Properties

Label 12-2205e6-1.1-c3e6-0-3
Degree $12$
Conductor $1.149\times 10^{20}$
Sign $1$
Analytic cond. $4.84895\times 10^{12}$
Root an. cond. $11.4061$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $6$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 14·4-s − 30·5-s + 83·16-s − 44·17-s + 420·20-s + 525·25-s + 380·37-s − 612·41-s − 328·43-s − 120·47-s − 136·59-s − 556·64-s + 1.11e3·67-s + 616·68-s + 1.40e3·79-s − 2.49e3·80-s − 2.91e3·83-s + 1.32e3·85-s − 372·89-s − 7.35e3·100-s − 3.19e3·101-s − 684·109-s − 4.47e3·121-s − 7.00e3·125-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  − 7/4·4-s − 2.68·5-s + 1.29·16-s − 0.627·17-s + 4.69·20-s + 21/5·25-s + 1.68·37-s − 2.33·41-s − 1.16·43-s − 0.372·47-s − 0.300·59-s − 1.08·64-s + 2.02·67-s + 1.09·68-s + 1.99·79-s − 3.47·80-s − 3.85·83-s + 1.68·85-s − 0.443·89-s − 7.34·100-s − 3.14·101-s − 0.601·109-s − 3.36·121-s − 5.00·125-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 5^{6} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 5^{6} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(3^{12} \cdot 5^{6} \cdot 7^{12}\)
Sign: $1$
Analytic conductor: \(4.84895\times 10^{12}\)
Root analytic conductor: \(11.4061\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2205} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(6\)
Selberg data: \((12,\ 3^{12} \cdot 5^{6} \cdot 7^{12} ,\ ( \ : [3/2]^{6} ),\ 1 )\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( ( 1 + p T )^{6} \)
7 \( 1 \)
good2 \( 1 + 7 p T^{2} + 113 T^{4} + 61 p^{4} T^{6} + 113 p^{6} T^{8} + 7 p^{13} T^{10} + p^{18} T^{12} \)
11 \( 1 + 4474 T^{2} + 10690519 T^{4} + 16986506732 T^{6} + 10690519 p^{6} T^{8} + 4474 p^{12} T^{10} + p^{18} T^{12} \)
13 \( 1 + 6438 T^{2} + 27862663 T^{4} + 71083547284 T^{6} + 27862663 p^{6} T^{8} + 6438 p^{12} T^{10} + p^{18} T^{12} \)
17 \( ( 1 + 22 T + 8879 T^{2} - 428 T^{3} + 8879 p^{3} T^{4} + 22 p^{6} T^{5} + p^{9} T^{6} )^{2} \)
19 \( 1 + 21978 T^{2} + 221297479 T^{4} + 1589592507244 T^{6} + 221297479 p^{6} T^{8} + 21978 p^{12} T^{10} + p^{18} T^{12} \)
23 \( 1 + 49178 T^{2} + 1213984799 T^{4} + 18399433839724 T^{6} + 1213984799 p^{6} T^{8} + 49178 p^{12} T^{10} + p^{18} T^{12} \)
29 \( 1 + 124478 T^{2} + 6905589367 T^{4} + 217717211232388 T^{6} + 6905589367 p^{6} T^{8} + 124478 p^{12} T^{10} + p^{18} T^{12} \)
31 \( 1 + 51250 T^{2} + 1551151871 T^{4} + 34382118836764 T^{6} + 1551151871 p^{6} T^{8} + 51250 p^{12} T^{10} + p^{18} T^{12} \)
37 \( ( 1 - 190 T + 99155 T^{2} - 19441940 T^{3} + 99155 p^{3} T^{4} - 190 p^{6} T^{5} + p^{9} T^{6} )^{2} \)
41 \( ( 1 + 306 T + 158503 T^{2} + 26678652 T^{3} + 158503 p^{3} T^{4} + 306 p^{6} T^{5} + p^{9} T^{6} )^{2} \)
43 \( ( 1 + 164 T + 238921 T^{2} + 25708696 T^{3} + 238921 p^{3} T^{4} + 164 p^{6} T^{5} + p^{9} T^{6} )^{2} \)
47 \( ( 1 + 60 T + 110269 T^{2} - 5181240 T^{3} + 110269 p^{3} T^{4} + 60 p^{6} T^{5} + p^{9} T^{6} )^{2} \)
53 \( 1 + 558230 T^{2} + 156874030999 T^{4} + 28183915930956148 T^{6} + 156874030999 p^{6} T^{8} + 558230 p^{12} T^{10} + p^{18} T^{12} \)
59 \( ( 1 + 68 T + 155897 T^{2} - 40876456 T^{3} + 155897 p^{3} T^{4} + 68 p^{6} T^{5} + p^{9} T^{6} )^{2} \)
61 \( 1 + 807486 T^{2} + 291238309815 T^{4} + 72313154579372420 T^{6} + 291238309815 p^{6} T^{8} + 807486 p^{12} T^{10} + p^{18} T^{12} \)
67 \( ( 1 - 556 T + 262353 T^{2} + 43010104 T^{3} + 262353 p^{3} T^{4} - 556 p^{6} T^{5} + p^{9} T^{6} )^{2} \)
71 \( 1 + 1993522 T^{2} + 1707171677167 T^{4} + 802969695744376412 T^{6} + 1707171677167 p^{6} T^{8} + 1993522 p^{12} T^{10} + p^{18} T^{12} \)
73 \( 1 + 361278 T^{2} - 19156046897 T^{4} - 93754135461399356 T^{6} - 19156046897 p^{6} T^{8} + 361278 p^{12} T^{10} + p^{18} T^{12} \)
79 \( ( 1 - 700 T + 1616525 T^{2} - 696310024 T^{3} + 1616525 p^{3} T^{4} - 700 p^{6} T^{5} + p^{9} T^{6} )^{2} \)
83 \( ( 1 + 1456 T + 1885361 T^{2} + 1479178144 T^{3} + 1885361 p^{3} T^{4} + 1456 p^{6} T^{5} + p^{9} T^{6} )^{2} \)
89 \( ( 1 + 186 T + 1188007 T^{2} + 166563468 T^{3} + 1188007 p^{3} T^{4} + 186 p^{6} T^{5} + p^{9} T^{6} )^{2} \)
97 \( 1 + 3688174 T^{2} + 6610702894847 T^{4} + 7408695484892503204 T^{6} + 6610702894847 p^{6} T^{8} + 3688174 p^{12} T^{10} + p^{18} T^{12} \)
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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

−4.66240244380236316200173159825, −4.49498894284600565048400957899, −4.36056506430712006469038177343, −4.24565769716126065642021892730, −4.23550126107631854688984874986, −4.13765931975179277074077718760, −4.09907671594196140534705611449, −3.57674629764516467688974770466, −3.54779288168015703562472691276, −3.54739137220826520531178891858, −3.32542477465447516412281628841, −3.30294695709990322095155745643, −2.99886870079497091349977612725, −2.88449123207816840220264240892, −2.67015811348104225091525886006, −2.43297761510611664611666088334, −2.34196454301172494322290304976, −1.99556019850539292409153997101, −1.97366932482756549266064292955, −1.73245690451864931881621611029, −1.28717877003608815556058401694, −1.04016133135528069436359703467, −1.00033382417056716184867423296, −0.973373150579459016674029112394, −0.953970971177634335085694098835, 0, 0, 0, 0, 0, 0, 0.953970971177634335085694098835, 0.973373150579459016674029112394, 1.00033382417056716184867423296, 1.04016133135528069436359703467, 1.28717877003608815556058401694, 1.73245690451864931881621611029, 1.97366932482756549266064292955, 1.99556019850539292409153997101, 2.34196454301172494322290304976, 2.43297761510611664611666088334, 2.67015811348104225091525886006, 2.88449123207816840220264240892, 2.99886870079497091349977612725, 3.30294695709990322095155745643, 3.32542477465447516412281628841, 3.54739137220826520531178891858, 3.54779288168015703562472691276, 3.57674629764516467688974770466, 4.09907671594196140534705611449, 4.13765931975179277074077718760, 4.23550126107631854688984874986, 4.24565769716126065642021892730, 4.36056506430712006469038177343, 4.49498894284600565048400957899, 4.66240244380236316200173159825

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

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