Properties

Degree 12
Conductor $ 2^{18} \cdot 7^{6} \cdot 11^{6} \cdot 13^{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-s + 5-s + 6·7-s − 4·9-s − 6·11-s − 6·13-s − 15-s + 2·19-s − 6·21-s + 11·23-s − 10·25-s + 5·27-s + 14·29-s + 21·31-s + 6·33-s + 6·35-s − 5·37-s + 6·39-s + 12·43-s − 4·45-s + 2·47-s + 21·49-s + 2·53-s − 6·55-s − 2·57-s + 3·59-s − 18·61-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s + 2.26·7-s − 4/3·9-s − 1.80·11-s − 1.66·13-s − 0.258·15-s + 0.458·19-s − 1.30·21-s + 2.29·23-s − 2·25-s + 0.962·27-s + 2.59·29-s + 3.77·31-s + 1.04·33-s + 1.01·35-s − 0.821·37-s + 0.960·39-s + 1.82·43-s − 0.596·45-s + 0.291·47-s + 3·49-s + 0.274·53-s − 0.809·55-s − 0.264·57-s + 0.390·59-s − 2.30·61-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{18} \cdot 7^{6} \cdot 11^{6} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr =\mathstrut & \,\Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{18} \cdot 7^{6} \cdot 11^{6} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr =\mathstrut & \,\Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(12\)
\( N \)  =  \(2^{18} \cdot 7^{6} \cdot 11^{6} \cdot 13^{6}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{8008} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(12,\ 2^{18} \cdot 7^{6} \cdot 11^{6} \cdot 13^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )$
$L(1)$  $\approx$  $29.36443803$
$L(\frac12)$  $\approx$  $29.36443803$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;7,\;11,\;13\}$, \(F_p\) is a polynomial of degree 12. If $p \in \{2,\;7,\;11,\;13\}$, then $F_p$ is a polynomial of degree at most 11.
$p$$F_p$
bad2 \( 1 \)
7 \( ( 1 - T )^{6} \)
11 \( ( 1 + T )^{6} \)
13 \( ( 1 + T )^{6} \)
good3 \( 1 + T + 5 T^{2} + 4 T^{3} + 8 T^{4} - 14 T^{5} + 8 T^{6} - 14 p T^{7} + 8 p^{2} T^{8} + 4 p^{3} T^{9} + 5 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} \)
5 \( 1 - T + 11 T^{2} - 16 T^{3} + 16 p T^{4} - 88 T^{5} + 502 T^{6} - 88 p T^{7} + 16 p^{3} T^{8} - 16 p^{3} T^{9} + 11 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 + 31 T^{2} - 20 T^{3} + 732 T^{4} - 820 T^{5} + 15004 T^{6} - 820 p T^{7} + 732 p^{2} T^{8} - 20 p^{3} T^{9} + 31 p^{4} T^{10} + p^{6} T^{12} \)
19 \( 1 - 2 T + 35 T^{2} - 6 T^{3} + 46 p T^{4} - 808 T^{5} + 22748 T^{6} - 808 p T^{7} + 46 p^{3} T^{8} - 6 p^{3} T^{9} + 35 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 - 11 T + 112 T^{2} - 685 T^{3} + 3951 T^{4} - 18154 T^{5} + 89568 T^{6} - 18154 p T^{7} + 3951 p^{2} T^{8} - 685 p^{3} T^{9} + 112 p^{4} T^{10} - 11 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 - 14 T + 198 T^{2} - 1774 T^{3} + 14615 T^{4} - 95404 T^{5} + 563924 T^{6} - 95404 p T^{7} + 14615 p^{2} T^{8} - 1774 p^{3} T^{9} + 198 p^{4} T^{10} - 14 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 - 21 T + 282 T^{2} - 2687 T^{3} + 20679 T^{4} - 134058 T^{5} + 786940 T^{6} - 134058 p T^{7} + 20679 p^{2} T^{8} - 2687 p^{3} T^{9} + 282 p^{4} T^{10} - 21 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 + 5 T + 128 T^{2} + 245 T^{3} + 6815 T^{4} + 1946 T^{5} + 269040 T^{6} + 1946 p T^{7} + 6815 p^{2} T^{8} + 245 p^{3} T^{9} + 128 p^{4} T^{10} + 5 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 + 162 T^{2} - 112 T^{3} + 12639 T^{4} - 304 p T^{5} + 627932 T^{6} - 304 p^{2} T^{7} + 12639 p^{2} T^{8} - 112 p^{3} T^{9} + 162 p^{4} T^{10} + p^{6} T^{12} \)
43 \( 1 - 12 T + 259 T^{2} - 2254 T^{3} + 27430 T^{4} - 181840 T^{5} + 1563900 T^{6} - 181840 p T^{7} + 27430 p^{2} T^{8} - 2254 p^{3} T^{9} + 259 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 - 2 T + 158 T^{2} - 222 T^{3} + 11087 T^{4} - 11004 T^{5} + 550756 T^{6} - 11004 p T^{7} + 11087 p^{2} T^{8} - 222 p^{3} T^{9} + 158 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 - 2 T + 87 T^{2} - 598 T^{3} + 134 p T^{4} - 44548 T^{5} + 450724 T^{6} - 44548 p T^{7} + 134 p^{3} T^{8} - 598 p^{3} T^{9} + 87 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 - 3 T + 62 T^{2} - 553 T^{3} + 6167 T^{4} - 20370 T^{5} + 412516 T^{6} - 20370 p T^{7} + 6167 p^{2} T^{8} - 553 p^{3} T^{9} + 62 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 + 18 T + 273 T^{2} + 3482 T^{3} + 34994 T^{4} + 313654 T^{5} + 2699140 T^{6} + 313654 p T^{7} + 34994 p^{2} T^{8} + 3482 p^{3} T^{9} + 273 p^{4} T^{10} + 18 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 + 25 T + 501 T^{2} + 6006 T^{3} + 65326 T^{4} + 531374 T^{5} + 4704996 T^{6} + 531374 p T^{7} + 65326 p^{2} T^{8} + 6006 p^{3} T^{9} + 501 p^{4} T^{10} + 25 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 + 11 T + 373 T^{2} + 2602 T^{3} + 53240 T^{4} + 259420 T^{5} + 4499622 T^{6} + 259420 p T^{7} + 53240 p^{2} T^{8} + 2602 p^{3} T^{9} + 373 p^{4} T^{10} + 11 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 - 14 T + 190 T^{2} - 1302 T^{3} + 18847 T^{4} - 163404 T^{5} + 1950180 T^{6} - 163404 p T^{7} + 18847 p^{2} T^{8} - 1302 p^{3} T^{9} + 190 p^{4} T^{10} - 14 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 - 24 T + 541 T^{2} - 7436 T^{3} + 100976 T^{4} - 1022182 T^{5} + 10348020 T^{6} - 1022182 p T^{7} + 100976 p^{2} T^{8} - 7436 p^{3} T^{9} + 541 p^{4} T^{10} - 24 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 - 24 T + 479 T^{2} - 5766 T^{3} + 65522 T^{4} - 554148 T^{5} + 5427132 T^{6} - 554148 p T^{7} + 65522 p^{2} T^{8} - 5766 p^{3} T^{9} + 479 p^{4} T^{10} - 24 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 - 55 T + 1703 T^{2} - 36314 T^{3} + 589608 T^{4} - 7583834 T^{5} + 79229360 T^{6} - 7583834 p T^{7} + 589608 p^{2} T^{8} - 36314 p^{3} T^{9} + 1703 p^{4} T^{10} - 55 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 - 37 T + 866 T^{2} - 14993 T^{3} + 216399 T^{4} - 2619306 T^{5} + 27731228 T^{6} - 2619306 p T^{7} + 216399 p^{2} T^{8} - 14993 p^{3} T^{9} + 866 p^{4} T^{10} - 37 p^{5} T^{11} + p^{6} T^{12} \)
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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

−4.22385966540770329732155044031, −3.70611252701762111099767429535, −3.54503679265529292431471970059, −3.43497087503946330687466578912, −3.43337663382470609415328164394, −3.27270323185389540936114166455, −3.25211220989511351791809790879, −2.94685296956336963033932472468, −2.80180844039832847671645037320, −2.77861199621897909463387438681, −2.64070180904266859657864631726, −2.32778293813320108339055355589, −2.29846856472839631272994129202, −2.21022460218798287254113095578, −1.99091529368464379479966126902, −1.89009025552844487052865830224, −1.87719523238672362857946798080, −1.69096114458614412316609926817, −1.25935154167695522133222100457, −1.05192609459721046642260147867, −0.833969023280542130836075593919, −0.63332007259287476634021488357, −0.56862514794612938016278200461, −0.56642332168074100064339288061, −0.53188743677340069151099207308, 0.53188743677340069151099207308, 0.56642332168074100064339288061, 0.56862514794612938016278200461, 0.63332007259287476634021488357, 0.833969023280542130836075593919, 1.05192609459721046642260147867, 1.25935154167695522133222100457, 1.69096114458614412316609926817, 1.87719523238672362857946798080, 1.89009025552844487052865830224, 1.99091529368464379479966126902, 2.21022460218798287254113095578, 2.29846856472839631272994129202, 2.32778293813320108339055355589, 2.64070180904266859657864631726, 2.77861199621897909463387438681, 2.80180844039832847671645037320, 2.94685296956336963033932472468, 3.25211220989511351791809790879, 3.27270323185389540936114166455, 3.43337663382470609415328164394, 3.43497087503946330687466578912, 3.54503679265529292431471970059, 3.70611252701762111099767429535, 4.22385966540770329732155044031

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

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