Properties

Degree 12
Conductor $ 2^{6} \cdot 5^{6} \cdot 13^{6} \cdot 31^{6} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 6

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 6·2-s − 3·3-s + 21·4-s − 6·5-s − 18·6-s − 2·7-s + 56·8-s − 5·9-s − 36·10-s − 4·11-s − 63·12-s + 6·13-s − 12·14-s + 18·15-s + 126·16-s − 8·17-s − 30·18-s − 9·19-s − 126·20-s + 6·21-s − 24·22-s − 7·23-s − 168·24-s + 21·25-s + 36·26-s + 30·27-s − 42·28-s + ⋯
L(s)  = 1  + 4.24·2-s − 1.73·3-s + 21/2·4-s − 2.68·5-s − 7.34·6-s − 0.755·7-s + 19.7·8-s − 5/3·9-s − 11.3·10-s − 1.20·11-s − 18.1·12-s + 1.66·13-s − 3.20·14-s + 4.64·15-s + 63/2·16-s − 1.94·17-s − 7.07·18-s − 2.06·19-s − 28.1·20-s + 1.30·21-s − 5.11·22-s − 1.45·23-s − 34.2·24-s + 21/5·25-s + 7.06·26-s + 5.77·27-s − 7.93·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{6} \cdot 5^{6} \cdot 13^{6} \cdot 31^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr =\mathstrut & \,\Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{6} \cdot 5^{6} \cdot 13^{6} \cdot 31^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr =\mathstrut & \,\Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(12\)
\( N \)  =  \(2^{6} \cdot 5^{6} \cdot 13^{6} \cdot 31^{6}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{4030} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  6
Selberg data  =  $(12,\ 2^{6} \cdot 5^{6} \cdot 13^{6} \cdot 31^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;5,\;13,\;31\}$, \(F_p\) is a polynomial of degree 12. If $p \in \{2,\;5,\;13,\;31\}$, then $F_p$ is a polynomial of degree at most 11.
$p$$F_p$
bad2 \( ( 1 - T )^{6} \)
5 \( ( 1 + T )^{6} \)
13 \( ( 1 - T )^{6} \)
31 \( ( 1 + T )^{6} \)
good3 \( 1 + p T + 14 T^{2} + p^{3} T^{3} + 25 p T^{4} + 110 T^{5} + 253 T^{6} + 110 p T^{7} + 25 p^{3} T^{8} + p^{6} T^{9} + 14 p^{4} T^{10} + p^{6} T^{11} + p^{6} T^{12} \)
7 \( 1 + 2 T + 30 T^{2} + 46 T^{3} + 415 T^{4} + 75 p T^{5} + 73 p^{2} T^{6} + 75 p^{2} T^{7} + 415 p^{2} T^{8} + 46 p^{3} T^{9} + 30 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 + 4 T + 37 T^{2} + 100 T^{3} + 697 T^{4} + 1796 T^{5} + 9721 T^{6} + 1796 p T^{7} + 697 p^{2} T^{8} + 100 p^{3} T^{9} + 37 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 + 8 T + 93 T^{2} + 598 T^{3} + 3836 T^{4} + 19017 T^{5} + 86373 T^{6} + 19017 p T^{7} + 3836 p^{2} T^{8} + 598 p^{3} T^{9} + 93 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 + 9 T + 101 T^{2} + 701 T^{3} + 4572 T^{4} + 24246 T^{5} + 113969 T^{6} + 24246 p T^{7} + 4572 p^{2} T^{8} + 701 p^{3} T^{9} + 101 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 + 7 T + 100 T^{2} + 495 T^{3} + 4146 T^{4} + 16185 T^{5} + 109519 T^{6} + 16185 p T^{7} + 4146 p^{2} T^{8} + 495 p^{3} T^{9} + 100 p^{4} T^{10} + 7 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 + 14 T + 149 T^{2} + 1306 T^{3} + 9448 T^{4} + 60477 T^{5} + 353623 T^{6} + 60477 p T^{7} + 9448 p^{2} T^{8} + 1306 p^{3} T^{9} + 149 p^{4} T^{10} + 14 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 + 125 T^{2} - 150 T^{3} + 8475 T^{4} - 10142 T^{5} + 389027 T^{6} - 10142 p T^{7} + 8475 p^{2} T^{8} - 150 p^{3} T^{9} + 125 p^{4} T^{10} + p^{6} T^{12} \)
41 \( 1 - 2 T + 183 T^{2} - 297 T^{3} + 15795 T^{4} - 21384 T^{5} + 816245 T^{6} - 21384 p T^{7} + 15795 p^{2} T^{8} - 297 p^{3} T^{9} + 183 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 + 7 T + 73 T^{2} + 227 T^{3} - 640 T^{4} - 17942 T^{5} - 154993 T^{6} - 17942 p T^{7} - 640 p^{2} T^{8} + 227 p^{3} T^{9} + 73 p^{4} T^{10} + 7 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 + 8 T + 240 T^{2} + 1462 T^{3} + 25055 T^{4} + 120015 T^{5} + 1501985 T^{6} + 120015 p T^{7} + 25055 p^{2} T^{8} + 1462 p^{3} T^{9} + 240 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 + 24 T + 385 T^{2} + 4565 T^{3} + 47881 T^{4} + 425376 T^{5} + 3362047 T^{6} + 425376 p T^{7} + 47881 p^{2} T^{8} + 4565 p^{3} T^{9} + 385 p^{4} T^{10} + 24 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 + 5 T + 182 T^{2} + 1283 T^{3} + 18686 T^{4} + 123043 T^{5} + 1356483 T^{6} + 123043 p T^{7} + 18686 p^{2} T^{8} + 1283 p^{3} T^{9} + 182 p^{4} T^{10} + 5 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 + 5 T + 273 T^{2} + 1441 T^{3} + 34337 T^{4} + 171791 T^{5} + 2610283 T^{6} + 171791 p T^{7} + 34337 p^{2} T^{8} + 1441 p^{3} T^{9} + 273 p^{4} T^{10} + 5 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 + 12 T + 188 T^{2} + 1692 T^{3} + 19871 T^{4} + 183775 T^{5} + 1664727 T^{6} + 183775 p T^{7} + 19871 p^{2} T^{8} + 1692 p^{3} T^{9} + 188 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 + 10 T + 237 T^{2} + 2002 T^{3} + 21842 T^{4} + 184106 T^{5} + 1433293 T^{6} + 184106 p T^{7} + 21842 p^{2} T^{8} + 2002 p^{3} T^{9} + 237 p^{4} T^{10} + 10 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 - 5 T + 157 T^{2} - 1015 T^{3} + 12753 T^{4} - 58937 T^{5} + 980785 T^{6} - 58937 p T^{7} + 12753 p^{2} T^{8} - 1015 p^{3} T^{9} + 157 p^{4} T^{10} - 5 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 + 16 T + 447 T^{2} + 4786 T^{3} + 77837 T^{4} + 629628 T^{5} + 7714993 T^{6} + 629628 p T^{7} + 77837 p^{2} T^{8} + 4786 p^{3} T^{9} + 447 p^{4} T^{10} + 16 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 + 22 T + 426 T^{2} + 6128 T^{3} + 74797 T^{4} + 776717 T^{5} + 7689041 T^{6} + 776717 p T^{7} + 74797 p^{2} T^{8} + 6128 p^{3} T^{9} + 426 p^{4} T^{10} + 22 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 - 14 T + 391 T^{2} - 4028 T^{3} + 71112 T^{4} - 592655 T^{5} + 7892149 T^{6} - 592655 p T^{7} + 71112 p^{2} T^{8} - 4028 p^{3} T^{9} + 391 p^{4} T^{10} - 14 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 + 9 T + 356 T^{2} + 2520 T^{3} + 61232 T^{4} + 351767 T^{5} + 6986163 T^{6} + 351767 p T^{7} + 61232 p^{2} T^{8} + 2520 p^{3} T^{9} + 356 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
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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

−4.86317830940873443169435975126, −4.51473091711273147743596962466, −4.46978052464605268275228356593, −4.35015115593125417879829886458, −4.21507603821217617716466636578, −4.15744713456309670116596082155, −3.92725483185964929202318235836, −3.83445539348733219093480712173, −3.58381008013633195530218147078, −3.49869889521612133208211770085, −3.45864112595461403444891569306, −3.42838338611249459719349813222, −3.34345103794946846890114531416, −2.85624532151780667063459694251, −2.73464016407768138782546903729, −2.67187855685231552205747493125, −2.64860878177217478320781577730, −2.57120832499572523432133051955, −2.47755763776379686511947491664, −2.00188938433334472994055470085, −1.74589457789337261977549738655, −1.53844168845950213150010032283, −1.50322302499238233980405843474, −1.30999270594204474368782989440, −1.22298998758999407745186170780, 0, 0, 0, 0, 0, 0, 1.22298998758999407745186170780, 1.30999270594204474368782989440, 1.50322302499238233980405843474, 1.53844168845950213150010032283, 1.74589457789337261977549738655, 2.00188938433334472994055470085, 2.47755763776379686511947491664, 2.57120832499572523432133051955, 2.64860878177217478320781577730, 2.67187855685231552205747493125, 2.73464016407768138782546903729, 2.85624532151780667063459694251, 3.34345103794946846890114531416, 3.42838338611249459719349813222, 3.45864112595461403444891569306, 3.49869889521612133208211770085, 3.58381008013633195530218147078, 3.83445539348733219093480712173, 3.92725483185964929202318235836, 4.15744713456309670116596082155, 4.21507603821217617716466636578, 4.35015115593125417879829886458, 4.46978052464605268275228356593, 4.51473091711273147743596962466, 4.86317830940873443169435975126

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

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