Properties

Degree $12$
Conductor $9.209\times 10^{16}$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s + 3·7-s + 3·9-s − 6·13-s − 6·17-s + 3·19-s + 9·21-s − 6·23-s + 6·25-s − 2·27-s + 24·29-s + 3·31-s − 3·37-s − 18·39-s − 12·41-s − 30·43-s + 12·47-s + 9·49-s − 18·51-s − 6·53-s + 9·57-s + 12·59-s + 18·61-s + 9·63-s + 9·67-s − 18·69-s − 33·73-s + ⋯
L(s)  = 1  + 1.73·3-s + 1.13·7-s + 9-s − 1.66·13-s − 1.45·17-s + 0.688·19-s + 1.96·21-s − 1.25·23-s + 6/5·25-s − 0.384·27-s + 4.45·29-s + 0.538·31-s − 0.493·37-s − 2.88·39-s − 1.87·41-s − 4.57·43-s + 1.75·47-s + 9/7·49-s − 2.52·51-s − 0.824·53-s + 1.19·57-s + 1.56·59-s + 2.30·61-s + 1.13·63-s + 1.09·67-s − 2.16·69-s − 3.86·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 3^{6} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 3^{6} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{30} \cdot 3^{6} \cdot 7^{6}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{672} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{30} \cdot 3^{6} \cdot 7^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(3.18153\)
\(L(\frac12)\) \(\approx\) \(3.18153\)
\(L(\frac{3}{2})\) 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 \( ( 1 - T + T^{2} )^{3} \)
7 \( 1 - 3 T + 5 T^{3} - 3 p^{2} T^{5} + p^{3} T^{6} \)
good5 \( 1 - 6 T^{2} + 8 T^{3} + 6 T^{4} - 24 T^{5} + 86 T^{6} - 24 p T^{7} + 6 p^{2} T^{8} + 8 p^{3} T^{9} - 6 p^{4} T^{10} + p^{6} T^{12} \)
11 \( 1 - 6 T^{2} - 76 T^{3} - 30 T^{4} + 228 T^{5} + 3710 T^{6} + 228 p T^{7} - 30 p^{2} T^{8} - 76 p^{3} T^{9} - 6 p^{4} T^{10} + p^{6} T^{12} \)
13 \( ( 1 + 3 T + 3 T^{2} - 34 T^{3} + 3 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
17 \( 1 + 6 T + 9 T^{2} - 54 T^{3} - 378 T^{4} - 858 T^{5} - 1307 T^{6} - 858 p T^{7} - 378 p^{2} T^{8} - 54 p^{3} T^{9} + 9 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 - 3 T - 12 T^{2} - 59 T^{3} + 36 T^{4} + 1269 T^{5} + 2094 T^{6} + 1269 p T^{7} + 36 p^{2} T^{8} - 59 p^{3} T^{9} - 12 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 + 6 T - 9 T^{2} - 90 T^{3} - 90 T^{4} - 1146 T^{5} - 8885 T^{6} - 1146 p T^{7} - 90 p^{2} T^{8} - 90 p^{3} T^{9} - 9 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
29 \( ( 1 - 12 T + 126 T^{2} - 728 T^{3} + 126 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
31 \( 1 - 3 T - 63 T^{2} + 2 p T^{3} + 2535 T^{4} + 501 T^{5} - 90450 T^{6} + 501 p T^{7} + 2535 p^{2} T^{8} + 2 p^{4} T^{9} - 63 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 + 3 T - 18 T^{2} + 373 T^{3} + 168 T^{4} - 5829 T^{5} + 83328 T^{6} - 5829 p T^{7} + 168 p^{2} T^{8} + 373 p^{3} T^{9} - 18 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
41 \( ( 1 + 6 T + 27 T^{2} - 20 T^{3} + 27 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
43 \( ( 1 + 15 T + 177 T^{2} + 1318 T^{3} + 177 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
47 \( 1 - 12 T - 9 T^{2} + 196 T^{3} + 4026 T^{4} - 9372 T^{5} - 178417 T^{6} - 9372 p T^{7} + 4026 p^{2} T^{8} + 196 p^{3} T^{9} - 9 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 + 6 T - 42 T^{2} - 1136 T^{3} - 54 p T^{4} + 29802 T^{5} + 517382 T^{6} + 29802 p T^{7} - 54 p^{3} T^{8} - 1136 p^{3} T^{9} - 42 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 - 12 T - 54 T^{2} + 292 T^{3} + 11514 T^{4} - 25536 T^{5} - 623986 T^{6} - 25536 p T^{7} + 11514 p^{2} T^{8} + 292 p^{3} T^{9} - 54 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 - 18 T + 129 T^{2} - 222 T^{3} - 3462 T^{4} + 40662 T^{5} - 368431 T^{6} + 40662 p T^{7} - 3462 p^{2} T^{8} - 222 p^{3} T^{9} + 129 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 - 9 T - 108 T^{2} + 383 T^{3} + 13680 T^{4} - 9405 T^{5} - 1069626 T^{6} - 9405 p T^{7} + 13680 p^{2} T^{8} + 383 p^{3} T^{9} - 108 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
71 \( ( 1 + 177 T^{2} + 32 T^{3} + 177 p T^{4} + p^{3} T^{6} )^{2} \)
73 \( 1 + 33 T + 534 T^{2} + 6687 T^{3} + 75648 T^{4} + 728637 T^{5} + 6291524 T^{6} + 728637 p T^{7} + 75648 p^{2} T^{8} + 6687 p^{3} T^{9} + 534 p^{4} T^{10} + 33 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 + 27 T + 297 T^{2} + 2426 T^{3} + 27783 T^{4} + 321003 T^{5} + 3028254 T^{6} + 321003 p T^{7} + 27783 p^{2} T^{8} + 2426 p^{3} T^{9} + 297 p^{4} T^{10} + 27 p^{5} T^{11} + p^{6} T^{12} \)
83 \( ( 1 + 18 T + 234 T^{2} + 2040 T^{3} + 234 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
89 \( 1 - 12 T - 135 T^{2} + 700 T^{3} + 28722 T^{4} - 63804 T^{5} - 2561959 T^{6} - 63804 p T^{7} + 28722 p^{2} T^{8} + 700 p^{3} T^{9} - 135 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
97 \( ( 1 + 264 T^{2} + 38 T^{3} + 264 p T^{4} + p^{3} T^{6} )^{2} \)
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

−5.51714148138722495166841102735, −5.36461961511056220078084620852, −5.14770163838071184466546251330, −5.05590272631229646629392714726, −4.95740265844783163784361505607, −4.71810992125832883959134260886, −4.48447240508059446606187707422, −4.39920482762315674455241326155, −4.38883090686921638903833796266, −4.04718165133076204803657424869, −3.94908938491063240132677140897, −3.51559420616996438489286187651, −3.31007562640876040853001415032, −3.30740317663983878250352593892, −2.88101373773014752042247051757, −2.74936477547043484423228849895, −2.74160861253324596500372140267, −2.65514470273238601080802599721, −2.12498169124565234063860487991, −2.03570098204310812243980361832, −1.83677340753415590168192409366, −1.54843430853644033002120335281, −1.23441540075090375389614989861, −0.893900328245166901811766946596, −0.26672962071162906737515845062, 0.26672962071162906737515845062, 0.893900328245166901811766946596, 1.23441540075090375389614989861, 1.54843430853644033002120335281, 1.83677340753415590168192409366, 2.03570098204310812243980361832, 2.12498169124565234063860487991, 2.65514470273238601080802599721, 2.74160861253324596500372140267, 2.74936477547043484423228849895, 2.88101373773014752042247051757, 3.30740317663983878250352593892, 3.31007562640876040853001415032, 3.51559420616996438489286187651, 3.94908938491063240132677140897, 4.04718165133076204803657424869, 4.38883090686921638903833796266, 4.39920482762315674455241326155, 4.48447240508059446606187707422, 4.71810992125832883959134260886, 4.95740265844783163784361505607, 5.05590272631229646629392714726, 5.14770163838071184466546251330, 5.36461961511056220078084620852, 5.51714148138722495166841102735

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

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