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Properties

Label 12-3024e6-1.1-c1e6-0-1
Degree $12$
Conductor $7.647\times 10^{20}$
Sign $1$
Analytic cond. $1.98222\times 10^{8}$
Root an. cond. $4.91393$
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  + 3·5-s + 3·7-s − 6·11-s + 3·13-s − 12·17-s + 6·19-s − 12·23-s + 15·25-s + 9·29-s − 3·31-s + 9·35-s − 6·37-s − 3·43-s − 3·47-s + 3·49-s − 12·53-s − 18·55-s + 3·59-s − 6·61-s + 9·65-s − 12·67-s + 18·71-s − 42·73-s − 18·77-s − 21·79-s + 18·83-s − 36·85-s + ⋯
L(s)  = 1  + 1.34·5-s + 1.13·7-s − 1.80·11-s + 0.832·13-s − 2.91·17-s + 1.37·19-s − 2.50·23-s + 3·25-s + 1.67·29-s − 0.538·31-s + 1.52·35-s − 0.986·37-s − 0.457·43-s − 0.437·47-s + 3/7·49-s − 1.64·53-s − 2.42·55-s + 0.390·59-s − 0.768·61-s + 1.11·65-s − 1.46·67-s + 2.13·71-s − 4.91·73-s − 2.05·77-s − 2.36·79-s + 1.97·83-s − 3.90·85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{18} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{18} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{24} \cdot 3^{18} \cdot 7^{6}\)
Sign: $1$
Analytic conductor: \(1.98222\times 10^{8}\)
Root analytic conductor: \(4.91393\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{24} \cdot 3^{18} \cdot 7^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.7792805142\)
\(L(\frac12)\) \(\approx\) \(0.7792805142\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( ( 1 - T + T^{2} )^{3} \)
good5 \( 1 - 3 T - 6 T^{2} + 9 T^{3} + 69 T^{4} - 6 p T^{5} - 371 T^{6} - 6 p^{2} T^{7} + 69 p^{2} T^{8} + 9 p^{3} T^{9} - 6 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 + 6 T - 6 T^{2} - 18 T^{3} + 492 T^{4} + 852 T^{5} - 2873 T^{6} + 852 p T^{7} + 492 p^{2} T^{8} - 18 p^{3} T^{9} - 6 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 - 3 T + 3 T^{2} - 76 T^{3} + 45 T^{4} + 135 T^{5} + 3246 T^{6} + 135 p T^{7} + 45 p^{2} T^{8} - 76 p^{3} T^{9} + 3 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
17 \( ( 1 + 6 T + 60 T^{2} + 207 T^{3} + 60 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
19 \( ( 1 - 3 T + 51 T^{2} - 97 T^{3} + 51 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
23 \( 1 + 12 T + 48 T^{2} + 54 T^{3} + 420 T^{4} + 6060 T^{5} + 37591 T^{6} + 6060 p T^{7} + 420 p^{2} T^{8} + 54 p^{3} T^{9} + 48 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 - 9 T + 30 T^{2} - 81 T^{3} - 579 T^{4} + 9414 T^{5} - 59051 T^{6} + 9414 p T^{7} - 579 p^{2} T^{8} - 81 p^{3} T^{9} + 30 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 + 3 T - 6 T^{2} + 319 T^{3} + 171 T^{4} - 1962 T^{5} + 62727 T^{6} - 1962 p T^{7} + 171 p^{2} T^{8} + 319 p^{3} T^{9} - 6 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
37 \( ( 1 + 3 T + 33 T^{2} - 101 T^{3} + 33 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
41 \( 1 - 114 T^{2} + 18 T^{3} + 8322 T^{4} - 1026 T^{5} - 394913 T^{6} - 1026 p T^{7} + 8322 p^{2} T^{8} + 18 p^{3} T^{9} - 114 p^{4} T^{10} + p^{6} T^{12} \)
43 \( 1 + 3 T - 114 T^{2} - 149 T^{3} + 9063 T^{4} + 5670 T^{5} - 441093 T^{6} + 5670 p T^{7} + 9063 p^{2} T^{8} - 149 p^{3} T^{9} - 114 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 + 3 T - 78 T^{2} - 405 T^{3} + 2481 T^{4} + 11064 T^{5} - 57089 T^{6} + 11064 p T^{7} + 2481 p^{2} T^{8} - 405 p^{3} T^{9} - 78 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
53 \( ( 1 + 6 T + 150 T^{2} + 639 T^{3} + 150 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
59 \( 1 - 3 T - 96 T^{2} + 495 T^{3} + 3615 T^{4} - 15798 T^{5} - 107021 T^{6} - 15798 p T^{7} + 3615 p^{2} T^{8} + 495 p^{3} T^{9} - 96 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 + 6 T - 132 T^{2} - 418 T^{3} + 13698 T^{4} + 19134 T^{5} - 893289 T^{6} + 19134 p T^{7} + 13698 p^{2} T^{8} - 418 p^{3} T^{9} - 132 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 + 12 T - 78 T^{2} - 518 T^{3} + 15318 T^{4} + 50094 T^{5} - 815637 T^{6} + 50094 p T^{7} + 15318 p^{2} T^{8} - 518 p^{3} T^{9} - 78 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
71 \( ( 1 - 9 T + 159 T^{2} - 1305 T^{3} + 159 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
73 \( ( 1 + 21 T + 303 T^{2} + 2797 T^{3} + 303 p T^{4} + 21 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
79 \( 1 + 21 T + 84 T^{2} + 499 T^{3} + 25767 T^{4} + 195678 T^{5} + 408327 T^{6} + 195678 p T^{7} + 25767 p^{2} T^{8} + 499 p^{3} T^{9} + 84 p^{4} T^{10} + 21 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 - 18 T + 30 T^{2} + 702 T^{3} + 8088 T^{4} - 126648 T^{5} + 719359 T^{6} - 126648 p T^{7} + 8088 p^{2} T^{8} + 702 p^{3} T^{9} + 30 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \)
89 \( ( 1 + 12 T + 204 T^{2} + 1323 T^{3} + 204 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
97 \( 1 - 3 T - 114 T^{2} + 149 T^{3} + 2421 T^{4} + 11502 T^{5} + 340233 T^{6} + 11502 p T^{7} + 2421 p^{2} T^{8} + 149 p^{3} T^{9} - 114 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
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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

−4.52589104318284472520102362456, −4.45624901871692434918379628275, −4.43166585441271062324164880376, −4.36127962847568121635989714073, −3.81229846274657484821625893350, −3.71264710572358999546153987999, −3.69333962510757171965602765239, −3.38660221420670639833856263815, −3.24183432595968349284447898071, −3.02724878692830172995125367380, −3.02456493576158690488866380694, −2.87631383231924023977783070511, −2.73345235586049779668365882675, −2.34283937817641426075497387896, −2.29328921904180811263251352858, −2.09219004006189121692826232082, −2.08717260733584341751018240730, −1.86591370332882636654850577885, −1.54669089170175427871290621770, −1.54203969038440297010019150756, −1.33965778957413149720389951765, −1.04712854471032150105736736375, −0.77225936527475558033759696815, −0.41994142237343550909924097011, −0.094830234442373545781719550628, 0.094830234442373545781719550628, 0.41994142237343550909924097011, 0.77225936527475558033759696815, 1.04712854471032150105736736375, 1.33965778957413149720389951765, 1.54203969038440297010019150756, 1.54669089170175427871290621770, 1.86591370332882636654850577885, 2.08717260733584341751018240730, 2.09219004006189121692826232082, 2.29328921904180811263251352858, 2.34283937817641426075497387896, 2.73345235586049779668365882675, 2.87631383231924023977783070511, 3.02456493576158690488866380694, 3.02724878692830172995125367380, 3.24183432595968349284447898071, 3.38660221420670639833856263815, 3.69333962510757171965602765239, 3.71264710572358999546153987999, 3.81229846274657484821625893350, 4.36127962847568121635989714073, 4.43166585441271062324164880376, 4.45624901871692434918379628275, 4.52589104318284472520102362456

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

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