Properties

Label 12-570e6-1.1-c1e6-0-10
Degree $12$
Conductor $3.430\times 10^{16}$
Sign $1$
Analytic cond. $8890.20$
Root an. cond. $2.13341$
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  − 3·7-s + 8-s − 3·11-s + 9·13-s + 9·17-s + 21·19-s + 15·23-s + 27-s − 15·29-s + 6·31-s + 18·37-s + 12·41-s + 21·43-s − 6·47-s + 21·49-s + 6·53-s − 3·56-s − 33·59-s − 9·61-s + 6·67-s + 30·71-s − 27·73-s + 9·77-s + 9·79-s − 3·83-s − 3·88-s − 21·89-s + ⋯
L(s)  = 1  − 1.13·7-s + 0.353·8-s − 0.904·11-s + 2.49·13-s + 2.18·17-s + 4.81·19-s + 3.12·23-s + 0.192·27-s − 2.78·29-s + 1.07·31-s + 2.95·37-s + 1.87·41-s + 3.20·43-s − 0.875·47-s + 3·49-s + 0.824·53-s − 0.400·56-s − 4.29·59-s − 1.15·61-s + 0.733·67-s + 3.56·71-s − 3.16·73-s + 1.02·77-s + 1.01·79-s − 0.329·83-s − 0.319·88-s − 2.22·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 5^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 5^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{6} \cdot 3^{6} \cdot 5^{6} \cdot 19^{6}\)
Sign: $1$
Analytic conductor: \(8890.20\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{6} \cdot 3^{6} \cdot 5^{6} \cdot 19^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(7.938719255\)
\(L(\frac12)\) \(\approx\) \(7.938719255\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T^{3} + T^{6} \)
3 \( 1 - T^{3} + T^{6} \)
5 \( 1 - T^{3} + T^{6} \)
19 \( ( 1 - 7 T + p T^{2} )^{3} \)
good7 \( 1 + 3 T - 12 T^{2} - 19 T^{3} + 171 T^{4} + 18 p T^{5} - 1161 T^{6} + 18 p^{2} T^{7} + 171 p^{2} T^{8} - 19 p^{3} T^{9} - 12 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 + 3 T - 6 T^{2} + 27 T^{3} + 3 p T^{4} - 492 T^{5} - 821 T^{6} - 492 p T^{7} + 3 p^{3} T^{8} + 27 p^{3} T^{9} - 6 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 - 9 T + 36 T^{2} - 72 T^{3} - 153 T^{4} + 2493 T^{5} - 12295 T^{6} + 2493 p T^{7} - 153 p^{2} T^{8} - 72 p^{3} T^{9} + 36 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 - 9 T + 54 T^{2} - 310 T^{3} + 1593 T^{4} - 6651 T^{5} + 27629 T^{6} - 6651 p T^{7} + 1593 p^{2} T^{8} - 310 p^{3} T^{9} + 54 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 - 15 T + 93 T^{2} - 13 p T^{3} + 1344 T^{4} - 14418 T^{5} + 97421 T^{6} - 14418 p T^{7} + 1344 p^{2} T^{8} - 13 p^{4} T^{9} + 93 p^{4} T^{10} - 15 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 + 15 T + 177 T^{2} + 1549 T^{3} + 12336 T^{4} + 79524 T^{5} + 464861 T^{6} + 79524 p T^{7} + 12336 p^{2} T^{8} + 1549 p^{3} T^{9} + 177 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 - 6 T - 42 T^{2} + 130 T^{3} + 1872 T^{4} - 72 T^{5} - 74217 T^{6} - 72 p T^{7} + 1872 p^{2} T^{8} + 130 p^{3} T^{9} - 42 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
37 \( ( 1 - 9 T + 117 T^{2} - 593 T^{3} + 117 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
41 \( 1 - 12 T + 84 T^{2} - 529 T^{3} + 4287 T^{4} - 29439 T^{5} + 175865 T^{6} - 29439 p T^{7} + 4287 p^{2} T^{8} - 529 p^{3} T^{9} + 84 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 - 21 T + 252 T^{2} - 1974 T^{3} + 10296 T^{4} - 35319 T^{5} + 109385 T^{6} - 35319 p T^{7} + 10296 p^{2} T^{8} - 1974 p^{3} T^{9} + 252 p^{4} T^{10} - 21 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 + 6 T + 126 T^{2} + 513 T^{3} + 9855 T^{4} + 38607 T^{5} + 570925 T^{6} + 38607 p T^{7} + 9855 p^{2} T^{8} + 513 p^{3} T^{9} + 126 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 - 6 T - 117 T^{2} + 999 T^{3} + 1053 T^{4} - 35709 T^{5} + 253630 T^{6} - 35709 p T^{7} + 1053 p^{2} T^{8} + 999 p^{3} T^{9} - 117 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 + 33 T + 624 T^{2} + 8696 T^{3} + 1662 p T^{4} + 930537 T^{5} + 7629125 T^{6} + 930537 p T^{7} + 1662 p^{3} T^{8} + 8696 p^{3} T^{9} + 624 p^{4} T^{10} + 33 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 + 9 T - 144 T^{2} - 1096 T^{3} + 10611 T^{4} + 36747 T^{5} - 623895 T^{6} + 36747 p T^{7} + 10611 p^{2} T^{8} - 1096 p^{3} T^{9} - 144 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 - 6 T - 24 T^{2} - 293 T^{3} - 4275 T^{4} + 44073 T^{5} + 247245 T^{6} + 44073 p T^{7} - 4275 p^{2} T^{8} - 293 p^{3} T^{9} - 24 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 - 30 T + 360 T^{2} - 1863 T^{3} - 4095 T^{4} + 178755 T^{5} - 1993067 T^{6} + 178755 p T^{7} - 4095 p^{2} T^{8} - 1863 p^{3} T^{9} + 360 p^{4} T^{10} - 30 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 + 27 T + 360 T^{2} + 2682 T^{3} + 2871 T^{4} - 197919 T^{5} - 2607499 T^{6} - 197919 p T^{7} + 2871 p^{2} T^{8} + 2682 p^{3} T^{9} + 360 p^{4} T^{10} + 27 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 - 9 T + 171 T^{2} - 1087 T^{3} + 8316 T^{4} - 39474 T^{5} + 126957 T^{6} - 39474 p T^{7} + 8316 p^{2} T^{8} - 1087 p^{3} T^{9} + 171 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 + 3 T - 216 T^{2} - 215 T^{3} + 30615 T^{4} + 8742 T^{5} - 2922277 T^{6} + 8742 p T^{7} + 30615 p^{2} T^{8} - 215 p^{3} T^{9} - 216 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 + 21 T + 204 T^{2} + 1718 T^{3} + 5424 T^{4} - 101313 T^{5} - 1492117 T^{6} - 101313 p T^{7} + 5424 p^{2} T^{8} + 1718 p^{3} T^{9} + 204 p^{4} T^{10} + 21 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 - 12 T + 18 T^{2} + 1878 T^{3} - 14454 T^{4} - 39558 T^{5} + 2111975 T^{6} - 39558 p T^{7} - 14454 p^{2} T^{8} + 1878 p^{3} T^{9} + 18 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
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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

−5.81106139982521118793544399893, −5.54468552280826147815457451191, −5.38322519595274537083847474547, −5.29228264960409890063957582794, −5.23877080928320762788931192669, −5.12095929337421819705732010965, −4.79903077711401749353940232714, −4.44447062198084891304307475936, −4.30714204472339485370212287771, −4.01178984363588441796927598759, −3.87267238234459201266877255823, −3.76173044058227277670842854520, −3.75083995730667298725444205468, −3.13772938463925988846464321131, −3.12423741796904091830245310372, −2.93478261384108184008126012817, −2.76565060929595089852085674446, −2.74381254230467785679306032049, −2.73150711522693493000033748506, −1.92770443811309515266417563416, −1.53963783770125931891956233506, −1.29277694890635790102848507713, −0.951181918214195893670420141669, −0.912509302716693228639213401554, −0.873647871330714978105554528877, 0.873647871330714978105554528877, 0.912509302716693228639213401554, 0.951181918214195893670420141669, 1.29277694890635790102848507713, 1.53963783770125931891956233506, 1.92770443811309515266417563416, 2.73150711522693493000033748506, 2.74381254230467785679306032049, 2.76565060929595089852085674446, 2.93478261384108184008126012817, 3.12423741796904091830245310372, 3.13772938463925988846464321131, 3.75083995730667298725444205468, 3.76173044058227277670842854520, 3.87267238234459201266877255823, 4.01178984363588441796927598759, 4.30714204472339485370212287771, 4.44447062198084891304307475936, 4.79903077711401749353940232714, 5.12095929337421819705732010965, 5.23877080928320762788931192669, 5.29228264960409890063957582794, 5.38322519595274537083847474547, 5.54468552280826147815457451191, 5.81106139982521118793544399893

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

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