↑

Properties

Label 12-2520e6-1.1-c1e6-0-6
Degree $12$
Conductor $2.561\times 10^{20}$
Sign $1$
Analytic cond. $6.63843\times 10^{7}$
Root an. cond. $4.48578$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3·5-s − 7-s + 2·11-s − 2·13-s − 2·17-s + 3·19-s + 2·23-s + 3·25-s + 16·29-s − 5·31-s + 3·35-s + 9·37-s − 28·41-s − 18·43-s + 6·47-s − 49-s − 6·55-s + 4·59-s − 10·61-s + 6·65-s + 13·67-s − 16·71-s − 9·73-s − 2·77-s + 17·79-s − 12·83-s + 6·85-s + ⋯
L(s)  = 1  − 1.34·5-s − 0.377·7-s + 0.603·11-s − 0.554·13-s − 0.485·17-s + 0.688·19-s + 0.417·23-s + 3/5·25-s + 2.97·29-s − 0.898·31-s + 0.507·35-s + 1.47·37-s − 4.37·41-s − 2.74·43-s + 0.875·47-s − 1/7·49-s − 0.809·55-s + 0.520·59-s − 1.28·61-s + 0.744·65-s + 1.58·67-s − 1.89·71-s − 1.05·73-s − 0.227·77-s + 1.91·79-s − 1.31·83-s + 0.650·85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{12} \cdot 5^{6} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{12} \cdot 5^{6} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{18} \cdot 3^{12} \cdot 5^{6} \cdot 7^{6}\)
Sign: $1$
Analytic conductor: \(6.63843\times 10^{7}\)
Root analytic conductor: \(4.48578\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{18} \cdot 3^{12} \cdot 5^{6} \cdot 7^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.3105518556\)
\(L(\frac12)\) \(\approx\) \(0.3105518556\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( ( 1 + T + T^{2} )^{3} \)
7 \( 1 + T + 2 T^{2} - 23 T^{3} + 2 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \)
good11 \( 1 - 2 T - 23 T^{2} + 38 T^{3} + 340 T^{4} - 296 T^{5} - 3725 T^{6} - 296 p T^{7} + 340 p^{2} T^{8} + 38 p^{3} T^{9} - 23 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
13 \( ( 1 + T + 20 T^{2} - 11 T^{3} + 20 p T^{4} + p^{2} T^{5} + p^{3} T^{6} )^{2} \)
17 \( 1 + 2 T - 43 T^{2} - 30 T^{3} + 1286 T^{4} + 296 T^{5} - 25039 T^{6} + 296 p T^{7} + 1286 p^{2} T^{8} - 30 p^{3} T^{9} - 43 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 - 3 T - 43 T^{2} + 66 T^{3} + 1383 T^{4} - 927 T^{5} - 28586 T^{6} - 927 p T^{7} + 1383 p^{2} T^{8} + 66 p^{3} T^{9} - 43 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 - 2 T - 61 T^{2} + 42 T^{3} + 2558 T^{4} - 668 T^{5} - 67561 T^{6} - 668 p T^{7} + 2558 p^{2} T^{8} + 42 p^{3} T^{9} - 61 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
29 \( ( 1 - 8 T + 101 T^{2} - 470 T^{3} + 101 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
31 \( 1 + 5 T - 55 T^{2} - 202 T^{3} + 2415 T^{4} + 4085 T^{5} - 72242 T^{6} + 4085 p T^{7} + 2415 p^{2} T^{8} - 202 p^{3} T^{9} - 55 p^{4} T^{10} + 5 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 - 9 T - 13 T^{2} + 264 T^{3} + 363 T^{4} - 51 p T^{5} - 770 p T^{6} - 51 p^{2} T^{7} + 363 p^{2} T^{8} + 264 p^{3} T^{9} - 13 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
41 \( ( 1 + 14 T + 181 T^{2} + 1222 T^{3} + 181 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
43 \( ( 1 + 9 T + 112 T^{2} + 675 T^{3} + 112 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
47 \( 1 - 6 T - 85 T^{2} + 354 T^{3} + 5714 T^{4} - 10350 T^{5} - 268405 T^{6} - 10350 p T^{7} + 5714 p^{2} T^{8} + 354 p^{3} T^{9} - 85 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 + 17 T^{2} - 96 T^{3} + 1190 T^{4} - 816 T^{5} + 284741 T^{6} - 816 p T^{7} + 1190 p^{2} T^{8} - 96 p^{3} T^{9} + 17 p^{4} T^{10} + p^{6} T^{12} \)
59 \( 1 - 4 T - 55 T^{2} - 512 T^{3} + 1308 T^{4} + 27682 T^{5} + 70639 T^{6} + 27682 p T^{7} + 1308 p^{2} T^{8} - 512 p^{3} T^{9} - 55 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 + 10 T - 59 T^{2} - 858 T^{3} + 3342 T^{4} + 37846 T^{5} + 37897 T^{6} + 37846 p T^{7} + 3342 p^{2} T^{8} - 858 p^{3} T^{9} - 59 p^{4} T^{10} + 10 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 - 13 T + 27 T^{2} + 620 T^{3} - 5379 T^{4} + 13921 T^{5} + 12186 T^{6} + 13921 p T^{7} - 5379 p^{2} T^{8} + 620 p^{3} T^{9} + 27 p^{4} T^{10} - 13 p^{5} T^{11} + p^{6} T^{12} \)
71 \( ( 1 + 8 T + 123 T^{2} + 1298 T^{3} + 123 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
73 \( 1 + 9 T - 49 T^{2} - 100 T^{3} + 1695 T^{4} - 40309 T^{5} - 484090 T^{6} - 40309 p T^{7} + 1695 p^{2} T^{8} - 100 p^{3} T^{9} - 49 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 - 17 T - 27 T^{2} + 198 T^{3} + 27547 T^{4} - 1739 p T^{5} - 949970 T^{6} - 1739 p^{2} T^{7} + 27547 p^{2} T^{8} + 198 p^{3} T^{9} - 27 p^{4} T^{10} - 17 p^{5} T^{11} + p^{6} T^{12} \)
83 \( ( 1 + 6 T + 213 T^{2} + 834 T^{3} + 213 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
89 \( 1 - 6 T - 169 T^{2} + 494 T^{3} + 18608 T^{4} - 5296 T^{5} - 1872399 T^{6} - 5296 p T^{7} + 18608 p^{2} T^{8} + 494 p^{3} T^{9} - 169 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
97 \( ( 1 + 8 T + 147 T^{2} + 1948 T^{3} + 147 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
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

−4.74092180175927647999247156253, −4.42481570027803861045708597930, −4.41537261181767672157363362899, −4.14599378368167986762858251291, −4.08901913136771167875429692695, −3.73487801052286142756429582382, −3.61805149493393404900687333382, −3.60000498371199962147983777630, −3.44790075170538555099699826705, −3.36440333598966397007213348540, −3.13674021937204485940851033389, −2.88235208928925118447373458220, −2.85975698824950682706594033732, −2.65873256871238723425753286810, −2.51212330871836388270639344437, −2.21232272078083636273967047572, −2.05534991660144585210842022381, −1.85934834341585773483932510348, −1.65343078019560456853600589381, −1.41990732902142818312419267488, −1.10359131543364394765915405232, −1.01907267330319941438785823670, −0.935517284310261240536855508449, −0.21829433145189585381084098099, −0.14515968393869151574594149802, 0.14515968393869151574594149802, 0.21829433145189585381084098099, 0.935517284310261240536855508449, 1.01907267330319941438785823670, 1.10359131543364394765915405232, 1.41990732902142818312419267488, 1.65343078019560456853600589381, 1.85934834341585773483932510348, 2.05534991660144585210842022381, 2.21232272078083636273967047572, 2.51212330871836388270639344437, 2.65873256871238723425753286810, 2.85975698824950682706594033732, 2.88235208928925118447373458220, 3.13674021937204485940851033389, 3.36440333598966397007213348540, 3.44790075170538555099699826705, 3.60000498371199962147983777630, 3.61805149493393404900687333382, 3.73487801052286142756429582382, 4.08901913136771167875429692695, 4.14599378368167986762858251291, 4.41537261181767672157363362899, 4.42481570027803861045708597930, 4.74092180175927647999247156253

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

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