e6-1.1-c1e6-0-0
Degree $12$
Conductor $3.222\times 10^{23}$
Sign $1$
Analytic cond. $8.35304\times 10^{10}$
Root an. cond. $8.13118$
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  + 6·5-s − 2·7-s − 4·11-s + 4·17-s + 8·19-s + 6·23-s + 21·25-s + 18·29-s − 8·31-s − 12·35-s + 6·37-s + 6·41-s + 8·43-s − 8·47-s − 14·49-s + 8·53-s − 24·55-s + 2·59-s − 8·61-s − 2·67-s + 2·71-s + 8·73-s + 8·77-s − 28·79-s + 24·83-s + 24·85-s + 4·89-s + ⋯
L(s)  = 1  + 2.68·5-s − 0.755·7-s − 1.20·11-s + 0.970·17-s + 1.83·19-s + 1.25·23-s + 21/5·25-s + 3.34·29-s − 1.43·31-s − 2.02·35-s + 0.986·37-s + 0.937·41-s + 1.21·43-s − 1.16·47-s − 2·49-s + 1.09·53-s − 3.23·55-s + 0.260·59-s − 1.02·61-s − 0.244·67-s + 0.237·71-s + 0.936·73-s + 0.911·77-s − 3.15·79-s + 2.63·83-s + 2.60·85-s + 0.423·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{12} \cdot 5^{6} \cdot 23^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{12} \cdot 5^{6} \cdot 23^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{18} \cdot 3^{12} \cdot 5^{6} \cdot 23^{6}\)
Sign: $1$
Analytic conductor: \(8.35304\times 10^{10}\)
Root analytic conductor: \(8.13118\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{18} \cdot 3^{12} \cdot 5^{6} \cdot 23^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(40.32141527\)
\(L(\frac12)\) \(\approx\) \(40.32141527\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( ( 1 - T )^{6} \)
23 \( ( 1 - T )^{6} \)
good7 \( 1 + 2 T + 18 T^{2} + 40 T^{3} + 234 T^{4} + 66 p T^{5} + 1842 T^{6} + 66 p^{2} T^{7} + 234 p^{2} T^{8} + 40 p^{3} T^{9} + 18 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 + 4 T + 28 T^{2} + 120 T^{3} + 507 T^{4} + 1700 T^{5} + 7056 T^{6} + 1700 p T^{7} + 507 p^{2} T^{8} + 120 p^{3} T^{9} + 28 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 + 36 T^{2} - 60 T^{3} + 411 T^{4} - 2292 T^{5} + 2880 T^{6} - 2292 p T^{7} + 411 p^{2} T^{8} - 60 p^{3} T^{9} + 36 p^{4} T^{10} + p^{6} T^{12} \)
17 \( 1 - 4 T + 30 T^{2} - 64 T^{3} + 678 T^{4} - 1332 T^{5} + 12158 T^{6} - 1332 p T^{7} + 678 p^{2} T^{8} - 64 p^{3} T^{9} + 30 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 - 8 T + 96 T^{2} - 556 T^{3} + 4019 T^{4} - 18404 T^{5} + 98408 T^{6} - 18404 p T^{7} + 4019 p^{2} T^{8} - 556 p^{3} T^{9} + 96 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 - 18 T + 164 T^{2} - 772 T^{3} + 1124 T^{4} + 13454 T^{5} - 112530 T^{6} + 13454 p T^{7} + 1124 p^{2} T^{8} - 772 p^{3} T^{9} + 164 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 + 8 T + 152 T^{2} + 832 T^{3} + 9816 T^{4} + 42568 T^{5} + 383406 T^{6} + 42568 p T^{7} + 9816 p^{2} T^{8} + 832 p^{3} T^{9} + 152 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 - 6 T + 66 T^{2} - 72 T^{3} + 2058 T^{4} - 150 T^{5} + 81618 T^{6} - 150 p T^{7} + 2058 p^{2} T^{8} - 72 p^{3} T^{9} + 66 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 - 6 T + 120 T^{2} - 636 T^{3} + 6324 T^{4} - 30510 T^{5} + 248138 T^{6} - 30510 p T^{7} + 6324 p^{2} T^{8} - 636 p^{3} T^{9} + 120 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 - 8 T + 214 T^{2} - 1416 T^{3} + 20631 T^{4} - 110816 T^{5} + 1143316 T^{6} - 110816 p T^{7} + 20631 p^{2} T^{8} - 1416 p^{3} T^{9} + 214 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 + 8 T + 160 T^{2} + 756 T^{3} + 10275 T^{4} + 33724 T^{5} + 489144 T^{6} + 33724 p T^{7} + 10275 p^{2} T^{8} + 756 p^{3} T^{9} + 160 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 - 8 T + 230 T^{2} - 1412 T^{3} + 24878 T^{4} - 124672 T^{5} + 1651414 T^{6} - 124672 p T^{7} + 24878 p^{2} T^{8} - 1412 p^{3} T^{9} + 230 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 - 2 T + 296 T^{2} - 612 T^{3} + 39164 T^{4} - 72854 T^{5} + 2970962 T^{6} - 72854 p T^{7} + 39164 p^{2} T^{8} - 612 p^{3} T^{9} + 296 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 + 8 T + 224 T^{2} + 812 T^{3} + 18411 T^{4} + 14780 T^{5} + 1072984 T^{6} + 14780 p T^{7} + 18411 p^{2} T^{8} + 812 p^{3} T^{9} + 224 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 + 2 T + 270 T^{2} + 796 T^{3} + 33714 T^{4} + 116958 T^{5} + 2694942 T^{6} + 116958 p T^{7} + 33714 p^{2} T^{8} + 796 p^{3} T^{9} + 270 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 - 2 T + 252 T^{2} - 940 T^{3} + 33916 T^{4} - 117734 T^{5} + 3025574 T^{6} - 117734 p T^{7} + 33916 p^{2} T^{8} - 940 p^{3} T^{9} + 252 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 - 8 T + 200 T^{2} - 1372 T^{3} + 25899 T^{4} - 149452 T^{5} + 2122008 T^{6} - 149452 p T^{7} + 25899 p^{2} T^{8} - 1372 p^{3} T^{9} + 200 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 + 28 T + 626 T^{2} + 9332 T^{3} + 122319 T^{4} + 1280600 T^{5} + 12451644 T^{6} + 1280600 p T^{7} + 122319 p^{2} T^{8} + 9332 p^{3} T^{9} + 626 p^{4} T^{10} + 28 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 - 24 T + 382 T^{2} - 3180 T^{3} + 21614 T^{4} - 768 p T^{5} + 548518 T^{6} - 768 p^{2} T^{7} + 21614 p^{2} T^{8} - 3180 p^{3} T^{9} + 382 p^{4} T^{10} - 24 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 - 4 T + 262 T^{2} - 1428 T^{3} + 43551 T^{4} - 210056 T^{5} + 4643796 T^{6} - 210056 p T^{7} + 43551 p^{2} T^{8} - 1428 p^{3} T^{9} + 262 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 - 24 T + 494 T^{2} - 6904 T^{3} + 85743 T^{4} - 915568 T^{5} + 9189188 T^{6} - 915568 p T^{7} + 85743 p^{2} T^{8} - 6904 p^{3} T^{9} + 494 p^{4} T^{10} - 24 p^{5} T^{11} + p^{6} T^{12} \)
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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

−4.09031111654381802645672123338, −3.56120644817169924573871883052, −3.45493870074482052559131606654, −3.39754294267740865089801568863, −3.39512106286656091814725679259, −3.37654742283708812921054201548, −3.26102459286128709439405811516, −2.87071965778548710600295564363, −2.76046535411553398857554480071, −2.75439491999202419768320496715, −2.58307139298571321093782278126, −2.54422616734848551979751937159, −2.50225248407168403152636984477, −2.13272666985618256259963074159, −1.83684426880189937497365115176, −1.80220197294988521264035880407, −1.76520432537489537850685469556, −1.61607583415162811502754656433, −1.50658520067359405612869638364, −1.02462340215201586864938219267, −0.871246311518717853347318950590, −0.817989701220327307007684618086, −0.805474589480996308870296806476, −0.55212206858907260519352753040, −0.32240760495662175513553040452, 0.32240760495662175513553040452, 0.55212206858907260519352753040, 0.805474589480996308870296806476, 0.817989701220327307007684618086, 0.871246311518717853347318950590, 1.02462340215201586864938219267, 1.50658520067359405612869638364, 1.61607583415162811502754656433, 1.76520432537489537850685469556, 1.80220197294988521264035880407, 1.83684426880189937497365115176, 2.13272666985618256259963074159, 2.50225248407168403152636984477, 2.54422616734848551979751937159, 2.58307139298571321093782278126, 2.75439491999202419768320496715, 2.76046535411553398857554480071, 2.87071965778548710600295564363, 3.26102459286128709439405811516, 3.37654742283708812921054201548, 3.39512106286656091814725679259, 3.39754294267740865089801568863, 3.45493870074482052559131606654, 3.56120644817169924573871883052, 4.09031111654381802645672123338

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

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