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Properties

Label 12-19e12-1.1-c1e6-0-1
Degree $12$
Conductor $2.213\times 10^{15}$
Sign $1$
Analytic cond. $573.727$
Root an. cond. $1.69782$
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 − 3·3-s + 6·4-s + 3·5-s + 9·6-s − 9·8-s + 9·9-s − 9·10-s − 18·12-s − 9·15-s + 12·16-s − 6·17-s − 27·18-s + 18·20-s + 6·23-s + 27·24-s + 15·25-s − 20·27-s − 15·29-s + 27·30-s + 18·31-s − 12·32-s + 18·34-s + 54·36-s − 27·40-s − 12·41-s + 27·45-s + ⋯
L(s)  = 1  − 2.12·2-s − 1.73·3-s + 3·4-s + 1.34·5-s + 3.67·6-s − 3.18·8-s + 3·9-s − 2.84·10-s − 5.19·12-s − 2.32·15-s + 3·16-s − 1.45·17-s − 6.36·18-s + 4.02·20-s + 1.25·23-s + 5.51·24-s + 3·25-s − 3.84·27-s − 2.78·29-s + 4.92·30-s + 3.23·31-s − 2.12·32-s + 3.08·34-s + 9·36-s − 4.26·40-s − 1.87·41-s + 4.02·45-s + ⋯

Functional equation

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

Invariants

Degree: \(12\)
Conductor: \(19^{12}\)
Sign: $1$
Analytic conductor: \(573.727\)
Root analytic conductor: \(1.69782\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 19^{12} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.5348838961\)
\(L(\frac12)\) \(\approx\) \(0.5348838961\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad19 \( 1 \)
good2 \( 1 + 3 T + 3 T^{2} - 3 T^{4} - 3 p T^{5} - 11 T^{6} - 3 p^{2} T^{7} - 3 p^{2} T^{8} + 3 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
3 \( 1 + p T - 7 T^{3} + p T^{4} + 2 p^{2} T^{5} + 19 T^{6} + 2 p^{3} T^{7} + p^{3} T^{8} - 7 p^{3} T^{9} + p^{6} T^{11} + p^{6} T^{12} \)
5 \( 1 - 3 T - 6 T^{2} + 9 T^{3} + 69 T^{4} - 6 p T^{5} - 371 T^{6} - 6 p^{2} T^{7} + 69 p^{2} T^{8} + 9 p^{3} T^{9} - 6 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
7 \( ( 1 + 18 T^{2} + T^{3} + 18 p T^{4} + p^{3} T^{6} )^{2} \)
11 \( ( 1 + 24 T^{2} - 9 T^{3} + 24 p T^{4} + p^{3} T^{6} )^{2} \)
13 \( 1 - 18 T^{2} - 74 T^{3} + 90 T^{4} + 666 T^{5} + 1551 T^{6} + 666 p T^{7} + 90 p^{2} T^{8} - 74 p^{3} T^{9} - 18 p^{4} T^{10} + p^{6} T^{12} \)
17 \( 1 + 6 T - 24 T^{2} - 54 T^{3} + 1338 T^{4} + 1914 T^{5} - 18929 T^{6} + 1914 p T^{7} + 1338 p^{2} T^{8} - 54 p^{3} T^{9} - 24 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 - 6 T - 33 T^{2} + 90 T^{3} + 1662 T^{4} - 66 p T^{5} - 40709 T^{6} - 66 p^{2} T^{7} + 1662 p^{2} T^{8} + 90 p^{3} T^{9} - 33 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 + 15 T + 66 T^{2} + 423 T^{3} + 5955 T^{4} + 30264 T^{5} + 88765 T^{6} + 30264 p T^{7} + 5955 p^{2} T^{8} + 423 p^{3} T^{9} + 66 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} \)
31 \( ( 1 - 9 T + 99 T^{2} - 505 T^{3} + 99 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
37 \( ( 1 + 90 T^{2} + 17 T^{3} + 90 p T^{4} + p^{3} T^{6} )^{2} \)
41 \( 1 + 12 T + 12 T^{2} - 162 T^{3} + 1536 T^{4} + 5520 T^{5} - 56477 T^{6} + 5520 p T^{7} + 1536 p^{2} T^{8} - 162 p^{3} T^{9} + 12 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 - 72 T^{2} + 326 T^{3} + 2088 T^{4} - 11736 T^{5} - 37329 T^{6} - 11736 p T^{7} + 2088 p^{2} T^{8} + 326 p^{3} T^{9} - 72 p^{4} T^{10} + p^{6} T^{12} \)
47 \( 1 - 6 T - 96 T^{2} + 342 T^{3} + 7818 T^{4} - 13650 T^{5} - 369317 T^{6} - 13650 p T^{7} + 7818 p^{2} T^{8} + 342 p^{3} T^{9} - 96 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 + 6 T - 114 T^{2} - 270 T^{3} + 11040 T^{4} + 9204 T^{5} - 653645 T^{6} + 9204 p T^{7} + 11040 p^{2} T^{8} - 270 p^{3} T^{9} - 114 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 + 21 T + 129 T^{2} + 18 p T^{3} + 21291 T^{4} + 156405 T^{5} + 667366 T^{6} + 156405 p T^{7} + 21291 p^{2} T^{8} + 18 p^{4} T^{9} + 129 p^{4} T^{10} + 21 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 + 9 T - 81 T^{2} - 376 T^{3} + 8109 T^{4} + 4167 T^{5} - 607434 T^{6} + 4167 p T^{7} + 8109 p^{2} T^{8} - 376 p^{3} T^{9} - 81 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 - 18 T + 99 T^{2} - 74 T^{3} - 234 T^{4} + 14598 T^{5} - 292641 T^{6} + 14598 p T^{7} - 234 p^{2} T^{8} - 74 p^{3} T^{9} + 99 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 + 30 T + 399 T^{2} + 4734 T^{3} + 60990 T^{4} + 596118 T^{5} + 4859755 T^{6} + 596118 p T^{7} + 60990 p^{2} T^{8} + 4734 p^{3} T^{9} + 399 p^{4} T^{10} + 30 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 - 171 T^{2} + 128 T^{3} + 16758 T^{4} - 10944 T^{5} - 1352463 T^{6} - 10944 p T^{7} + 16758 p^{2} T^{8} + 128 p^{3} T^{9} - 171 p^{4} T^{10} + p^{6} T^{12} \)
79 \( 1 + 9 T - 54 T^{2} - 11 T^{3} + 2043 T^{4} - 53046 T^{5} - 583449 T^{6} - 53046 p T^{7} + 2043 p^{2} T^{8} - 11 p^{3} T^{9} - 54 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
83 \( ( 1 + 60 T^{2} + 459 T^{3} + 60 p T^{4} + p^{3} T^{6} )^{2} \)
89 \( 1 + 15 T - 96 T^{2} - 639 T^{3} + 33567 T^{4} + 137112 T^{5} - 2106407 T^{6} + 137112 p T^{7} + 33567 p^{2} T^{8} - 639 p^{3} T^{9} - 96 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 - 15 T - 105 T^{2} + 616 T^{3} + 35145 T^{4} - 99225 T^{5} - 3109890 T^{6} - 99225 p T^{7} + 35145 p^{2} T^{8} + 616 p^{3} T^{9} - 105 p^{4} T^{10} - 15 p^{5} T^{11} + p^{6} T^{12} \)
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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

−6.29509012405236765834163679057, −6.13676541557135144327713029794, −6.11998716761066261889294908575, −6.00318846831061244120932032466, −5.44076485295806675484023674404, −5.22652724312943165311891020933, −5.22087292663419402273064393518, −5.04781520362783428027811079596, −4.74465598166483357304056886974, −4.66774918571958173674546271606, −4.41738843391220657581987999745, −4.29351881543810221085817425288, −4.15128647879574296946410012581, −3.45963469748350464469398619784, −3.35442375070765790366979271527, −3.11603123310430425953184073363, −2.90016522155440990288185943735, −2.82068740628152992517793540383, −2.29029808350549722361078127814, −1.90208274629775313726054877896, −1.60710616291669339673987143838, −1.56282004942595339634312974596, −1.54976932439311631903015362400, −0.835799331045518671959280107681, −0.41154943998556114603173191184, 0.41154943998556114603173191184, 0.835799331045518671959280107681, 1.54976932439311631903015362400, 1.56282004942595339634312974596, 1.60710616291669339673987143838, 1.90208274629775313726054877896, 2.29029808350549722361078127814, 2.82068740628152992517793540383, 2.90016522155440990288185943735, 3.11603123310430425953184073363, 3.35442375070765790366979271527, 3.45963469748350464469398619784, 4.15128647879574296946410012581, 4.29351881543810221085817425288, 4.41738843391220657581987999745, 4.66774918571958173674546271606, 4.74465598166483357304056886974, 5.04781520362783428027811079596, 5.22087292663419402273064393518, 5.22652724312943165311891020933, 5.44076485295806675484023674404, 6.00318846831061244120932032466, 6.11998716761066261889294908575, 6.13676541557135144327713029794, 6.29509012405236765834163679057

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

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