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Properties

Label 12-855e6-1.1-c1e6-0-2
Degree $12$
Conductor $3.907\times 10^{17}$
Sign $1$
Analytic cond. $101264.$
Root an. cond. $2.61289$
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  + 2-s − 3·5-s + 4·7-s + 5·8-s − 3·10-s + 10·11-s + 15·13-s + 4·14-s + 8·16-s + 17-s + 10·22-s + 4·23-s + 3·25-s + 15·26-s − 2·29-s − 2·31-s + 4·32-s + 34-s − 12·35-s − 4·37-s − 15·40-s − 2·41-s + 43-s + 4·46-s + 6·47-s − 20·49-s + 3·50-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.34·5-s + 1.51·7-s + 1.76·8-s − 0.948·10-s + 3.01·11-s + 4.16·13-s + 1.06·14-s + 2·16-s + 0.242·17-s + 2.13·22-s + 0.834·23-s + 3/5·25-s + 2.94·26-s − 0.371·29-s − 0.359·31-s + 0.707·32-s + 0.171·34-s − 2.02·35-s − 0.657·37-s − 2.37·40-s − 0.312·41-s + 0.152·43-s + 0.589·46-s + 0.875·47-s − 2.85·49-s + 0.424·50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 5^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 5^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(3^{12} \cdot 5^{6} \cdot 19^{6}\)
Sign: $1$
Analytic conductor: \(101264.\)
Root analytic conductor: \(2.61289\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 3^{12} \cdot 5^{6} \cdot 19^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(21.48508403\)
\(L(\frac12)\) \(\approx\) \(21.48508403\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( ( 1 + T + T^{2} )^{3} \)
19 \( 1 + 7 p T^{3} + p^{3} T^{6} \)
good2 \( 1 - T + T^{2} - 3 p T^{3} + 3 T^{4} - p^{2} T^{5} + 21 T^{6} - p^{3} T^{7} + 3 p^{2} T^{8} - 3 p^{4} T^{9} + p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \)
7 \( ( 1 - 2 T + 16 T^{2} - 29 T^{3} + 16 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
11 \( ( 1 - 5 T + 35 T^{2} - 109 T^{3} + 35 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
13 \( ( 1 - 7 T + p T^{2} )^{3}( 1 + 2 T + p T^{2} )^{3} \)
17 \( 1 - T - 6 T^{2} + 47 T^{3} - 97 T^{4} - 240 T^{5} + 9433 T^{6} - 240 p T^{7} - 97 p^{2} T^{8} + 47 p^{3} T^{9} - 6 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 - 4 T - 14 T^{2} + 150 T^{3} - 330 T^{4} - 646 T^{5} + 13395 T^{6} - 646 p T^{7} - 330 p^{2} T^{8} + 150 p^{3} T^{9} - 14 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 + 2 T - 78 T^{2} - 70 T^{3} + 4112 T^{4} + 1764 T^{5} - 135893 T^{6} + 1764 p T^{7} + 4112 p^{2} T^{8} - 70 p^{3} T^{9} - 78 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
31 \( ( 1 + T + 87 T^{2} + 55 T^{3} + 87 p T^{4} + p^{2} T^{5} + p^{3} T^{6} )^{2} \)
37 \( ( 1 + 2 T - 8 T^{2} - 79 T^{3} - 8 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
41 \( 1 + 2 T - 76 T^{2} - 94 T^{3} + 2866 T^{4} + 402 T^{5} - 115153 T^{6} + 402 p T^{7} + 2866 p^{2} T^{8} - 94 p^{3} T^{9} - 76 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 - T - 84 T^{2} - 155 T^{3} + 3605 T^{4} + 8436 T^{5} - 152285 T^{6} + 8436 p T^{7} + 3605 p^{2} T^{8} - 155 p^{3} T^{9} - 84 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 - 6 T - 98 T^{2} + 226 T^{3} + 8568 T^{4} - 6688 T^{5} - 450585 T^{6} - 6688 p T^{7} + 8568 p^{2} T^{8} + 226 p^{3} T^{9} - 98 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 - 11 T + 4 T^{2} + 423 T^{3} - 1917 T^{4} + 5488 T^{5} - 36627 T^{6} + 5488 p T^{7} - 1917 p^{2} T^{8} + 423 p^{3} T^{9} + 4 p^{4} T^{10} - 11 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 - 6 T - 134 T^{2} + 298 T^{3} + 14916 T^{4} - 12556 T^{5} - 985377 T^{6} - 12556 p T^{7} + 14916 p^{2} T^{8} + 298 p^{3} T^{9} - 134 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 - 9 T - 53 T^{2} + 892 T^{3} + 341 T^{4} - 26923 T^{5} + 157646 T^{6} - 26923 p T^{7} + 341 p^{2} T^{8} + 892 p^{3} T^{9} - 53 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 - 20 T + 91 T^{2} - 644 T^{3} + 19418 T^{4} - 116972 T^{5} + 70523 T^{6} - 116972 p T^{7} + 19418 p^{2} T^{8} - 644 p^{3} T^{9} + 91 p^{4} T^{10} - 20 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 + 29 T + 392 T^{2} + 3851 T^{3} + 36757 T^{4} + 355872 T^{5} + 3184895 T^{6} + 355872 p T^{7} + 36757 p^{2} T^{8} + 3851 p^{3} T^{9} + 392 p^{4} T^{10} + 29 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 - 22 T + 148 T^{2} - 814 T^{3} + 16010 T^{4} - 143110 T^{5} + 774911 T^{6} - 143110 p T^{7} + 16010 p^{2} T^{8} - 814 p^{3} T^{9} + 148 p^{4} T^{10} - 22 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 - 24 T + 223 T^{2} - 1384 T^{3} + 11666 T^{4} - 62240 T^{5} + 107087 T^{6} - 62240 p T^{7} + 11666 p^{2} T^{8} - 1384 p^{3} T^{9} + 223 p^{4} T^{10} - 24 p^{5} T^{11} + p^{6} T^{12} \)
83 \( ( 1 - 3 T + 195 T^{2} - 575 T^{3} + 195 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
89 \( 1 + 14 T - 35 T^{2} - 1638 T^{3} - 1302 T^{4} + 97958 T^{5} + 1042389 T^{6} + 97958 p T^{7} - 1302 p^{2} T^{8} - 1638 p^{3} T^{9} - 35 p^{4} T^{10} + 14 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 + 7 T - 176 T^{2} - 899 T^{3} + 20141 T^{4} + 38638 T^{5} - 2016151 T^{6} + 38638 p T^{7} + 20141 p^{2} T^{8} - 899 p^{3} T^{9} - 176 p^{4} T^{10} + 7 p^{5} T^{11} + p^{6} T^{12} \)
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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

−5.19207595269564458102058354266, −5.15893207963018382639306160436, −5.13559357468523693885187638810, −5.05418663496903355850203714350, −4.66272543541607611111443837977, −4.40579250954443567283355052516, −4.35297532357895347315084722523, −4.11036052168669984667063970235, −3.94483291136481067690134085594, −3.88144490551443204033296595464, −3.80721491750881589511302228519, −3.76632719722354185379734238878, −3.55506685921991911433188923144, −3.28139273191288228166320499463, −3.17455059096047881226755744440, −2.77413244391896021600820961797, −2.65661221440307688990045431490, −2.14093075211920399384381005937, −1.83195136494491115977167745430, −1.67560568202253393793191788166, −1.47748021961720739342696918665, −1.44696843022817216828331123756, −1.14564056617295789954246641625, −0.870998846931753068844988891859, −0.72094563299972330615224213059, 0.72094563299972330615224213059, 0.870998846931753068844988891859, 1.14564056617295789954246641625, 1.44696843022817216828331123756, 1.47748021961720739342696918665, 1.67560568202253393793191788166, 1.83195136494491115977167745430, 2.14093075211920399384381005937, 2.65661221440307688990045431490, 2.77413244391896021600820961797, 3.17455059096047881226755744440, 3.28139273191288228166320499463, 3.55506685921991911433188923144, 3.76632719722354185379734238878, 3.80721491750881589511302228519, 3.88144490551443204033296595464, 3.94483291136481067690134085594, 4.11036052168669984667063970235, 4.35297532357895347315084722523, 4.40579250954443567283355052516, 4.66272543541607611111443837977, 5.05418663496903355850203714350, 5.13559357468523693885187638810, 5.15893207963018382639306160436, 5.19207595269564458102058354266

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

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