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Properties

Label 12-5054e6-1.1-c1e6-0-3
Degree $12$
Conductor $1.667\times 10^{22}$
Sign $1$
Analytic cond. $4.31990\times 10^{9}$
Root an. cond. $6.35266$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $6$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 6·2-s + 21·4-s − 5-s + 6·7-s − 56·8-s − 5·9-s + 6·10-s + 4·11-s − 15·13-s − 36·14-s + 126·16-s − 9·17-s + 30·18-s − 21·20-s − 24·22-s + 4·23-s − 14·25-s + 90·26-s + 4·27-s + 126·28-s − 7·29-s − 4·31-s − 252·32-s + 54·34-s − 6·35-s − 105·36-s − 3·37-s + ⋯
L(s)  = 1  − 4.24·2-s + 21/2·4-s − 0.447·5-s + 2.26·7-s − 19.7·8-s − 5/3·9-s + 1.89·10-s + 1.20·11-s − 4.16·13-s − 9.62·14-s + 63/2·16-s − 2.18·17-s + 7.07·18-s − 4.69·20-s − 5.11·22-s + 0.834·23-s − 2.79·25-s + 17.6·26-s + 0.769·27-s + 23.8·28-s − 1.29·29-s − 0.718·31-s − 44.5·32-s + 9.26·34-s − 1.01·35-s − 17.5·36-s − 0.493·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 7^{6} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 7^{6} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{6} \cdot 7^{6} \cdot 19^{12}\)
Sign: $1$
Analytic conductor: \(4.31990\times 10^{9}\)
Root analytic conductor: \(6.35266\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(6\)
Selberg data: \((12,\ 2^{6} \cdot 7^{6} \cdot 19^{12} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 + T )^{6} \)
7 \( ( 1 - T )^{6} \)
19 \( 1 \)
good3 \( 1 + 5 T^{2} - 4 T^{3} + 20 T^{4} - 20 T^{5} + 68 T^{6} - 20 p T^{7} + 20 p^{2} T^{8} - 4 p^{3} T^{9} + 5 p^{4} T^{10} + p^{6} T^{12} \)
5 \( 1 + T + 3 p T^{2} + 19 T^{3} + 142 T^{4} + 153 T^{5} + 859 T^{6} + 153 p T^{7} + 142 p^{2} T^{8} + 19 p^{3} T^{9} + 3 p^{5} T^{10} + p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 - 4 T + 36 T^{2} - 124 T^{3} + 708 T^{4} - 2028 T^{5} + 9542 T^{6} - 2028 p T^{7} + 708 p^{2} T^{8} - 124 p^{3} T^{9} + 36 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 + 15 T + 148 T^{2} + 1048 T^{3} + 5925 T^{4} + 27672 T^{5} + 108311 T^{6} + 27672 p T^{7} + 5925 p^{2} T^{8} + 1048 p^{3} T^{9} + 148 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 + 9 T + 76 T^{2} + 400 T^{3} + 141 p T^{4} + 10920 T^{5} + 52371 T^{6} + 10920 p T^{7} + 141 p^{3} T^{8} + 400 p^{3} T^{9} + 76 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 - 4 T + 112 T^{2} - 404 T^{3} + 5696 T^{4} - 17316 T^{5} + 167838 T^{6} - 17316 p T^{7} + 5696 p^{2} T^{8} - 404 p^{3} T^{9} + 112 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 + 7 T + 107 T^{2} + 597 T^{3} + 6294 T^{4} + 28379 T^{5} + 221231 T^{6} + 28379 p T^{7} + 6294 p^{2} T^{8} + 597 p^{3} T^{9} + 107 p^{4} T^{10} + 7 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 + 4 T + 96 T^{2} + 444 T^{3} + 5788 T^{4} + 22308 T^{5} + 7082 p T^{6} + 22308 p T^{7} + 5788 p^{2} T^{8} + 444 p^{3} T^{9} + 96 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 + 3 T + 171 T^{2} + 453 T^{3} + 13142 T^{4} + 29767 T^{5} + 605575 T^{6} + 29767 p T^{7} + 13142 p^{2} T^{8} + 453 p^{3} T^{9} + 171 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 + 11 T + 188 T^{2} + 1472 T^{3} + 16213 T^{4} + 99408 T^{5} + 823803 T^{6} + 99408 p T^{7} + 16213 p^{2} T^{8} + 1472 p^{3} T^{9} + 188 p^{4} T^{10} + 11 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 + 10 T + 185 T^{2} + 1324 T^{3} + 15020 T^{4} + 86250 T^{5} + 768828 T^{6} + 86250 p T^{7} + 15020 p^{2} T^{8} + 1324 p^{3} T^{9} + 185 p^{4} T^{10} + 10 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 - 12 T + 192 T^{2} - 852 T^{3} + 6572 T^{4} + 24772 T^{5} - 11786 T^{6} + 24772 p T^{7} + 6572 p^{2} T^{8} - 852 p^{3} T^{9} + 192 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 - 5 T + 219 T^{2} - 1203 T^{3} + 24286 T^{4} - 113921 T^{5} + 1643839 T^{6} - 113921 p T^{7} + 24286 p^{2} T^{8} - 1203 p^{3} T^{9} + 219 p^{4} T^{10} - 5 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 + 4 T + 176 T^{2} + 956 T^{3} + 19308 T^{4} + 88132 T^{5} + 1404670 T^{6} + 88132 p T^{7} + 19308 p^{2} T^{8} + 956 p^{3} T^{9} + 176 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 + 21 T + 179 T^{2} + 599 T^{3} + 4942 T^{4} + 117501 T^{5} + 22139 p T^{6} + 117501 p T^{7} + 4942 p^{2} T^{8} + 599 p^{3} T^{9} + 179 p^{4} T^{10} + 21 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 + 14 T + 229 T^{2} + 2448 T^{3} + 32716 T^{4} + 268646 T^{5} + 2539284 T^{6} + 268646 p T^{7} + 32716 p^{2} T^{8} + 2448 p^{3} T^{9} + 229 p^{4} T^{10} + 14 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 + 24 T + 614 T^{2} + 8936 T^{3} + 128127 T^{4} + 1295504 T^{5} + 12719764 T^{6} + 1295504 p T^{7} + 128127 p^{2} T^{8} + 8936 p^{3} T^{9} + 614 p^{4} T^{10} + 24 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 + 21 T + 472 T^{2} + 5656 T^{3} + 72441 T^{4} + 625024 T^{5} + 6266543 T^{6} + 625024 p T^{7} + 72441 p^{2} T^{8} + 5656 p^{3} T^{9} + 472 p^{4} T^{10} + 21 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 + 58 T + 1777 T^{2} + 36768 T^{3} + 569544 T^{4} + 6942346 T^{5} + 68385596 T^{6} + 6942346 p T^{7} + 569544 p^{2} T^{8} + 36768 p^{3} T^{9} + 1777 p^{4} T^{10} + 58 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 - 20 T + 310 T^{2} - 1836 T^{3} + 5415 T^{4} + 163640 T^{5} - 1578732 T^{6} + 163640 p T^{7} + 5415 p^{2} T^{8} - 1836 p^{3} T^{9} + 310 p^{4} T^{10} - 20 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 + 7 T + 132 T^{2} + 2332 T^{3} + 17469 T^{4} + 192764 T^{5} + 2578631 T^{6} + 192764 p T^{7} + 17469 p^{2} T^{8} + 2332 p^{3} T^{9} + 132 p^{4} T^{10} + 7 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 + 7 T + 79 T^{2} - 2035 T^{3} - 446 T^{4} + 9927 T^{5} + 3506719 T^{6} + 9927 p T^{7} - 446 p^{2} T^{8} - 2035 p^{3} T^{9} + 79 p^{4} T^{10} + 7 p^{5} T^{11} + p^{6} T^{12} \)
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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

−4.64608912162753996438333453330, −4.46148748519345126704262201496, −4.28436804831562626579064699806, −4.23170034792204102252373250946, −4.00226524568640285686444744099, −3.99858702730375594914655330859, −3.99065861828001880082134384176, −3.32285321306006299571845873218, −3.23658438021552824838533835799, −3.16934309209095985262356226341, −3.14678830049853493771963219010, −3.00685092914363888936097714454, −2.75358582630456436709925923939, −2.43291788811713198651229709014, −2.42442817386586942248987765254, −2.35285095124620922989002308146, −2.33613291328013782488075808886, −1.99080183900560126931728672124, −1.78961552068974456284501372531, −1.71618700748961108962819219767, −1.69503179221821627293882996823, −1.42697099610727802334058262190, −1.17965341171357015550276668293, −1.16653713555866172778529786745, −0.905865542393033865215375786262, 0, 0, 0, 0, 0, 0, 0.905865542393033865215375786262, 1.16653713555866172778529786745, 1.17965341171357015550276668293, 1.42697099610727802334058262190, 1.69503179221821627293882996823, 1.71618700748961108962819219767, 1.78961552068974456284501372531, 1.99080183900560126931728672124, 2.33613291328013782488075808886, 2.35285095124620922989002308146, 2.42442817386586942248987765254, 2.43291788811713198651229709014, 2.75358582630456436709925923939, 3.00685092914363888936097714454, 3.14678830049853493771963219010, 3.16934309209095985262356226341, 3.23658438021552824838533835799, 3.32285321306006299571845873218, 3.99065861828001880082134384176, 3.99858702730375594914655330859, 4.00226524568640285686444744099, 4.23170034792204102252373250946, 4.28436804831562626579064699806, 4.46148748519345126704262201496, 4.64608912162753996438333453330

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

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