↑

Properties

Label 12-7616e6-1.1-c1e6-0-6
Degree $12$
Conductor $1.951\times 10^{23}$
Sign $1$
Analytic cond. $5.05852\times 10^{10}$
Root an. cond. $7.79833$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $6$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 4·3-s + 4·5-s − 6·7-s + 3·9-s − 8·11-s + 4·13-s − 16·15-s + 6·17-s − 18·19-s + 24·21-s + 6·23-s − 25-s + 10·27-s − 4·29-s − 8·31-s + 32·33-s − 24·35-s + 8·37-s − 16·39-s − 4·41-s − 20·43-s + 12·45-s + 6·47-s + 21·49-s − 24·51-s − 14·53-s − 32·55-s + ⋯
L(s)  = 1  − 2.30·3-s + 1.78·5-s − 2.26·7-s + 9-s − 2.41·11-s + 1.10·13-s − 4.13·15-s + 1.45·17-s − 4.12·19-s + 5.23·21-s + 1.25·23-s − 1/5·25-s + 1.92·27-s − 0.742·29-s − 1.43·31-s + 5.57·33-s − 4.05·35-s + 1.31·37-s − 2.56·39-s − 0.624·41-s − 3.04·43-s + 1.78·45-s + 0.875·47-s + 3·49-s − 3.36·51-s − 1.92·53-s − 4.31·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 7^{6} \cdot 17^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 7^{6} \cdot 17^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{36} \cdot 7^{6} \cdot 17^{6}\)
Sign: $1$
Analytic conductor: \(5.05852\times 10^{10}\)
Root analytic conductor: \(7.79833\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(6\)
Selberg data: \((12,\ 2^{36} \cdot 7^{6} \cdot 17^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

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 \( 1 \)
7 \( ( 1 + T )^{6} \)
17 \( ( 1 - T )^{6} \)
good3 \( 1 + 4 T + 13 T^{2} + 10 p T^{3} + 58 T^{4} + 32 p T^{5} + 172 T^{6} + 32 p^{2} T^{7} + 58 p^{2} T^{8} + 10 p^{4} T^{9} + 13 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
5 \( 1 - 4 T + 17 T^{2} - 38 T^{3} + 108 T^{4} - 36 p T^{5} + 4 p^{3} T^{6} - 36 p^{2} T^{7} + 108 p^{2} T^{8} - 38 p^{3} T^{9} + 17 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 + 8 T + 52 T^{2} + 20 p T^{3} + 81 p T^{4} + 3084 T^{5} + 10848 T^{6} + 3084 p T^{7} + 81 p^{3} T^{8} + 20 p^{4} T^{9} + 52 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 - 4 T + 56 T^{2} - 200 T^{3} + 1507 T^{4} - 4660 T^{5} + 24584 T^{6} - 4660 p T^{7} + 1507 p^{2} T^{8} - 200 p^{3} T^{9} + 56 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 + 18 T + 202 T^{2} + 1618 T^{3} + 10435 T^{4} + 56040 T^{5} + 261988 T^{6} + 56040 p T^{7} + 10435 p^{2} T^{8} + 1618 p^{3} T^{9} + 202 p^{4} T^{10} + 18 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 - 6 T + 58 T^{2} - 270 T^{3} + 2451 T^{4} - 8744 T^{5} + 57028 T^{6} - 8744 p T^{7} + 2451 p^{2} T^{8} - 270 p^{3} T^{9} + 58 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 + 4 T + 124 T^{2} + 528 T^{3} + 7603 T^{4} + 27764 T^{5} + 281776 T^{6} + 27764 p T^{7} + 7603 p^{2} T^{8} + 528 p^{3} T^{9} + 124 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 + 8 T + 135 T^{2} + 1006 T^{3} + 8330 T^{4} + 55450 T^{5} + 316652 T^{6} + 55450 p T^{7} + 8330 p^{2} T^{8} + 1006 p^{3} T^{9} + 135 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 - 8 T + 118 T^{2} - 476 T^{3} + 4883 T^{4} - 10836 T^{5} + 160764 T^{6} - 10836 p T^{7} + 4883 p^{2} T^{8} - 476 p^{3} T^{9} + 118 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 + 4 T + 149 T^{2} + 286 T^{3} + 9140 T^{4} + 1718 T^{5} + 386364 T^{6} + 1718 p T^{7} + 9140 p^{2} T^{8} + 286 p^{3} T^{9} + 149 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 + 20 T + 397 T^{2} + 4688 T^{3} + 51994 T^{4} + 419784 T^{5} + 3162376 T^{6} + 419784 p T^{7} + 51994 p^{2} T^{8} + 4688 p^{3} T^{9} + 397 p^{4} T^{10} + 20 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 - 6 T + 176 T^{2} - 866 T^{3} + 14819 T^{4} - 63532 T^{5} + 827864 T^{6} - 63532 p T^{7} + 14819 p^{2} T^{8} - 866 p^{3} T^{9} + 176 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 + 14 T + 261 T^{2} + 2956 T^{3} + 32512 T^{4} + 277990 T^{5} + 2263724 T^{6} + 277990 p T^{7} + 32512 p^{2} T^{8} + 2956 p^{3} T^{9} + 261 p^{4} T^{10} + 14 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 + 20 T + 366 T^{2} + 4204 T^{3} + 47463 T^{4} + 408168 T^{5} + 3531140 T^{6} + 408168 p T^{7} + 47463 p^{2} T^{8} + 4204 p^{3} T^{9} + 366 p^{4} T^{10} + 20 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 + 2 T + 269 T^{2} + 562 T^{3} + 34032 T^{4} + 63610 T^{5} + 2601044 T^{6} + 63610 p T^{7} + 34032 p^{2} T^{8} + 562 p^{3} T^{9} + 269 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 + 28 T + 647 T^{2} + 10040 T^{3} + 132218 T^{4} + 1384784 T^{5} + 12507300 T^{6} + 1384784 p T^{7} + 132218 p^{2} T^{8} + 10040 p^{3} T^{9} + 647 p^{4} T^{10} + 28 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 - 18 T + 342 T^{2} - 4106 T^{3} + 50939 T^{4} - 484328 T^{5} + 4569916 T^{6} - 484328 p T^{7} + 50939 p^{2} T^{8} - 4106 p^{3} T^{9} + 342 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 - 4 T + 245 T^{2} - 458 T^{3} + 26548 T^{4} + 7374 T^{5} + 2042124 T^{6} + 7374 p T^{7} + 26548 p^{2} T^{8} - 458 p^{3} T^{9} + 245 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 + 2 T + 312 T^{2} + 646 T^{3} + 46403 T^{4} + 83092 T^{5} + 4426632 T^{6} + 83092 p T^{7} + 46403 p^{2} T^{8} + 646 p^{3} T^{9} + 312 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 + 20 T + 416 T^{2} + 4760 T^{3} + 61547 T^{4} + 531412 T^{5} + 5714376 T^{6} + 531412 p T^{7} + 61547 p^{2} T^{8} + 4760 p^{3} T^{9} + 416 p^{4} T^{10} + 20 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 + 6 T + 452 T^{2} + 1886 T^{3} + 87475 T^{4} + 263996 T^{5} + 9819472 T^{6} + 263996 p T^{7} + 87475 p^{2} T^{8} + 1886 p^{3} T^{9} + 452 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 - 30 T + 669 T^{2} - 9782 T^{3} + 130940 T^{4} - 1420356 T^{5} + 15225068 T^{6} - 1420356 p T^{7} + 130940 p^{2} T^{8} - 9782 p^{3} T^{9} + 669 p^{4} T^{10} - 30 p^{5} T^{11} + p^{6} T^{12} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−4.36003957850684604320387796901, −4.33882415184372465088764412797, −4.11776338073238077240993276533, −4.04081028011911900899804640386, −3.85008685479583154456366448547, −3.77400102757382766672779215646, −3.61300884634249185435004296941, −3.31131709141352298586343146777, −3.24224800651309192710236481864, −3.12411936873718029694890683947, −3.12065196594263637404570009235, −2.96192669349876259734167893998, −2.91496724587242005961427728592, −2.45951179862655491352689991950, −2.31491246289632490673065205884, −2.30093048811850210155586237302, −2.28852458658223760108746048702, −2.23686335763951118861384284553, −1.80362041593907853443372415476, −1.69325025738536999329434632300, −1.37069486131719391480357467943, −1.36711150677578575764008292589, −1.33862233878362576704371610683, −0.859530195714047423430091273106, −0.796694187450468642999656181153, 0, 0, 0, 0, 0, 0, 0.796694187450468642999656181153, 0.859530195714047423430091273106, 1.33862233878362576704371610683, 1.36711150677578575764008292589, 1.37069486131719391480357467943, 1.69325025738536999329434632300, 1.80362041593907853443372415476, 2.23686335763951118861384284553, 2.28852458658223760108746048702, 2.30093048811850210155586237302, 2.31491246289632490673065205884, 2.45951179862655491352689991950, 2.91496724587242005961427728592, 2.96192669349876259734167893998, 3.12065196594263637404570009235, 3.12411936873718029694890683947, 3.24224800651309192710236481864, 3.31131709141352298586343146777, 3.61300884634249185435004296941, 3.77400102757382766672779215646, 3.85008685479583154456366448547, 4.04081028011911900899804640386, 4.11776338073238077240993276533, 4.33882415184372465088764412797, 4.36003957850684604320387796901

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

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