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Properties

Label 12-9800e6-1.1-c1e6-0-0
Degree $12$
Conductor $8.858\times 10^{23}$
Sign $1$
Analytic cond. $2.29624\times 10^{11}$
Root an. cond. $8.84609$
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-s − 5·9-s + 11-s + 7·13-s + 4·17-s − 5·19-s − 6·23-s − 4·27-s − 3·29-s − 2·31-s + 33-s − 9·37-s + 7·39-s + 6·41-s + 3·43-s + 27·47-s + 4·51-s + 5·53-s − 5·57-s − 24·59-s + 9·61-s − 17·67-s − 6·69-s + 4·71-s + 18·73-s − 22·79-s + 7·81-s + ⋯
L(s)  = 1  + 0.577·3-s − 5/3·9-s + 0.301·11-s + 1.94·13-s + 0.970·17-s − 1.14·19-s − 1.25·23-s − 0.769·27-s − 0.557·29-s − 0.359·31-s + 0.174·33-s − 1.47·37-s + 1.12·39-s + 0.937·41-s + 0.457·43-s + 3.93·47-s + 0.560·51-s + 0.686·53-s − 0.662·57-s − 3.12·59-s + 1.15·61-s − 2.07·67-s − 0.722·69-s + 0.474·71-s + 2.10·73-s − 2.47·79-s + 7/9·81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 5^{12} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 5^{12} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{18} \cdot 5^{12} \cdot 7^{12}\)
Sign: $1$
Analytic conductor: \(2.29624\times 10^{11}\)
Root analytic conductor: \(8.84609\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{18} \cdot 5^{12} \cdot 7^{12} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(10.94831994\)
\(L(\frac12)\) \(\approx\) \(10.94831994\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good3 \( 1 - T + 2 p T^{2} - 7 T^{3} + 26 T^{4} - 11 p T^{5} + 94 T^{6} - 11 p^{2} T^{7} + 26 p^{2} T^{8} - 7 p^{3} T^{9} + 2 p^{5} T^{10} - p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 - T + 29 T^{2} + 14 T^{3} + 483 T^{4} + 235 T^{5} + 6782 T^{6} + 235 p T^{7} + 483 p^{2} T^{8} + 14 p^{3} T^{9} + 29 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 - 7 T + 68 T^{2} - 355 T^{3} + 2007 T^{4} - 8250 T^{5} + 33848 T^{6} - 8250 p T^{7} + 2007 p^{2} T^{8} - 355 p^{3} T^{9} + 68 p^{4} T^{10} - 7 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 - 4 T + 27 T^{2} - 96 T^{3} + 891 T^{4} - 2668 T^{5} + 14962 T^{6} - 2668 p T^{7} + 891 p^{2} T^{8} - 96 p^{3} T^{9} + 27 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 + 5 T + 67 T^{2} + 174 T^{3} + 1749 T^{4} + 2193 T^{5} + 32374 T^{6} + 2193 p T^{7} + 1749 p^{2} T^{8} + 174 p^{3} T^{9} + 67 p^{4} T^{10} + 5 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 + 6 T + 119 T^{2} + 582 T^{3} + 6310 T^{4} + 24766 T^{5} + 188355 T^{6} + 24766 p T^{7} + 6310 p^{2} T^{8} + 582 p^{3} T^{9} + 119 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 + 3 T + 69 T^{2} + 400 T^{3} + 3087 T^{4} + 19485 T^{5} + 104422 T^{6} + 19485 p T^{7} + 3087 p^{2} T^{8} + 400 p^{3} T^{9} + 69 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 + 2 T + 71 T^{2} + 6 p T^{3} + 2171 T^{4} + 5960 T^{5} + 55754 T^{6} + 5960 p T^{7} + 2171 p^{2} T^{8} + 6 p^{4} T^{9} + 71 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 + 9 T + 147 T^{2} + 1118 T^{3} + 10827 T^{4} + 70557 T^{5} + 496562 T^{6} + 70557 p T^{7} + 10827 p^{2} T^{8} + 1118 p^{3} T^{9} + 147 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 - 6 T + 146 T^{2} - 964 T^{3} + 11686 T^{4} - 62770 T^{5} + 608006 T^{6} - 62770 p T^{7} + 11686 p^{2} T^{8} - 964 p^{3} T^{9} + 146 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 - 3 T + 107 T^{2} - 442 T^{3} + 8753 T^{4} - 31575 T^{5} + 420350 T^{6} - 31575 p T^{7} + 8753 p^{2} T^{8} - 442 p^{3} T^{9} + 107 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 - 27 T + 555 T^{2} - 7616 T^{3} + 87063 T^{4} - 776961 T^{5} + 5938186 T^{6} - 776961 p T^{7} + 87063 p^{2} T^{8} - 7616 p^{3} T^{9} + 555 p^{4} T^{10} - 27 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 - 5 T + 241 T^{2} - 1058 T^{3} + 26841 T^{4} - 99265 T^{5} + 1789306 T^{6} - 99265 p T^{7} + 26841 p^{2} T^{8} - 1058 p^{3} T^{9} + 241 p^{4} T^{10} - 5 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 + 24 T + 447 T^{2} + 5038 T^{3} + 50343 T^{4} + 380622 T^{5} + 3114514 T^{6} + 380622 p T^{7} + 50343 p^{2} T^{8} + 5038 p^{3} T^{9} + 447 p^{4} T^{10} + 24 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 - 9 T + 194 T^{2} - 1195 T^{3} + 19136 T^{4} - 87421 T^{5} + 1239000 T^{6} - 87421 p T^{7} + 19136 p^{2} T^{8} - 1195 p^{3} T^{9} + 194 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 + 17 T + 440 T^{2} + 5299 T^{3} + 76318 T^{4} + 685417 T^{5} + 6881660 T^{6} + 685417 p T^{7} + 76318 p^{2} T^{8} + 5299 p^{3} T^{9} + 440 p^{4} T^{10} + 17 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 - 4 T + 154 T^{2} - 140 T^{3} + 10351 T^{4} + 37464 T^{5} + 449740 T^{6} + 37464 p T^{7} + 10351 p^{2} T^{8} - 140 p^{3} T^{9} + 154 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 - 18 T + 359 T^{2} - 2572 T^{3} + 21527 T^{4} + 46910 T^{5} - 84318 T^{6} + 46910 p T^{7} + 21527 p^{2} T^{8} - 2572 p^{3} T^{9} + 359 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 + 22 T + 535 T^{2} + 7818 T^{3} + 111171 T^{4} + 1168164 T^{5} + 11872954 T^{6} + 1168164 p T^{7} + 111171 p^{2} T^{8} + 7818 p^{3} T^{9} + 535 p^{4} T^{10} + 22 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 - 9 T + 395 T^{2} - 2346 T^{3} + 64909 T^{4} - 272593 T^{5} + 6476022 T^{6} - 272593 p T^{7} + 64909 p^{2} T^{8} - 2346 p^{3} T^{9} + 395 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 - 15 T + 266 T^{2} - 2177 T^{3} + 12040 T^{4} - 30051 T^{5} - 545360 T^{6} - 30051 p T^{7} + 12040 p^{2} T^{8} - 2177 p^{3} T^{9} + 266 p^{4} T^{10} - 15 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 - 12 T + 374 T^{2} - 4220 T^{3} + 74255 T^{4} - 677816 T^{5} + 9184692 T^{6} - 677816 p T^{7} + 74255 p^{2} T^{8} - 4220 p^{3} T^{9} + 374 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
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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

−3.82672412853379936380622838352, −3.65311313843751847830063211754, −3.58843118780486980598245209940, −3.52764249510956316197444901384, −3.39306543923914837306330635943, −3.29035274361866566801281533240, −3.22141153301856326708082343301, −2.79989688454218267124986271297, −2.77247637055391242724221043165, −2.60228835615253553536827342674, −2.57183620645572856956383250896, −2.54027741889267015151983403070, −2.40748642071179087829087950150, −2.08349928801170605978572734147, −1.86180008463591049493442720155, −1.74981321061645388456096564715, −1.63828607174108005733678228687, −1.61299758324732156866673862805, −1.47199401770242192238868269450, −1.06789274073402295565954056364, −0.949168978223663473875550432631, −0.67359813377073765500979997014, −0.57860111157560968954837595020, −0.43793864572741948138550324650, −0.24373056081250074976232457507, 0.24373056081250074976232457507, 0.43793864572741948138550324650, 0.57860111157560968954837595020, 0.67359813377073765500979997014, 0.949168978223663473875550432631, 1.06789274073402295565954056364, 1.47199401770242192238868269450, 1.61299758324732156866673862805, 1.63828607174108005733678228687, 1.74981321061645388456096564715, 1.86180008463591049493442720155, 2.08349928801170605978572734147, 2.40748642071179087829087950150, 2.54027741889267015151983403070, 2.57183620645572856956383250896, 2.60228835615253553536827342674, 2.77247637055391242724221043165, 2.79989688454218267124986271297, 3.22141153301856326708082343301, 3.29035274361866566801281533240, 3.39306543923914837306330635943, 3.52764249510956316197444901384, 3.58843118780486980598245209940, 3.65311313843751847830063211754, 3.82672412853379936380622838352

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

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