Properties

Degree 12
Conductor $ 2^{24} \cdot 3^{12} \cdot 7^{6} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·5-s + 3·7-s + 6·11-s + 3·13-s + 12·17-s + 6·19-s + 12·23-s + 15·25-s − 9·27-s − 9·29-s − 3·31-s − 9·35-s − 6·37-s − 3·43-s + 3·47-s + 3·49-s + 12·53-s − 18·55-s − 3·59-s − 6·61-s − 9·65-s − 12·67-s − 18·71-s − 42·73-s + 18·77-s − 21·79-s − 18·83-s + ⋯
L(s)  = 1  − 1.34·5-s + 1.13·7-s + 1.80·11-s + 0.832·13-s + 2.91·17-s + 1.37·19-s + 2.50·23-s + 3·25-s − 1.73·27-s − 1.67·29-s − 0.538·31-s − 1.52·35-s − 0.986·37-s − 0.457·43-s + 0.437·47-s + 3/7·49-s + 1.64·53-s − 2.42·55-s − 0.390·59-s − 0.768·61-s − 1.11·65-s − 1.46·67-s − 2.13·71-s − 4.91·73-s + 2.05·77-s − 2.36·79-s − 1.97·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(12\)
\( N \)  =  \(2^{24} \cdot 3^{12} \cdot 7^{6}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{1008} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((12,\ 2^{24} \cdot 3^{12} \cdot 7^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)
\(L(1)\)  \(\approx\)  \(3.676497791\)
\(L(\frac12)\)  \(\approx\)  \(3.676497791\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7\}$,\(F_p(T)\) is a polynomial of degree 12. If $p \in \{2,\;3,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 11.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + p^{2} T^{3} + p^{3} T^{6} \)
7 \( ( 1 - T + T^{2} )^{3} \)
good5 \( 1 + 3 T - 6 T^{2} - 9 T^{3} + 69 T^{4} + 6 p T^{5} - 371 T^{6} + 6 p^{2} T^{7} + 69 p^{2} T^{8} - 9 p^{3} T^{9} - 6 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 - 6 T - 6 T^{2} + 18 T^{3} + 492 T^{4} - 852 T^{5} - 2873 T^{6} - 852 p T^{7} + 492 p^{2} T^{8} + 18 p^{3} T^{9} - 6 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 - 3 T + 3 T^{2} - 76 T^{3} + 45 T^{4} + 135 T^{5} + 3246 T^{6} + 135 p T^{7} + 45 p^{2} T^{8} - 76 p^{3} T^{9} + 3 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
17 \( ( 1 - 6 T + 60 T^{2} - 207 T^{3} + 60 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
19 \( ( 1 - 3 T + 51 T^{2} - 97 T^{3} + 51 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
23 \( 1 - 12 T + 48 T^{2} - 54 T^{3} + 420 T^{4} - 6060 T^{5} + 37591 T^{6} - 6060 p T^{7} + 420 p^{2} T^{8} - 54 p^{3} T^{9} + 48 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 + 9 T + 30 T^{2} + 81 T^{3} - 579 T^{4} - 9414 T^{5} - 59051 T^{6} - 9414 p T^{7} - 579 p^{2} T^{8} + 81 p^{3} T^{9} + 30 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 + 3 T - 6 T^{2} + 319 T^{3} + 171 T^{4} - 1962 T^{5} + 62727 T^{6} - 1962 p T^{7} + 171 p^{2} T^{8} + 319 p^{3} T^{9} - 6 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
37 \( ( 1 + 3 T + 33 T^{2} - 101 T^{3} + 33 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
41 \( 1 - 114 T^{2} - 18 T^{3} + 8322 T^{4} + 1026 T^{5} - 394913 T^{6} + 1026 p T^{7} + 8322 p^{2} T^{8} - 18 p^{3} T^{9} - 114 p^{4} T^{10} + p^{6} T^{12} \)
43 \( 1 + 3 T - 114 T^{2} - 149 T^{3} + 9063 T^{4} + 5670 T^{5} - 441093 T^{6} + 5670 p T^{7} + 9063 p^{2} T^{8} - 149 p^{3} T^{9} - 114 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 - 3 T - 78 T^{2} + 405 T^{3} + 2481 T^{4} - 11064 T^{5} - 57089 T^{6} - 11064 p T^{7} + 2481 p^{2} T^{8} + 405 p^{3} T^{9} - 78 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
53 \( ( 1 - 6 T + 150 T^{2} - 639 T^{3} + 150 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
59 \( 1 + 3 T - 96 T^{2} - 495 T^{3} + 3615 T^{4} + 15798 T^{5} - 107021 T^{6} + 15798 p T^{7} + 3615 p^{2} T^{8} - 495 p^{3} T^{9} - 96 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 + 6 T - 132 T^{2} - 418 T^{3} + 13698 T^{4} + 19134 T^{5} - 893289 T^{6} + 19134 p T^{7} + 13698 p^{2} T^{8} - 418 p^{3} T^{9} - 132 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 + 12 T - 78 T^{2} - 518 T^{3} + 15318 T^{4} + 50094 T^{5} - 815637 T^{6} + 50094 p T^{7} + 15318 p^{2} T^{8} - 518 p^{3} T^{9} - 78 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
71 \( ( 1 + 9 T + 159 T^{2} + 1305 T^{3} + 159 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
73 \( ( 1 + 21 T + 303 T^{2} + 2797 T^{3} + 303 p T^{4} + 21 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
79 \( 1 + 21 T + 84 T^{2} + 499 T^{3} + 25767 T^{4} + 195678 T^{5} + 408327 T^{6} + 195678 p T^{7} + 25767 p^{2} T^{8} + 499 p^{3} T^{9} + 84 p^{4} T^{10} + 21 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 + 18 T + 30 T^{2} - 702 T^{3} + 8088 T^{4} + 126648 T^{5} + 719359 T^{6} + 126648 p T^{7} + 8088 p^{2} T^{8} - 702 p^{3} T^{9} + 30 p^{4} T^{10} + 18 p^{5} T^{11} + p^{6} T^{12} \)
89 \( ( 1 - 12 T + 204 T^{2} - 1323 T^{3} + 204 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
97 \( 1 - 3 T - 114 T^{2} + 149 T^{3} + 2421 T^{4} + 11502 T^{5} + 340233 T^{6} + 11502 p T^{7} + 2421 p^{2} T^{8} + 149 p^{3} T^{9} - 114 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−5.25951120974827262824113446218, −5.22970001412823381162898142837, −4.89601658221015665136827576233, −4.79418099229561396168914779269, −4.47930790051873363644347740068, −4.47119294362224353242174637234, −4.26717112675753584662117407788, −4.24306421836378867503753801782, −3.86373547275153675005352696058, −3.81649840934088093443857162008, −3.45148047845483548542918218645, −3.29895697477240429701127352211, −3.28058416340649369229655902088, −3.18910708892034705590491890274, −3.06546615981756344861273065414, −2.87379612677785301862097154924, −2.42286205617761853709066996569, −2.24280527903032127388952740519, −1.78617183941398144140853609023, −1.54739200221319459854699463644, −1.40612509920672158778229391881, −1.31806526928197970526707385381, −1.07557105603813125082445035523, −0.912704431766251368778684771770, −0.27427708269657519555789863141, 0.27427708269657519555789863141, 0.912704431766251368778684771770, 1.07557105603813125082445035523, 1.31806526928197970526707385381, 1.40612509920672158778229391881, 1.54739200221319459854699463644, 1.78617183941398144140853609023, 2.24280527903032127388952740519, 2.42286205617761853709066996569, 2.87379612677785301862097154924, 3.06546615981756344861273065414, 3.18910708892034705590491890274, 3.28058416340649369229655902088, 3.29895697477240429701127352211, 3.45148047845483548542918218645, 3.81649840934088093443857162008, 3.86373547275153675005352696058, 4.24306421836378867503753801782, 4.26717112675753584662117407788, 4.47119294362224353242174637234, 4.47930790051873363644347740068, 4.79418099229561396168914779269, 4.89601658221015665136827576233, 5.22970001412823381162898142837, 5.25951120974827262824113446218

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

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