Properties

Label 22-8034e11-1.1-c1e11-0-2
Degree $22$
Conductor $9.000\times 10^{42}$
Sign $-1$
Analytic cond. $7.57352\times 10^{19}$
Root an. cond. $8.00948$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $11$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 11·2-s − 11·3-s + 66·4-s − 2·5-s − 121·6-s − 2·7-s + 286·8-s + 66·9-s − 22·10-s − 10·11-s − 726·12-s − 11·13-s − 22·14-s + 22·15-s + 1.00e3·16-s + 6·17-s + 726·18-s − 3·19-s − 132·20-s + 22·21-s − 110·22-s + 3·23-s − 3.14e3·24-s − 26·25-s − 121·26-s − 286·27-s − 132·28-s + ⋯
L(s)  = 1  + 7.77·2-s − 6.35·3-s + 33·4-s − 0.894·5-s − 49.3·6-s − 0.755·7-s + 101.·8-s + 22·9-s − 6.95·10-s − 3.01·11-s − 209.·12-s − 3.05·13-s − 5.87·14-s + 5.68·15-s + 250.·16-s + 1.45·17-s + 171.·18-s − 0.688·19-s − 29.5·20-s + 4.80·21-s − 23.4·22-s + 0.625·23-s − 642.·24-s − 5.19·25-s − 23.7·26-s − 55.0·27-s − 24.9·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{11} \cdot 3^{11} \cdot 13^{11} \cdot 103^{11}\right)^{s/2} \, \Gamma_{\C}(s)^{11} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{11} \cdot 3^{11} \cdot 13^{11} \cdot 103^{11}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{11} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(22\)
Conductor: \(2^{11} \cdot 3^{11} \cdot 13^{11} \cdot 103^{11}\)
Sign: $-1$
Analytic conductor: \(7.57352\times 10^{19}\)
Root analytic conductor: \(8.00948\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(11\)
Selberg data: \((22,\ 2^{11} \cdot 3^{11} \cdot 13^{11} \cdot 103^{11} ,\ ( \ : [1/2]^{11} ),\ -1 )\)

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)$
bad2 \( ( 1 - T )^{11} \)
3 \( ( 1 + T )^{11} \)
13 \( ( 1 + T )^{11} \)
103 \( ( 1 - T )^{11} \)
good5 \( 1 + 2 T + 6 p T^{2} + 67 T^{3} + 471 T^{4} + 1071 T^{5} + 1011 p T^{6} + 2217 p T^{7} + 40648 T^{8} + 16657 p T^{9} + 255323 T^{10} + 475132 T^{11} + 255323 p T^{12} + 16657 p^{3} T^{13} + 40648 p^{3} T^{14} + 2217 p^{5} T^{15} + 1011 p^{6} T^{16} + 1071 p^{6} T^{17} + 471 p^{7} T^{18} + 67 p^{8} T^{19} + 6 p^{10} T^{20} + 2 p^{10} T^{21} + p^{11} T^{22} \)
7 \( 1 + 2 T + 43 T^{2} + 82 T^{3} + 970 T^{4} + 1728 T^{5} + 14860 T^{6} + 24539 T^{7} + 169492 T^{8} + 256250 T^{9} + 1501676 T^{10} + 2041294 T^{11} + 1501676 p T^{12} + 256250 p^{2} T^{13} + 169492 p^{3} T^{14} + 24539 p^{4} T^{15} + 14860 p^{5} T^{16} + 1728 p^{6} T^{17} + 970 p^{7} T^{18} + 82 p^{8} T^{19} + 43 p^{9} T^{20} + 2 p^{10} T^{21} + p^{11} T^{22} \)
11 \( 1 + 10 T + 101 T^{2} + 685 T^{3} + 4503 T^{4} + 24247 T^{5} + 125440 T^{6} + 565963 T^{7} + 2453623 T^{8} + 9530464 T^{9} + 35599326 T^{10} + 120402496 T^{11} + 35599326 p T^{12} + 9530464 p^{2} T^{13} + 2453623 p^{3} T^{14} + 565963 p^{4} T^{15} + 125440 p^{5} T^{16} + 24247 p^{6} T^{17} + 4503 p^{7} T^{18} + 685 p^{8} T^{19} + 101 p^{9} T^{20} + 10 p^{10} T^{21} + p^{11} T^{22} \)
17 \( 1 - 6 T + 9 p T^{2} - 773 T^{3} + 10899 T^{4} - 47519 T^{5} + 483464 T^{6} - 1841343 T^{7} + 14940230 T^{8} - 49860261 T^{9} + 338801625 T^{10} - 985794764 T^{11} + 338801625 p T^{12} - 49860261 p^{2} T^{13} + 14940230 p^{3} T^{14} - 1841343 p^{4} T^{15} + 483464 p^{5} T^{16} - 47519 p^{6} T^{17} + 10899 p^{7} T^{18} - 773 p^{8} T^{19} + 9 p^{10} T^{20} - 6 p^{10} T^{21} + p^{11} T^{22} \)
19 \( 1 + 3 T + 147 T^{2} + 26 p T^{3} + 10378 T^{4} + 36802 T^{5} + 470428 T^{6} + 1662087 T^{7} + 15325338 T^{8} + 51183015 T^{9} + 377523214 T^{10} + 1135877742 T^{11} + 377523214 p T^{12} + 51183015 p^{2} T^{13} + 15325338 p^{3} T^{14} + 1662087 p^{4} T^{15} + 470428 p^{5} T^{16} + 36802 p^{6} T^{17} + 10378 p^{7} T^{18} + 26 p^{9} T^{19} + 147 p^{9} T^{20} + 3 p^{10} T^{21} + p^{11} T^{22} \)
23 \( 1 - 3 T + 128 T^{2} - 336 T^{3} + 9146 T^{4} - 21852 T^{5} + 447249 T^{6} - 959985 T^{7} + 16471669 T^{8} - 31995863 T^{9} + 475003423 T^{10} - 825435562 T^{11} + 475003423 p T^{12} - 31995863 p^{2} T^{13} + 16471669 p^{3} T^{14} - 959985 p^{4} T^{15} + 447249 p^{5} T^{16} - 21852 p^{6} T^{17} + 9146 p^{7} T^{18} - 336 p^{8} T^{19} + 128 p^{9} T^{20} - 3 p^{10} T^{21} + p^{11} T^{22} \)
29 \( 1 + 22 T + 409 T^{2} + 5193 T^{3} + 58002 T^{4} + 530702 T^{5} + 4411942 T^{6} + 32023646 T^{7} + 217127200 T^{8} + 1334856936 T^{9} + 7862310242 T^{10} + 43001382826 T^{11} + 7862310242 p T^{12} + 1334856936 p^{2} T^{13} + 217127200 p^{3} T^{14} + 32023646 p^{4} T^{15} + 4411942 p^{5} T^{16} + 530702 p^{6} T^{17} + 58002 p^{7} T^{18} + 5193 p^{8} T^{19} + 409 p^{9} T^{20} + 22 p^{10} T^{21} + p^{11} T^{22} \)
31 \( 1 + 5 T + 201 T^{2} + 1154 T^{3} + 21418 T^{4} + 124918 T^{5} + 1544180 T^{6} + 8550937 T^{7} + 81914778 T^{8} + 412754431 T^{9} + 3302157742 T^{10} + 14763067030 T^{11} + 3302157742 p T^{12} + 412754431 p^{2} T^{13} + 81914778 p^{3} T^{14} + 8550937 p^{4} T^{15} + 1544180 p^{5} T^{16} + 124918 p^{6} T^{17} + 21418 p^{7} T^{18} + 1154 p^{8} T^{19} + 201 p^{9} T^{20} + 5 p^{10} T^{21} + p^{11} T^{22} \)
37 \( 1 + 26 T + 489 T^{2} + 6595 T^{3} + 76490 T^{4} + 754780 T^{5} + 6795368 T^{6} + 55178850 T^{7} + 419200900 T^{8} + 2943564386 T^{9} + 19593332962 T^{10} + 122071212918 T^{11} + 19593332962 p T^{12} + 2943564386 p^{2} T^{13} + 419200900 p^{3} T^{14} + 55178850 p^{4} T^{15} + 6795368 p^{5} T^{16} + 754780 p^{6} T^{17} + 76490 p^{7} T^{18} + 6595 p^{8} T^{19} + 489 p^{9} T^{20} + 26 p^{10} T^{21} + p^{11} T^{22} \)
41 \( 1 + 6 T + 154 T^{2} + 1292 T^{3} + 16131 T^{4} + 133430 T^{5} + 1259185 T^{6} + 9487247 T^{7} + 78059976 T^{8} + 521336780 T^{9} + 3892073849 T^{10} + 23498933434 T^{11} + 3892073849 p T^{12} + 521336780 p^{2} T^{13} + 78059976 p^{3} T^{14} + 9487247 p^{4} T^{15} + 1259185 p^{5} T^{16} + 133430 p^{6} T^{17} + 16131 p^{7} T^{18} + 1292 p^{8} T^{19} + 154 p^{9} T^{20} + 6 p^{10} T^{21} + p^{11} T^{22} \)
43 \( 1 + 8 T + 257 T^{2} + 1922 T^{3} + 34431 T^{4} + 229570 T^{5} + 3057642 T^{6} + 18257673 T^{7} + 202337237 T^{8} + 1086725122 T^{9} + 10622872958 T^{10} + 51749930850 T^{11} + 10622872958 p T^{12} + 1086725122 p^{2} T^{13} + 202337237 p^{3} T^{14} + 18257673 p^{4} T^{15} + 3057642 p^{5} T^{16} + 229570 p^{6} T^{17} + 34431 p^{7} T^{18} + 1922 p^{8} T^{19} + 257 p^{9} T^{20} + 8 p^{10} T^{21} + p^{11} T^{22} \)
47 \( 1 - 6 T + 337 T^{2} - 1402 T^{3} + 52230 T^{4} - 144994 T^{5} + 5153116 T^{6} - 9092673 T^{7} + 374217553 T^{8} - 422895210 T^{9} + 21532147431 T^{10} - 18757789270 T^{11} + 21532147431 p T^{12} - 422895210 p^{2} T^{13} + 374217553 p^{3} T^{14} - 9092673 p^{4} T^{15} + 5153116 p^{5} T^{16} - 144994 p^{6} T^{17} + 52230 p^{7} T^{18} - 1402 p^{8} T^{19} + 337 p^{9} T^{20} - 6 p^{10} T^{21} + p^{11} T^{22} \)
53 \( 1 + 25 T + 619 T^{2} + 9855 T^{3} + 149099 T^{4} + 1801485 T^{5} + 20817330 T^{6} + 206810763 T^{7} + 1981169116 T^{8} + 16841786942 T^{9} + 138651743555 T^{10} + 1025154356308 T^{11} + 138651743555 p T^{12} + 16841786942 p^{2} T^{13} + 1981169116 p^{3} T^{14} + 206810763 p^{4} T^{15} + 20817330 p^{5} T^{16} + 1801485 p^{6} T^{17} + 149099 p^{7} T^{18} + 9855 p^{8} T^{19} + 619 p^{9} T^{20} + 25 p^{10} T^{21} + p^{11} T^{22} \)
59 \( 1 - 7 T + 512 T^{2} - 2885 T^{3} + 122284 T^{4} - 568155 T^{5} + 18296575 T^{6} - 71677749 T^{7} + 1924958012 T^{8} - 6476214306 T^{9} + 150095875310 T^{10} - 438638498468 T^{11} + 150095875310 p T^{12} - 6476214306 p^{2} T^{13} + 1924958012 p^{3} T^{14} - 71677749 p^{4} T^{15} + 18296575 p^{5} T^{16} - 568155 p^{6} T^{17} + 122284 p^{7} T^{18} - 2885 p^{8} T^{19} + 512 p^{9} T^{20} - 7 p^{10} T^{21} + p^{11} T^{22} \)
61 \( 1 + 36 T + 927 T^{2} + 17038 T^{3} + 258799 T^{4} + 3230440 T^{5} + 35002066 T^{6} + 326929975 T^{7} + 2749040241 T^{8} + 20994526136 T^{9} + 156817014258 T^{10} + 1183809324718 T^{11} + 156817014258 p T^{12} + 20994526136 p^{2} T^{13} + 2749040241 p^{3} T^{14} + 326929975 p^{4} T^{15} + 35002066 p^{5} T^{16} + 3230440 p^{6} T^{17} + 258799 p^{7} T^{18} + 17038 p^{8} T^{19} + 927 p^{9} T^{20} + 36 p^{10} T^{21} + p^{11} T^{22} \)
67 \( 1 + 12 T + 450 T^{2} + 4618 T^{3} + 99845 T^{4} + 882603 T^{5} + 14367617 T^{6} + 111359650 T^{7} + 22585423 p T^{8} + 10459786639 T^{9} + 124986111174 T^{10} + 778694940968 T^{11} + 124986111174 p T^{12} + 10459786639 p^{2} T^{13} + 22585423 p^{4} T^{14} + 111359650 p^{4} T^{15} + 14367617 p^{5} T^{16} + 882603 p^{6} T^{17} + 99845 p^{7} T^{18} + 4618 p^{8} T^{19} + 450 p^{9} T^{20} + 12 p^{10} T^{21} + p^{11} T^{22} \)
71 \( 1 + 15 T + 460 T^{2} + 5709 T^{3} + 101331 T^{4} + 1059170 T^{5} + 14426384 T^{6} + 132046005 T^{7} + 1533650249 T^{8} + 12679228673 T^{9} + 131483903629 T^{10} + 991178313608 T^{11} + 131483903629 p T^{12} + 12679228673 p^{2} T^{13} + 1533650249 p^{3} T^{14} + 132046005 p^{4} T^{15} + 14426384 p^{5} T^{16} + 1059170 p^{6} T^{17} + 101331 p^{7} T^{18} + 5709 p^{8} T^{19} + 460 p^{9} T^{20} + 15 p^{10} T^{21} + p^{11} T^{22} \)
73 \( 1 + 12 T + 269 T^{2} + 2810 T^{3} + 34616 T^{4} + 276559 T^{5} + 2814362 T^{6} + 17330908 T^{7} + 165152680 T^{8} + 868190313 T^{9} + 9233347874 T^{10} + 47442325372 T^{11} + 9233347874 p T^{12} + 868190313 p^{2} T^{13} + 165152680 p^{3} T^{14} + 17330908 p^{4} T^{15} + 2814362 p^{5} T^{16} + 276559 p^{6} T^{17} + 34616 p^{7} T^{18} + 2810 p^{8} T^{19} + 269 p^{9} T^{20} + 12 p^{10} T^{21} + p^{11} T^{22} \)
79 \( 1 + 15 T + 656 T^{2} + 8710 T^{3} + 203361 T^{4} + 2420570 T^{5} + 39740037 T^{6} + 425366961 T^{7} + 5486053746 T^{8} + 52595388347 T^{9} + 565914477663 T^{10} + 4801330205914 T^{11} + 565914477663 p T^{12} + 52595388347 p^{2} T^{13} + 5486053746 p^{3} T^{14} + 425366961 p^{4} T^{15} + 39740037 p^{5} T^{16} + 2420570 p^{6} T^{17} + 203361 p^{7} T^{18} + 8710 p^{8} T^{19} + 656 p^{9} T^{20} + 15 p^{10} T^{21} + p^{11} T^{22} \)
83 \( 1 + 16 T + 472 T^{2} + 4401 T^{3} + 88401 T^{4} + 562166 T^{5} + 11222094 T^{6} + 50531917 T^{7} + 1137435267 T^{8} + 3270833838 T^{9} + 97068555563 T^{10} + 205318968620 T^{11} + 97068555563 p T^{12} + 3270833838 p^{2} T^{13} + 1137435267 p^{3} T^{14} + 50531917 p^{4} T^{15} + 11222094 p^{5} T^{16} + 562166 p^{6} T^{17} + 88401 p^{7} T^{18} + 4401 p^{8} T^{19} + 472 p^{9} T^{20} + 16 p^{10} T^{21} + p^{11} T^{22} \)
89 \( 1 + 2 T + 373 T^{2} + 1730 T^{3} + 59328 T^{4} + 386819 T^{5} + 5352366 T^{6} + 29357950 T^{7} + 303165090 T^{8} - 1063103067 T^{9} + 12325808272 T^{10} - 302795014824 T^{11} + 12325808272 p T^{12} - 1063103067 p^{2} T^{13} + 303165090 p^{3} T^{14} + 29357950 p^{4} T^{15} + 5352366 p^{5} T^{16} + 386819 p^{6} T^{17} + 59328 p^{7} T^{18} + 1730 p^{8} T^{19} + 373 p^{9} T^{20} + 2 p^{10} T^{21} + p^{11} T^{22} \)
97 \( 1 + 10 T + 668 T^{2} + 5964 T^{3} + 209863 T^{4} + 1666711 T^{5} + 42169765 T^{6} + 297153066 T^{7} + 6236322519 T^{8} + 39154729327 T^{9} + 733840332424 T^{10} + 4162266502532 T^{11} + 733840332424 p T^{12} + 39154729327 p^{2} T^{13} + 6236322519 p^{3} T^{14} + 297153066 p^{4} T^{15} + 42169765 p^{5} T^{16} + 1666711 p^{6} T^{17} + 209863 p^{7} T^{18} + 5964 p^{8} T^{19} + 668 p^{9} T^{20} + 10 p^{10} T^{21} + p^{11} T^{22} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{22} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−3.01869706729069337872557284394, −2.99747411698235781414868507446, −2.99515740248746697949709939939, −2.67336309263929383748080997621, −2.66287937139959167494658779357, −2.48440293351693844847980025565, −2.34935331469920232871820061221, −2.34695765209119808724520174452, −2.27109549342622252103139416466, −2.17974634778983866313494296360, −2.17149545501427469266534759467, −2.14470652192325889387839791027, −2.10035994361565140606792791422, −2.07095594288929562344988755458, −1.68524919142995241822381682709, −1.56433582282428894712971783978, −1.51868004871217650689192651200, −1.48330135607175936103693692391, −1.35934981427904192063501542386, −1.33592593104068917491414567706, −1.33413944209964897072141055424, −1.31527434709106754142443480091, −1.26134295463816242756421831972, −1.21952918416460719160640352707, −0.979063179060467373095483688087, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.979063179060467373095483688087, 1.21952918416460719160640352707, 1.26134295463816242756421831972, 1.31527434709106754142443480091, 1.33413944209964897072141055424, 1.33592593104068917491414567706, 1.35934981427904192063501542386, 1.48330135607175936103693692391, 1.51868004871217650689192651200, 1.56433582282428894712971783978, 1.68524919142995241822381682709, 2.07095594288929562344988755458, 2.10035994361565140606792791422, 2.14470652192325889387839791027, 2.17149545501427469266534759467, 2.17974634778983866313494296360, 2.27109549342622252103139416466, 2.34695765209119808724520174452, 2.34935331469920232871820061221, 2.48440293351693844847980025565, 2.66287937139959167494658779357, 2.67336309263929383748080997621, 2.99515740248746697949709939939, 2.99747411698235781414868507446, 3.01869706729069337872557284394

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

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