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  + 4·3-s + 5·5-s − 3·7-s + 6·9-s − 2·11-s − 3·13-s + 20·15-s − 24·17-s + 6·19-s − 12·21-s + 17·25-s + 5·27-s − 29-s − 3·31-s − 8·33-s − 15·35-s − 6·37-s − 12·39-s + 22·41-s − 3·43-s + 30·45-s − 9·47-s + 3·49-s − 96·51-s − 36·53-s − 10·55-s + 24·57-s + ⋯
L(s)  = 1  + 2.30·3-s + 2.23·5-s − 1.13·7-s + 2·9-s − 0.603·11-s − 0.832·13-s + 5.16·15-s − 5.82·17-s + 1.37·19-s − 2.61·21-s + 17/5·25-s + 0.962·27-s − 0.185·29-s − 0.538·31-s − 1.39·33-s − 2.53·35-s − 0.986·37-s − 1.92·39-s + 3.43·41-s − 0.457·43-s + 4.47·45-s − 1.31·47-s + 3/7·49-s − 13.4·51-s − 4.94·53-s − 1.34·55-s + 3.17·57-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  =  \((12,\ 2^{24} \cdot 3^{12} \cdot 7^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)
\(L(1)\)  \(\approx\)  \(1.538833339\)
\(L(\frac12)\)  \(\approx\)  \(1.538833339\)
\(L(\frac{3}{2})\)   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 \( 1 - 4 T + 10 T^{2} - 7 p T^{3} + 10 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
7 \( ( 1 + T + T^{2} )^{3} \)
good5 \( 1 - p T + 8 T^{2} - 7 T^{3} + 9 T^{4} + 62 T^{5} - 299 T^{6} + 62 p T^{7} + 9 p^{2} T^{8} - 7 p^{3} T^{9} + 8 p^{4} T^{10} - p^{6} T^{11} + p^{6} T^{12} \)
11 \( 1 + 2 T - 10 T^{2} + 34 T^{3} + 48 T^{4} - 416 T^{5} + 31 T^{6} - 416 p T^{7} + 48 p^{2} T^{8} + 34 p^{3} T^{9} - 10 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
13 \( ( 1 + T - 12 T^{2} + p T^{3} + p^{2} T^{4} )^{3} \)
17 \( ( 1 + 12 T + 90 T^{2} + 435 T^{3} + 90 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
19 \( ( 1 - 3 T + 51 T^{2} - 107 T^{3} + 51 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
23 \( 1 - 36 T^{2} + 18 T^{3} + 468 T^{4} - 324 T^{5} - 5393 T^{6} - 324 p T^{7} + 468 p^{2} T^{8} + 18 p^{3} T^{9} - 36 p^{4} T^{10} + p^{6} T^{12} \)
29 \( 1 + T - 82 T^{2} - 31 T^{3} + 4425 T^{4} + 758 T^{5} - 148595 T^{6} + 758 p T^{7} + 4425 p^{2} T^{8} - 31 p^{3} T^{9} - 82 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 + 3 T - 60 T^{2} - 219 T^{3} + 1983 T^{4} + 4746 T^{5} - 51289 T^{6} + 4746 p T^{7} + 1983 p^{2} T^{8} - 219 p^{3} T^{9} - 60 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
37 \( ( 1 + 3 T + 57 T^{2} + 303 T^{3} + 57 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
41 \( 1 - 22 T + 206 T^{2} - 1802 T^{3} + 18432 T^{4} - 135116 T^{5} + 808243 T^{6} - 135116 p T^{7} + 18432 p^{2} T^{8} - 1802 p^{3} T^{9} + 206 p^{4} T^{10} - 22 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 + 3 T - 54 T^{2} - 569 T^{3} + 123 T^{4} + 13170 T^{5} + 115347 T^{6} + 13170 p T^{7} + 123 p^{2} T^{8} - 569 p^{3} T^{9} - 54 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 + 9 T - 6 T^{2} - 531 T^{3} - 2433 T^{4} + 3438 T^{5} + 104623 T^{6} + 3438 p T^{7} - 2433 p^{2} T^{8} - 531 p^{3} T^{9} - 6 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
53 \( ( 1 + 18 T + 234 T^{2} + 1917 T^{3} + 234 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
59 \( 1 + 9 T - 90 T^{2} - 459 T^{3} + 10161 T^{4} + 20556 T^{5} - 598421 T^{6} + 20556 p T^{7} + 10161 p^{2} T^{8} - 459 p^{3} T^{9} - 90 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 - 6 T - 126 T^{2} + 358 T^{3} + 12372 T^{4} - 11472 T^{5} - 838653 T^{6} - 11472 p T^{7} + 12372 p^{2} T^{8} + 358 p^{3} T^{9} - 126 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 + 6 T^{2} - 1366 T^{3} + 438 T^{4} - 4098 T^{5} + 1065603 T^{6} - 4098 p T^{7} + 438 p^{2} T^{8} - 1366 p^{3} T^{9} + 6 p^{4} T^{10} + p^{6} T^{12} \)
71 \( ( 1 + 9 T + 207 T^{2} + 1197 T^{3} + 207 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
73 \( ( 1 - 3 T + 51 T^{2} - 681 T^{3} + 51 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
79 \( 1 - 15 T + 36 T^{2} + 367 T^{3} - 3225 T^{4} + 51726 T^{5} - 676905 T^{6} + 51726 p T^{7} - 3225 p^{2} T^{8} + 367 p^{3} T^{9} + 36 p^{4} T^{10} - 15 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 + 12 T - 144 T^{2} - 582 T^{3} + 34812 T^{4} + 90444 T^{5} - 2656433 T^{6} + 90444 p T^{7} + 34812 p^{2} T^{8} - 582 p^{3} T^{9} - 144 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
89 \( ( 1 + 2 T + 116 T^{2} + 735 T^{3} + 116 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
97 \( 1 + 3 T - 168 T^{2} + 573 T^{3} + 14223 T^{4} - 78504 T^{5} - 1297807 T^{6} - 78504 p T^{7} + 14223 p^{2} T^{8} + 573 p^{3} T^{9} - 168 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
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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

−5.30216392900746924784244509286, −5.12905990001648044963211064024, −5.08933983818160914651623539277, −4.65730624942320504115911071673, −4.51209605173735546994510245801, −4.50707828129654069646566518550, −4.20494174126303139163850103735, −4.16624888868048008405007585781, −4.10224613653032528539750990557, −3.71041576967219043348630829496, −3.34937144359613095821118184064, −3.19964595470058327890713143638, −3.01239372225652208495778763560, −2.97943582233612922475835676594, −2.83495541100542828969457357475, −2.72532738441254203151386483255, −2.28294469373731382316932961188, −2.28103040173823964805981785659, −2.27743933639922358566135441405, −1.93874669782375860681415157286, −1.66669006410357282136629872329, −1.52231272305937092667384380036, −1.33067511568237591796028561530, −0.54438454433668087146121881903, −0.15615558218268396706815106302, 0.15615558218268396706815106302, 0.54438454433668087146121881903, 1.33067511568237591796028561530, 1.52231272305937092667384380036, 1.66669006410357282136629872329, 1.93874669782375860681415157286, 2.27743933639922358566135441405, 2.28103040173823964805981785659, 2.28294469373731382316932961188, 2.72532738441254203151386483255, 2.83495541100542828969457357475, 2.97943582233612922475835676594, 3.01239372225652208495778763560, 3.19964595470058327890713143638, 3.34937144359613095821118184064, 3.71041576967219043348630829496, 4.10224613653032528539750990557, 4.16624888868048008405007585781, 4.20494174126303139163850103735, 4.50707828129654069646566518550, 4.51209605173735546994510245801, 4.65730624942320504115911071673, 5.08933983818160914651623539277, 5.12905990001648044963211064024, 5.30216392900746924784244509286

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

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