Properties

Degree 22
Conductor $ 2^{33} \cdot 7^{11} \cdot 11^{11} \cdot 13^{11} $
Sign $-1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 11

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 2·5-s − 11·7-s − 10·9-s + 11·11-s − 11·13-s − 4·15-s − 4·17-s − 16·19-s + 22·21-s − 3·23-s − 20·25-s + 21·27-s + 29-s + 14·31-s − 22·33-s − 22·35-s − 8·37-s + 22·39-s + 4·41-s − 30·43-s − 20·45-s − 3·47-s + 66·49-s + 8·51-s − 5·53-s + 22·55-s + ⋯
L(s)  = 1  − 1.15·3-s + 0.894·5-s − 4.15·7-s − 3.33·9-s + 3.31·11-s − 3.05·13-s − 1.03·15-s − 0.970·17-s − 3.67·19-s + 4.80·21-s − 0.625·23-s − 4·25-s + 4.04·27-s + 0.185·29-s + 2.51·31-s − 3.82·33-s − 3.71·35-s − 1.31·37-s + 3.52·39-s + 0.624·41-s − 4.57·43-s − 2.98·45-s − 0.437·47-s + 66/7·49-s + 1.12·51-s − 0.686·53-s + 2.96·55-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{33} \cdot 7^{11} \cdot 11^{11} \cdot 13^{11}\right)^{s/2} \, \Gamma_{\C}(s)^{11} \, L(s)\cr =\mathstrut & -\,\Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{33} \cdot 7^{11} \cdot 11^{11} \cdot 13^{11}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{11} \, L(s)\cr =\mathstrut & -\,\Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(22\)
\( N \)  =  \(2^{33} \cdot 7^{11} \cdot 11^{11} \cdot 13^{11}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{8008} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  11
Selberg data  =  $(22,\ 2^{33} \cdot 7^{11} \cdot 11^{11} \cdot 13^{11} ,\ ( \ : [1/2]^{11} ),\ -1 )$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;7,\;11,\;13\}$, \(F_p\) is a polynomial of degree 22. If $p \in \{2,\;7,\;11,\;13\}$, then $F_p$ is a polynomial of degree at most 21.
$p$$F_p$
bad2 \( 1 \)
7 \( ( 1 + T )^{11} \)
11 \( ( 1 - T )^{11} \)
13 \( ( 1 + T )^{11} \)
good3 \( 1 + 2 T + 14 T^{2} + p^{3} T^{3} + 34 p T^{4} + 196 T^{5} + 523 T^{6} + 988 T^{7} + 2158 T^{8} + 3889 T^{9} + 2515 p T^{10} + 12668 T^{11} + 2515 p^{2} T^{12} + 3889 p^{2} T^{13} + 2158 p^{3} T^{14} + 988 p^{4} T^{15} + 523 p^{5} T^{16} + 196 p^{6} T^{17} + 34 p^{8} T^{18} + p^{11} T^{19} + 14 p^{9} T^{20} + 2 p^{10} T^{21} + p^{11} T^{22} \)
5 \( 1 - 2 T + 24 T^{2} - 41 T^{3} + 252 T^{4} - 288 T^{5} + 1337 T^{6} - 22 T^{7} + 2086 T^{8} + 13787 T^{9} - 16063 T^{10} + 105706 T^{11} - 16063 p T^{12} + 13787 p^{2} T^{13} + 2086 p^{3} T^{14} - 22 p^{4} T^{15} + 1337 p^{5} T^{16} - 288 p^{6} T^{17} + 252 p^{7} T^{18} - 41 p^{8} T^{19} + 24 p^{9} T^{20} - 2 p^{10} T^{21} + p^{11} T^{22} \)
17 \( 1 + 4 T + 112 T^{2} + 399 T^{3} + 6538 T^{4} + 21364 T^{5} + 254625 T^{6} + 759042 T^{7} + 7270384 T^{8} + 19632831 T^{9} + 158781009 T^{10} + 382181206 T^{11} + 158781009 p T^{12} + 19632831 p^{2} T^{13} + 7270384 p^{3} T^{14} + 759042 p^{4} T^{15} + 254625 p^{5} T^{16} + 21364 p^{6} T^{17} + 6538 p^{7} T^{18} + 399 p^{8} T^{19} + 112 p^{9} T^{20} + 4 p^{10} T^{21} + p^{11} T^{22} \)
19 \( 1 + 16 T + 216 T^{2} + 2141 T^{3} + 18674 T^{4} + 140308 T^{5} + 955767 T^{6} + 5889704 T^{7} + 33407142 T^{8} + 174970235 T^{9} + 849975391 T^{10} + 3842050048 T^{11} + 849975391 p T^{12} + 174970235 p^{2} T^{13} + 33407142 p^{3} T^{14} + 5889704 p^{4} T^{15} + 955767 p^{5} T^{16} + 140308 p^{6} T^{17} + 18674 p^{7} T^{18} + 2141 p^{8} T^{19} + 216 p^{9} T^{20} + 16 p^{10} T^{21} + p^{11} T^{22} \)
23 \( 1 + 3 T + 145 T^{2} + 160 T^{3} + 9212 T^{4} - 7673 T^{5} + 352856 T^{6} - 47800 p T^{7} + 9550075 T^{8} - 54376154 T^{9} + 216513103 T^{10} - 1575350576 T^{11} + 216513103 p T^{12} - 54376154 p^{2} T^{13} + 9550075 p^{3} T^{14} - 47800 p^{5} T^{15} + 352856 p^{5} T^{16} - 7673 p^{6} T^{17} + 9212 p^{7} T^{18} + 160 p^{8} T^{19} + 145 p^{9} T^{20} + 3 p^{10} T^{21} + p^{11} T^{22} \)
29 \( 1 - T + 137 T^{2} - 90 T^{3} + 9528 T^{4} + 4979 T^{5} + 424978 T^{6} + 977800 T^{7} + 13642225 T^{8} + 62749830 T^{9} + 374654387 T^{10} + 2314314212 T^{11} + 374654387 p T^{12} + 62749830 p^{2} T^{13} + 13642225 p^{3} T^{14} + 977800 p^{4} T^{15} + 424978 p^{5} T^{16} + 4979 p^{6} T^{17} + 9528 p^{7} T^{18} - 90 p^{8} T^{19} + 137 p^{9} T^{20} - p^{10} T^{21} + p^{11} T^{22} \)
31 \( 1 - 14 T + 163 T^{2} - 1460 T^{3} + 12282 T^{4} - 94416 T^{5} + 709108 T^{6} - 4968372 T^{7} + 32760639 T^{8} - 204496002 T^{9} + 1211833827 T^{10} - 6877502064 T^{11} + 1211833827 p T^{12} - 204496002 p^{2} T^{13} + 32760639 p^{3} T^{14} - 4968372 p^{4} T^{15} + 709108 p^{5} T^{16} - 94416 p^{6} T^{17} + 12282 p^{7} T^{18} - 1460 p^{8} T^{19} + 163 p^{9} T^{20} - 14 p^{10} T^{21} + p^{11} T^{22} \)
37 \( 1 + 8 T + 197 T^{2} + 1579 T^{3} + 21296 T^{4} + 165610 T^{5} + 1619742 T^{6} + 12033248 T^{7} + 93804021 T^{8} + 652027294 T^{9} + 4308623407 T^{10} + 27269000138 T^{11} + 4308623407 p T^{12} + 652027294 p^{2} T^{13} + 93804021 p^{3} T^{14} + 12033248 p^{4} T^{15} + 1619742 p^{5} T^{16} + 165610 p^{6} T^{17} + 21296 p^{7} T^{18} + 1579 p^{8} T^{19} + 197 p^{9} T^{20} + 8 p^{10} T^{21} + p^{11} T^{22} \)
41 \( 1 - 4 T + 275 T^{2} - 1143 T^{3} + 38766 T^{4} - 158458 T^{5} + 3644434 T^{6} - 14144600 T^{7} + 250935913 T^{8} - 899641506 T^{9} + 13215630467 T^{10} - 42526117282 T^{11} + 13215630467 p T^{12} - 899641506 p^{2} T^{13} + 250935913 p^{3} T^{14} - 14144600 p^{4} T^{15} + 3644434 p^{5} T^{16} - 158458 p^{6} T^{17} + 38766 p^{7} T^{18} - 1143 p^{8} T^{19} + 275 p^{9} T^{20} - 4 p^{10} T^{21} + p^{11} T^{22} \)
43 \( 1 + 30 T + 17 p T^{2} + 12239 T^{3} + 179360 T^{4} + 2166645 T^{5} + 23769869 T^{6} + 228499151 T^{7} + 2033334848 T^{8} + 16268540337 T^{9} + 121507108082 T^{10} + 823769045648 T^{11} + 121507108082 p T^{12} + 16268540337 p^{2} T^{13} + 2033334848 p^{3} T^{14} + 228499151 p^{4} T^{15} + 23769869 p^{5} T^{16} + 2166645 p^{6} T^{17} + 179360 p^{7} T^{18} + 12239 p^{8} T^{19} + 17 p^{10} T^{20} + 30 p^{10} T^{21} + p^{11} T^{22} \)
47 \( 1 + 3 T + 409 T^{2} + 880 T^{3} + 78284 T^{4} + 117367 T^{5} + 9359720 T^{6} + 9619768 T^{7} + 783709483 T^{8} + 571072870 T^{9} + 48580055095 T^{10} + 602973776 p T^{11} + 48580055095 p T^{12} + 571072870 p^{2} T^{13} + 783709483 p^{3} T^{14} + 9619768 p^{4} T^{15} + 9359720 p^{5} T^{16} + 117367 p^{6} T^{17} + 78284 p^{7} T^{18} + 880 p^{8} T^{19} + 409 p^{9} T^{20} + 3 p^{10} T^{21} + p^{11} T^{22} \)
53 \( 1 + 5 T + 276 T^{2} + 1510 T^{3} + 42137 T^{4} + 233300 T^{5} + 4536915 T^{6} + 24053034 T^{7} + 374104317 T^{8} + 1846750136 T^{9} + 24522279021 T^{10} + 110391555996 T^{11} + 24522279021 p T^{12} + 1846750136 p^{2} T^{13} + 374104317 p^{3} T^{14} + 24053034 p^{4} T^{15} + 4536915 p^{5} T^{16} + 233300 p^{6} T^{17} + 42137 p^{7} T^{18} + 1510 p^{8} T^{19} + 276 p^{9} T^{20} + 5 p^{10} T^{21} + p^{11} T^{22} \)
59 \( 1 - 11 T + 512 T^{2} - 4749 T^{3} + 122537 T^{4} - 977559 T^{5} + 18325788 T^{6} - 127351140 T^{7} + 1921247727 T^{8} - 11702824390 T^{9} + 149293481449 T^{10} - 796492462910 T^{11} + 149293481449 p T^{12} - 11702824390 p^{2} T^{13} + 1921247727 p^{3} T^{14} - 127351140 p^{4} T^{15} + 18325788 p^{5} T^{16} - 977559 p^{6} T^{17} + 122537 p^{7} T^{18} - 4749 p^{8} T^{19} + 512 p^{9} T^{20} - 11 p^{10} T^{21} + p^{11} T^{22} \)
61 \( 1 - 15 T + 533 T^{2} - 6666 T^{3} + 134981 T^{4} - 1436831 T^{5} + 21448049 T^{6} - 197344763 T^{7} + 2379699357 T^{8} - 19060272832 T^{9} + 193787624380 T^{10} - 1349725783396 T^{11} + 193787624380 p T^{12} - 19060272832 p^{2} T^{13} + 2379699357 p^{3} T^{14} - 197344763 p^{4} T^{15} + 21448049 p^{5} T^{16} - 1436831 p^{6} T^{17} + 134981 p^{7} T^{18} - 6666 p^{8} T^{19} + 533 p^{9} T^{20} - 15 p^{10} T^{21} + p^{11} T^{22} \)
67 \( 1 + 41 T + 1283 T^{2} + 28647 T^{3} + 544892 T^{4} + 8698625 T^{5} + 123454265 T^{6} + 1547239523 T^{7} + 17591283388 T^{8} + 180388257753 T^{9} + 1694120695490 T^{10} + 14469404196866 T^{11} + 1694120695490 p T^{12} + 180388257753 p^{2} T^{13} + 17591283388 p^{3} T^{14} + 1547239523 p^{4} T^{15} + 123454265 p^{5} T^{16} + 8698625 p^{6} T^{17} + 544892 p^{7} T^{18} + 28647 p^{8} T^{19} + 1283 p^{9} T^{20} + 41 p^{10} T^{21} + p^{11} T^{22} \)
71 \( 1 - T + 235 T^{2} - 1295 T^{3} + 34735 T^{4} - 241618 T^{5} + 4237546 T^{6} - 31825546 T^{7} + 390365867 T^{8} - 3226920813 T^{9} + 32243910302 T^{10} - 247104099150 T^{11} + 32243910302 p T^{12} - 3226920813 p^{2} T^{13} + 390365867 p^{3} T^{14} - 31825546 p^{4} T^{15} + 4237546 p^{5} T^{16} - 241618 p^{6} T^{17} + 34735 p^{7} T^{18} - 1295 p^{8} T^{19} + 235 p^{9} T^{20} - p^{10} T^{21} + p^{11} T^{22} \)
73 \( 1 + 8 T + 353 T^{2} + 1571 T^{3} + 58426 T^{4} + 131772 T^{5} + 7058980 T^{6} + 8016780 T^{7} + 725969247 T^{8} + 545803068 T^{9} + 869990185 p T^{10} + 40525991010 T^{11} + 869990185 p^{2} T^{12} + 545803068 p^{2} T^{13} + 725969247 p^{3} T^{14} + 8016780 p^{4} T^{15} + 7058980 p^{5} T^{16} + 131772 p^{6} T^{17} + 58426 p^{7} T^{18} + 1571 p^{8} T^{19} + 353 p^{9} T^{20} + 8 p^{10} T^{21} + p^{11} T^{22} \)
79 \( 1 + 26 T + 713 T^{2} + 13585 T^{3} + 236394 T^{4} + 3511029 T^{5} + 48072505 T^{6} + 590102751 T^{7} + 6773619566 T^{8} + 71211867659 T^{9} + 704931052524 T^{10} + 6449584232316 T^{11} + 704931052524 p T^{12} + 71211867659 p^{2} T^{13} + 6773619566 p^{3} T^{14} + 590102751 p^{4} T^{15} + 48072505 p^{5} T^{16} + 3511029 p^{6} T^{17} + 236394 p^{7} T^{18} + 13585 p^{8} T^{19} + 713 p^{9} T^{20} + 26 p^{10} T^{21} + p^{11} T^{22} \)
83 \( 1 + 31 T + 611 T^{2} + 10730 T^{3} + 164653 T^{4} + 2177051 T^{5} + 26401659 T^{6} + 300406681 T^{7} + 3183266353 T^{8} + 31741364834 T^{9} + 306185009862 T^{10} + 2852573314610 T^{11} + 306185009862 p T^{12} + 31741364834 p^{2} T^{13} + 3183266353 p^{3} T^{14} + 300406681 p^{4} T^{15} + 26401659 p^{5} T^{16} + 2177051 p^{6} T^{17} + 164653 p^{7} T^{18} + 10730 p^{8} T^{19} + 611 p^{9} T^{20} + 31 p^{10} T^{21} + p^{11} T^{22} \)
89 \( 1 - 2 T + 596 T^{2} - 1695 T^{3} + 172400 T^{4} - 654480 T^{5} + 32360761 T^{6} - 150913266 T^{7} + 4465378150 T^{8} - 23067477511 T^{9} + 486095075877 T^{10} - 2447511368550 T^{11} + 486095075877 p T^{12} - 23067477511 p^{2} T^{13} + 4465378150 p^{3} T^{14} - 150913266 p^{4} T^{15} + 32360761 p^{5} T^{16} - 654480 p^{6} T^{17} + 172400 p^{7} T^{18} - 1695 p^{8} T^{19} + 596 p^{9} T^{20} - 2 p^{10} T^{21} + p^{11} T^{22} \)
97 \( 1 + 6 T + 157 T^{2} - 903 T^{3} + 8316 T^{4} - 139260 T^{5} + 1894990 T^{6} + 1885736 T^{7} + 293559165 T^{8} + 300116246 T^{9} + 9525182171 T^{10} - 132594826882 T^{11} + 9525182171 p T^{12} + 300116246 p^{2} T^{13} + 293559165 p^{3} T^{14} + 1885736 p^{4} T^{15} + 1894990 p^{5} T^{16} - 139260 p^{6} T^{17} + 8316 p^{7} T^{18} - 903 p^{8} T^{19} + 157 p^{9} T^{20} + 6 p^{10} T^{21} + p^{11} T^{22} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{22} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−2.98580043346744665196858015975, −2.85972697727490256506850838814, −2.84318681768069752678120844869, −2.57886791464118657542958795947, −2.37637450964850415531118444975, −2.34820392271169808717030094735, −2.34281524909761272755011259917, −2.33179910158826060240977524737, −2.28451496244898141683457338095, −2.26183415164715995717098064441, −2.24097641318711136695959758480, −2.16213816027521598313372587846, −2.06554818872864209692774015771, −1.90466366762016866848537460151, −1.61065428373010396336765740614, −1.59396526528548520263779594839, −1.57781969362899684004733331654, −1.52745346946479709337911809789, −1.40119889344467291967683755532, −1.21182295029578173048264851115, −1.00703752937106978641895816237, −1.00016009783383279374137696164, −0.982913487393907003824736734847, −0.982738684798633869226936440534, −0.865145367915235029067758848789, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.865145367915235029067758848789, 0.982738684798633869226936440534, 0.982913487393907003824736734847, 1.00016009783383279374137696164, 1.00703752937106978641895816237, 1.21182295029578173048264851115, 1.40119889344467291967683755532, 1.52745346946479709337911809789, 1.57781969362899684004733331654, 1.59396526528548520263779594839, 1.61065428373010396336765740614, 1.90466366762016866848537460151, 2.06554818872864209692774015771, 2.16213816027521598313372587846, 2.24097641318711136695959758480, 2.26183415164715995717098064441, 2.28451496244898141683457338095, 2.33179910158826060240977524737, 2.34281524909761272755011259917, 2.34820392271169808717030094735, 2.37637450964850415531118444975, 2.57886791464118657542958795947, 2.84318681768069752678120844869, 2.85972697727490256506850838814, 2.98580043346744665196858015975

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

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