Properties

Degree $22$
Conductor $3.648\times 10^{41}$
Sign $-1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $11$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 2·5-s + 3·7-s − 2·8-s + 4·10-s − 11·11-s − 5·13-s − 6·14-s + 4·16-s − 15·17-s − 6·19-s + 22·22-s − 11·23-s − 24·25-s + 10·26-s + 11·29-s + 35·31-s + 8·32-s + 30·34-s − 6·35-s − 28·37-s + 12·38-s + 4·40-s − 10·41-s − 6·43-s + 22·46-s − 15·47-s + ⋯
L(s)  = 1  − 1.41·2-s − 0.894·5-s + 1.13·7-s − 0.707·8-s + 1.26·10-s − 3.31·11-s − 1.38·13-s − 1.60·14-s + 16-s − 3.63·17-s − 1.37·19-s + 4.69·22-s − 2.29·23-s − 4.79·25-s + 1.96·26-s + 2.04·29-s + 6.28·31-s + 1.41·32-s + 5.14·34-s − 1.01·35-s − 4.60·37-s + 1.94·38-s + 0.632·40-s − 1.56·41-s − 0.914·43-s + 3.24·46-s − 2.18·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{22} \cdot 23^{11} \cdot 29^{11}\right)^{s/2} \, \Gamma_{\C}(s)^{11} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{22} \cdot 23^{11} \cdot 29^{11}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{11} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(22\)
Conductor: \(3^{22} \cdot 23^{11} \cdot 29^{11}\)
Sign: $-1$
Motivic weight: \(1\)
Character: induced by $\chi_{6003} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(11\)
Selberg data: \((22,\ 3^{22} \cdot 23^{11} \cdot 29^{11} ,\ ( \ : [1/2]^{11} ),\ -1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( ( 1 + T )^{11} \)
29 \( ( 1 - T )^{11} \)
good2 \( 1 + p T + p^{2} T^{2} + 5 p T^{3} + 5 p^{2} T^{4} + p^{5} T^{5} + 7 p^{3} T^{6} + 99 T^{7} + 77 p T^{8} + 213 T^{9} + 331 T^{10} + 61 p^{3} T^{11} + 331 p T^{12} + 213 p^{2} T^{13} + 77 p^{4} T^{14} + 99 p^{4} T^{15} + 7 p^{8} T^{16} + p^{11} T^{17} + 5 p^{9} T^{18} + 5 p^{9} T^{19} + p^{11} T^{20} + p^{11} T^{21} + p^{11} T^{22} \)
5 \( 1 + 2 T + 28 T^{2} + 47 T^{3} + 393 T^{4} + 517 T^{5} + 3618 T^{6} + 3614 T^{7} + 25024 T^{8} + 767 p^{2} T^{9} + 143096 T^{10} + 94058 T^{11} + 143096 p T^{12} + 767 p^{4} T^{13} + 25024 p^{3} T^{14} + 3614 p^{4} T^{15} + 3618 p^{5} T^{16} + 517 p^{6} T^{17} + 393 p^{7} T^{18} + 47 p^{8} T^{19} + 28 p^{9} T^{20} + 2 p^{10} T^{21} + p^{11} T^{22} \)
7 \( 1 - 3 T + 32 T^{2} - 86 T^{3} + 459 T^{4} - 1013 T^{5} + 4113 T^{6} - 6717 T^{7} + 27974 T^{8} - 34932 T^{9} + 175069 T^{10} - 213418 T^{11} + 175069 p T^{12} - 34932 p^{2} T^{13} + 27974 p^{3} T^{14} - 6717 p^{4} T^{15} + 4113 p^{5} T^{16} - 1013 p^{6} T^{17} + 459 p^{7} T^{18} - 86 p^{8} T^{19} + 32 p^{9} T^{20} - 3 p^{10} T^{21} + p^{11} T^{22} \)
11 \( 1 + p T + 107 T^{2} + 665 T^{3} + 3853 T^{4} + 17357 T^{5} + 77380 T^{6} + 26597 p T^{7} + 1161868 T^{8} + 4068001 T^{9} + 15108568 T^{10} + 48975106 T^{11} + 15108568 p T^{12} + 4068001 p^{2} T^{13} + 1161868 p^{3} T^{14} + 26597 p^{5} T^{15} + 77380 p^{5} T^{16} + 17357 p^{6} T^{17} + 3853 p^{7} T^{18} + 665 p^{8} T^{19} + 107 p^{9} T^{20} + p^{11} T^{21} + p^{11} T^{22} \)
13 \( 1 + 5 T + 84 T^{2} + 350 T^{3} + 3206 T^{4} + 11152 T^{5} + 72699 T^{6} + 212361 T^{7} + 86289 p T^{8} + 2859915 T^{9} + 14089132 T^{10} + 35296207 T^{11} + 14089132 p T^{12} + 2859915 p^{2} T^{13} + 86289 p^{4} T^{14} + 212361 p^{4} T^{15} + 72699 p^{5} T^{16} + 11152 p^{6} T^{17} + 3206 p^{7} T^{18} + 350 p^{8} T^{19} + 84 p^{9} T^{20} + 5 p^{10} T^{21} + p^{11} T^{22} \)
17 \( 1 + 15 T + 11 p T^{2} + 1517 T^{3} + 10634 T^{4} + 56023 T^{5} + 250369 T^{6} + 755250 T^{7} + 1190337 T^{8} - 6597342 T^{9} - 60323812 T^{10} - 321154542 T^{11} - 60323812 p T^{12} - 6597342 p^{2} T^{13} + 1190337 p^{3} T^{14} + 755250 p^{4} T^{15} + 250369 p^{5} T^{16} + 56023 p^{6} T^{17} + 10634 p^{7} T^{18} + 1517 p^{8} T^{19} + 11 p^{10} T^{20} + 15 p^{10} T^{21} + p^{11} T^{22} \)
19 \( 1 + 6 T + 80 T^{2} + 576 T^{3} + 4431 T^{4} + 26764 T^{5} + 174475 T^{6} + 923308 T^{7} + 5030336 T^{8} + 24438270 T^{9} + 118842117 T^{10} + 508708424 T^{11} + 118842117 p T^{12} + 24438270 p^{2} T^{13} + 5030336 p^{3} T^{14} + 923308 p^{4} T^{15} + 174475 p^{5} T^{16} + 26764 p^{6} T^{17} + 4431 p^{7} T^{18} + 576 p^{8} T^{19} + 80 p^{9} T^{20} + 6 p^{10} T^{21} + p^{11} T^{22} \)
31 \( 1 - 35 T + 740 T^{2} - 374 p T^{3} + 147509 T^{4} - 1594455 T^{5} + 15077700 T^{6} - 127032607 T^{7} + 965714594 T^{8} - 6680895592 T^{9} + 42287192376 T^{10} - 245648352538 T^{11} + 42287192376 p T^{12} - 6680895592 p^{2} T^{13} + 965714594 p^{3} T^{14} - 127032607 p^{4} T^{15} + 15077700 p^{5} T^{16} - 1594455 p^{6} T^{17} + 147509 p^{7} T^{18} - 374 p^{9} T^{19} + 740 p^{9} T^{20} - 35 p^{10} T^{21} + p^{11} T^{22} \)
37 \( 1 + 28 T + 617 T^{2} + 9207 T^{3} + 118856 T^{4} + 1241639 T^{5} + 11728670 T^{6} + 95688335 T^{7} + 730296367 T^{8} + 5015228404 T^{9} + 33274323831 T^{10} + 204290941550 T^{11} + 33274323831 p T^{12} + 5015228404 p^{2} T^{13} + 730296367 p^{3} T^{14} + 95688335 p^{4} T^{15} + 11728670 p^{5} T^{16} + 1241639 p^{6} T^{17} + 118856 p^{7} T^{18} + 9207 p^{8} T^{19} + 617 p^{9} T^{20} + 28 p^{10} T^{21} + p^{11} T^{22} \)
41 \( 1 + 10 T + 228 T^{2} + 1439 T^{3} + 21527 T^{4} + 100809 T^{5} + 1421858 T^{6} + 5953934 T^{7} + 80939232 T^{8} + 313060821 T^{9} + 3868196546 T^{10} + 13739665526 T^{11} + 3868196546 p T^{12} + 313060821 p^{2} T^{13} + 80939232 p^{3} T^{14} + 5953934 p^{4} T^{15} + 1421858 p^{5} T^{16} + 100809 p^{6} T^{17} + 21527 p^{7} T^{18} + 1439 p^{8} T^{19} + 228 p^{9} T^{20} + 10 p^{10} T^{21} + p^{11} T^{22} \)
43 \( 1 + 6 T + 249 T^{2} + 1421 T^{3} + 32256 T^{4} + 174058 T^{5} + 2861763 T^{6} + 14515904 T^{7} + 192474612 T^{8} + 905291064 T^{9} + 10258637493 T^{10} + 43907837686 T^{11} + 10258637493 p T^{12} + 905291064 p^{2} T^{13} + 192474612 p^{3} T^{14} + 14515904 p^{4} T^{15} + 2861763 p^{5} T^{16} + 174058 p^{6} T^{17} + 32256 p^{7} T^{18} + 1421 p^{8} T^{19} + 249 p^{9} T^{20} + 6 p^{10} T^{21} + p^{11} T^{22} \)
47 \( 1 + 15 T + 243 T^{2} + 2292 T^{3} + 28624 T^{4} + 260365 T^{5} + 2746148 T^{6} + 21501459 T^{7} + 190723757 T^{8} + 1362389332 T^{9} + 11104441367 T^{10} + 72604894898 T^{11} + 11104441367 p T^{12} + 1362389332 p^{2} T^{13} + 190723757 p^{3} T^{14} + 21501459 p^{4} T^{15} + 2746148 p^{5} T^{16} + 260365 p^{6} T^{17} + 28624 p^{7} T^{18} + 2292 p^{8} T^{19} + 243 p^{9} T^{20} + 15 p^{10} T^{21} + p^{11} T^{22} \)
53 \( 1 - 7 T + 252 T^{2} - 1046 T^{3} + 30049 T^{4} - 56128 T^{5} + 2444244 T^{6} - 432858 T^{7} + 168007971 T^{8} + 112638471 T^{9} + 10281805295 T^{10} + 8391601472 T^{11} + 10281805295 p T^{12} + 112638471 p^{2} T^{13} + 168007971 p^{3} T^{14} - 432858 p^{4} T^{15} + 2444244 p^{5} T^{16} - 56128 p^{6} T^{17} + 30049 p^{7} T^{18} - 1046 p^{8} T^{19} + 252 p^{9} T^{20} - 7 p^{10} T^{21} + p^{11} T^{22} \)
59 \( 1 - 20 T + 459 T^{2} - 6128 T^{3} + 81930 T^{4} - 851924 T^{5} + 8552108 T^{6} - 76180464 T^{7} + 645277215 T^{8} - 5291221496 T^{9} + 40997839179 T^{10} - 324105866432 T^{11} + 40997839179 p T^{12} - 5291221496 p^{2} T^{13} + 645277215 p^{3} T^{14} - 76180464 p^{4} T^{15} + 8552108 p^{5} T^{16} - 851924 p^{6} T^{17} + 81930 p^{7} T^{18} - 6128 p^{8} T^{19} + 459 p^{9} T^{20} - 20 p^{10} T^{21} + p^{11} T^{22} \)
61 \( 1 + 20 T + 539 T^{2} + 8475 T^{3} + 137126 T^{4} + 1767407 T^{5} + 21877875 T^{6} + 237530632 T^{7} + 2435554719 T^{8} + 22657355053 T^{9} + 198679093064 T^{10} + 1597886764642 T^{11} + 198679093064 p T^{12} + 22657355053 p^{2} T^{13} + 2435554719 p^{3} T^{14} + 237530632 p^{4} T^{15} + 21877875 p^{5} T^{16} + 1767407 p^{6} T^{17} + 137126 p^{7} T^{18} + 8475 p^{8} T^{19} + 539 p^{9} T^{20} + 20 p^{10} T^{21} + p^{11} T^{22} \)
67 \( 1 + 39 T + 16 p T^{2} + 20960 T^{3} + 340805 T^{4} + 4604354 T^{5} + 54800454 T^{6} + 574727854 T^{7} + 5531614723 T^{8} + 49152332927 T^{9} + 419703286294 T^{10} + 3456693252772 T^{11} + 419703286294 p T^{12} + 49152332927 p^{2} T^{13} + 5531614723 p^{3} T^{14} + 574727854 p^{4} T^{15} + 54800454 p^{5} T^{16} + 4604354 p^{6} T^{17} + 340805 p^{7} T^{18} + 20960 p^{8} T^{19} + 16 p^{10} T^{20} + 39 p^{10} T^{21} + p^{11} T^{22} \)
71 \( 1 + 49 T + 1561 T^{2} + 35931 T^{3} + 670365 T^{4} + 10461619 T^{5} + 141823527 T^{6} + 1699994043 T^{7} + 18416834909 T^{8} + 182866016817 T^{9} + 1690497985987 T^{10} + 14666905550090 T^{11} + 1690497985987 p T^{12} + 182866016817 p^{2} T^{13} + 18416834909 p^{3} T^{14} + 1699994043 p^{4} T^{15} + 141823527 p^{5} T^{16} + 10461619 p^{6} T^{17} + 670365 p^{7} T^{18} + 35931 p^{8} T^{19} + 1561 p^{9} T^{20} + 49 p^{10} T^{21} + p^{11} T^{22} \)
73 \( 1 + 3 T + 196 T^{2} - 159 T^{3} + 31858 T^{4} - 26254 T^{5} + 3900187 T^{6} - 6625858 T^{7} + 402069132 T^{8} - 632169893 T^{9} + 34172795610 T^{10} - 62902441758 T^{11} + 34172795610 p T^{12} - 632169893 p^{2} T^{13} + 402069132 p^{3} T^{14} - 6625858 p^{4} T^{15} + 3900187 p^{5} T^{16} - 26254 p^{6} T^{17} + 31858 p^{7} T^{18} - 159 p^{8} T^{19} + 196 p^{9} T^{20} + 3 p^{10} T^{21} + p^{11} T^{22} \)
79 \( 1 - 41 T + 1189 T^{2} - 26193 T^{3} + 483626 T^{4} - 7709044 T^{5} + 109460517 T^{6} - 1399418794 T^{7} + 16364772946 T^{8} - 175822723067 T^{9} + 1749124610179 T^{10} - 16126404005754 T^{11} + 1749124610179 p T^{12} - 175822723067 p^{2} T^{13} + 16364772946 p^{3} T^{14} - 1399418794 p^{4} T^{15} + 109460517 p^{5} T^{16} - 7709044 p^{6} T^{17} + 483626 p^{7} T^{18} - 26193 p^{8} T^{19} + 1189 p^{9} T^{20} - 41 p^{10} T^{21} + p^{11} T^{22} \)
83 \( 1 + 13 T + 502 T^{2} + 5300 T^{3} + 117049 T^{4} + 987588 T^{5} + 16724796 T^{6} + 110612382 T^{7} + 1685329611 T^{8} + 8817785535 T^{9} + 140832263751 T^{10} + 668486929788 T^{11} + 140832263751 p T^{12} + 8817785535 p^{2} T^{13} + 1685329611 p^{3} T^{14} + 110612382 p^{4} T^{15} + 16724796 p^{5} T^{16} + 987588 p^{6} T^{17} + 117049 p^{7} T^{18} + 5300 p^{8} T^{19} + 502 p^{9} T^{20} + 13 p^{10} T^{21} + p^{11} T^{22} \)
89 \( 1 + 34 T + 1201 T^{2} + 26387 T^{3} + 559512 T^{4} + 9300775 T^{5} + 147852867 T^{6} + 1992121814 T^{7} + 25687764507 T^{8} + 290370740809 T^{9} + 3147129733888 T^{10} + 30308852282314 T^{11} + 3147129733888 p T^{12} + 290370740809 p^{2} T^{13} + 25687764507 p^{3} T^{14} + 1992121814 p^{4} T^{15} + 147852867 p^{5} T^{16} + 9300775 p^{6} T^{17} + 559512 p^{7} T^{18} + 26387 p^{8} T^{19} + 1201 p^{9} T^{20} + 34 p^{10} T^{21} + p^{11} T^{22} \)
97 \( 1 + 11 T + 727 T^{2} + 8119 T^{3} + 252528 T^{4} + 2751124 T^{5} + 56605078 T^{6} + 574477674 T^{7} + 9226377413 T^{8} + 84145091209 T^{9} + 1150431401949 T^{10} + 9286762453998 T^{11} + 1150431401949 p T^{12} + 84145091209 p^{2} T^{13} + 9226377413 p^{3} T^{14} + 574477674 p^{4} T^{15} + 56605078 p^{5} T^{16} + 2751124 p^{6} T^{17} + 252528 p^{7} T^{18} + 8119 p^{8} T^{19} + 727 p^{9} T^{20} + 11 p^{10} T^{21} + p^{11} T^{22} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{22} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−2.91798540377670219941827839406, −2.88663363061383700615911942346, −2.85447807563427750916350283076, −2.70250782972099157010735829662, −2.53545760016929561265167301832, −2.50488563267297068479498272611, −2.49351217249105565231915563362, −2.47641730216395080393316494665, −2.37562826492444915867159441044, −2.24772796489840050780177183489, −2.21249200135021101925451740005, −2.01234635076835686333223910967, −1.98066885143951786097632502445, −1.97584307673730374560100598883, −1.92542415422370635278339264144, −1.76668711289421790111450920079, −1.53681641000682454756543949449, −1.48696621665579096522315409130, −1.45662568297017411635798652251, −1.31300442791953564075274872658, −1.18090658188474038834695081915, −1.11166536771958539267378662538, −1.06848718877325941063721111358, −1.03515048346275653796417884093, −0.77069239605553558716634987342, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.77069239605553558716634987342, 1.03515048346275653796417884093, 1.06848718877325941063721111358, 1.11166536771958539267378662538, 1.18090658188474038834695081915, 1.31300442791953564075274872658, 1.45662568297017411635798652251, 1.48696621665579096522315409130, 1.53681641000682454756543949449, 1.76668711289421790111450920079, 1.92542415422370635278339264144, 1.97584307673730374560100598883, 1.98066885143951786097632502445, 2.01234635076835686333223910967, 2.21249200135021101925451740005, 2.24772796489840050780177183489, 2.37562826492444915867159441044, 2.47641730216395080393316494665, 2.49351217249105565231915563362, 2.50488563267297068479498272611, 2.53545760016929561265167301832, 2.70250782972099157010735829662, 2.85447807563427750916350283076, 2.88663363061383700615911942346, 2.91798540377670219941827839406

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

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