Properties

Label 22-2013e11-1.1-c1e11-0-0
Degree $22$
Conductor $2.199\times 10^{36}$
Sign $-1$
Analytic cond. $1.85067\times 10^{13}$
Root an. cond. $4.00922$
Motivic weight $1$
Arithmetic yes
Rational yes
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 − 11·3-s − 4·4-s − 5-s + 22·6-s − 11·7-s + 11·8-s + 66·9-s + 2·10-s + 11·11-s + 44·12-s − 13·13-s + 22·14-s + 11·15-s + 2·16-s − 13·17-s − 132·18-s − 12·19-s + 4·20-s + 121·21-s − 22·22-s − 3·23-s − 121·24-s − 21·25-s + 26·26-s − 286·27-s + 44·28-s + ⋯
L(s)  = 1  − 1.41·2-s − 6.35·3-s − 2·4-s − 0.447·5-s + 8.98·6-s − 4.15·7-s + 3.88·8-s + 22·9-s + 0.632·10-s + 3.31·11-s + 12.7·12-s − 3.60·13-s + 5.87·14-s + 2.84·15-s + 1/2·16-s − 3.15·17-s − 31.1·18-s − 2.75·19-s + 0.894·20-s + 26.4·21-s − 4.69·22-s − 0.625·23-s − 24.6·24-s − 4.19·25-s + 5.09·26-s − 55.0·27-s + 8.31·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{11} \cdot 11^{11} \cdot 61^{11}\right)^{s/2} \, \Gamma_{\C}(s)^{11} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{11} \cdot 11^{11} \cdot 61^{11}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{11} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(22\)
Conductor: \(3^{11} \cdot 11^{11} \cdot 61^{11}\)
Sign: $-1$
Analytic conductor: \(1.85067\times 10^{13}\)
Root analytic conductor: \(4.00922\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(11\)
Selberg data: \((22,\ 3^{11} \cdot 11^{11} \cdot 61^{11} ,\ ( \ : [1/2]^{11} ),\ -1 )\)

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 \( ( 1 + T )^{11} \)
11 \( ( 1 - T )^{11} \)
61 \( ( 1 - T )^{11} \)
good2 \( 1 + p T + p^{3} T^{2} + 13 T^{3} + 17 p T^{4} + 53 T^{5} + 113 T^{6} + 169 T^{7} + 153 p T^{8} + 437 T^{9} + 353 p T^{10} + 957 T^{11} + 353 p^{2} T^{12} + 437 p^{2} T^{13} + 153 p^{4} T^{14} + 169 p^{4} T^{15} + 113 p^{5} T^{16} + 53 p^{6} T^{17} + 17 p^{8} T^{18} + 13 p^{8} T^{19} + p^{12} T^{20} + p^{11} T^{21} + p^{11} T^{22} \)
5 \( 1 + T + 22 T^{2} + 2 p T^{3} + 2 p^{3} T^{4} + 92 T^{5} + 2061 T^{6} + 941 T^{7} + 13377 T^{8} + 8028 T^{9} + 75394 T^{10} + 50051 T^{11} + 75394 p T^{12} + 8028 p^{2} T^{13} + 13377 p^{3} T^{14} + 941 p^{4} T^{15} + 2061 p^{5} T^{16} + 92 p^{6} T^{17} + 2 p^{10} T^{18} + 2 p^{9} T^{19} + 22 p^{9} T^{20} + p^{10} T^{21} + p^{11} T^{22} \)
7 \( 1 + 11 T + 108 T^{2} + 729 T^{3} + 4381 T^{4} + 21873 T^{5} + 98878 T^{6} + 393345 T^{7} + 1434035 T^{8} + 4707756 T^{9} + 14259573 T^{10} + 39234779 T^{11} + 14259573 p T^{12} + 4707756 p^{2} T^{13} + 1434035 p^{3} T^{14} + 393345 p^{4} T^{15} + 98878 p^{5} T^{16} + 21873 p^{6} T^{17} + 4381 p^{7} T^{18} + 729 p^{8} T^{19} + 108 p^{9} T^{20} + 11 p^{10} T^{21} + p^{11} T^{22} \)
13 \( 1 + p T + 150 T^{2} + 1197 T^{3} + 8477 T^{4} + 50644 T^{5} + 275926 T^{6} + 1352043 T^{7} + 6176229 T^{8} + 26038009 T^{9} + 7969160 p T^{10} + 383806651 T^{11} + 7969160 p^{2} T^{12} + 26038009 p^{2} T^{13} + 6176229 p^{3} T^{14} + 1352043 p^{4} T^{15} + 275926 p^{5} T^{16} + 50644 p^{6} T^{17} + 8477 p^{7} T^{18} + 1197 p^{8} T^{19} + 150 p^{9} T^{20} + p^{11} T^{21} + p^{11} T^{22} \)
17 \( 1 + 13 T + 156 T^{2} + 80 p T^{3} + 10974 T^{4} + 74673 T^{5} + 474033 T^{6} + 2699445 T^{7} + 846979 p T^{8} + 70304093 T^{9} + 323291325 T^{10} + 1374162515 T^{11} + 323291325 p T^{12} + 70304093 p^{2} T^{13} + 846979 p^{4} T^{14} + 2699445 p^{4} T^{15} + 474033 p^{5} T^{16} + 74673 p^{6} T^{17} + 10974 p^{7} T^{18} + 80 p^{9} T^{19} + 156 p^{9} T^{20} + 13 p^{10} T^{21} + p^{11} T^{22} \)
19 \( 1 + 12 T + 192 T^{2} + 1600 T^{3} + 14502 T^{4} + 4952 p T^{5} + 619600 T^{6} + 3372332 T^{7} + 17991486 T^{8} + 4585136 p T^{9} + 405441617 T^{10} + 1808732257 T^{11} + 405441617 p T^{12} + 4585136 p^{3} T^{13} + 17991486 p^{3} T^{14} + 3372332 p^{4} T^{15} + 619600 p^{5} T^{16} + 4952 p^{7} T^{17} + 14502 p^{7} T^{18} + 1600 p^{8} T^{19} + 192 p^{9} T^{20} + 12 p^{10} T^{21} + p^{11} T^{22} \)
23 \( 1 + 3 T + 145 T^{2} + 198 T^{3} + 9728 T^{4} + 454 T^{5} + 429546 T^{6} - 17251 p T^{7} + 14564661 T^{8} - 21122959 T^{9} + 403088352 T^{10} - 613049799 T^{11} + 403088352 p T^{12} - 21122959 p^{2} T^{13} + 14564661 p^{3} T^{14} - 17251 p^{5} T^{15} + 429546 p^{5} T^{16} + 454 p^{6} T^{17} + 9728 p^{7} T^{18} + 198 p^{8} T^{19} + 145 p^{9} T^{20} + 3 p^{10} T^{21} + p^{11} T^{22} \)
29 \( 1 - 2 T + 89 T^{2} - 122 T^{3} + 4647 T^{4} - 7017 T^{5} + 218100 T^{6} - 326437 T^{7} + 8599199 T^{8} - 11638759 T^{9} + 284538994 T^{10} - 368929043 T^{11} + 284538994 p T^{12} - 11638759 p^{2} T^{13} + 8599199 p^{3} T^{14} - 326437 p^{4} T^{15} + 218100 p^{5} T^{16} - 7017 p^{6} T^{17} + 4647 p^{7} T^{18} - 122 p^{8} T^{19} + 89 p^{9} T^{20} - 2 p^{10} T^{21} + p^{11} T^{22} \)
31 \( 1 - T + 222 T^{2} - 230 T^{3} + 24244 T^{4} - 24897 T^{5} + 1733170 T^{6} - 1711674 T^{7} + 90227039 T^{8} - 83314829 T^{9} + 3588768283 T^{10} - 2991531987 T^{11} + 3588768283 p T^{12} - 83314829 p^{2} T^{13} + 90227039 p^{3} T^{14} - 1711674 p^{4} T^{15} + 1733170 p^{5} T^{16} - 24897 p^{6} T^{17} + 24244 p^{7} T^{18} - 230 p^{8} T^{19} + 222 p^{9} T^{20} - p^{10} T^{21} + p^{11} T^{22} \)
37 \( 1 + 14 T + 291 T^{2} + 3245 T^{3} + 40744 T^{4} + 377581 T^{5} + 3641975 T^{6} + 28907272 T^{7} + 232333899 T^{8} + 1607659582 T^{9} + 11160234151 T^{10} + 67840750825 T^{11} + 11160234151 p T^{12} + 1607659582 p^{2} T^{13} + 232333899 p^{3} T^{14} + 28907272 p^{4} T^{15} + 3641975 p^{5} T^{16} + 377581 p^{6} T^{17} + 40744 p^{7} T^{18} + 3245 p^{8} T^{19} + 291 p^{9} T^{20} + 14 p^{10} T^{21} + p^{11} T^{22} \)
41 \( 1 - 3 T + 200 T^{2} - 948 T^{3} + 21068 T^{4} - 116950 T^{5} + 1665137 T^{6} - 8954847 T^{7} + 104162277 T^{8} - 528618594 T^{9} + 5171339938 T^{10} - 24698308195 T^{11} + 5171339938 p T^{12} - 528618594 p^{2} T^{13} + 104162277 p^{3} T^{14} - 8954847 p^{4} T^{15} + 1665137 p^{5} T^{16} - 116950 p^{6} T^{17} + 21068 p^{7} T^{18} - 948 p^{8} T^{19} + 200 p^{9} T^{20} - 3 p^{10} T^{21} + p^{11} T^{22} \)
43 \( 1 + 21 T + 411 T^{2} + 5608 T^{3} + 71396 T^{4} + 760537 T^{5} + 7616040 T^{6} + 68041789 T^{7} + 573679600 T^{8} + 4423116819 T^{9} + 32274274640 T^{10} + 217535136353 T^{11} + 32274274640 p T^{12} + 4423116819 p^{2} T^{13} + 573679600 p^{3} T^{14} + 68041789 p^{4} T^{15} + 7616040 p^{5} T^{16} + 760537 p^{6} T^{17} + 71396 p^{7} T^{18} + 5608 p^{8} T^{19} + 411 p^{9} T^{20} + 21 p^{10} T^{21} + p^{11} T^{22} \)
47 \( 1 + 16 T + 408 T^{2} + 4882 T^{3} + 75334 T^{4} + 741371 T^{5} + 8694921 T^{6} + 73191812 T^{7} + 709299677 T^{8} + 5211390271 T^{9} + 43378149237 T^{10} + 280028273333 T^{11} + 43378149237 p T^{12} + 5211390271 p^{2} T^{13} + 709299677 p^{3} T^{14} + 73191812 p^{4} T^{15} + 8694921 p^{5} T^{16} + 741371 p^{6} T^{17} + 75334 p^{7} T^{18} + 4882 p^{8} T^{19} + 408 p^{9} T^{20} + 16 p^{10} T^{21} + p^{11} T^{22} \)
53 \( 1 + 171 T^{2} - 447 T^{3} + 19622 T^{4} - 55268 T^{5} + 1874340 T^{6} - 4981360 T^{7} + 138786662 T^{8} - 376151679 T^{9} + 8696488649 T^{10} - 21797865799 T^{11} + 8696488649 p T^{12} - 376151679 p^{2} T^{13} + 138786662 p^{3} T^{14} - 4981360 p^{4} T^{15} + 1874340 p^{5} T^{16} - 55268 p^{6} T^{17} + 19622 p^{7} T^{18} - 447 p^{8} T^{19} + 171 p^{9} T^{20} + p^{11} T^{22} \)
59 \( 1 - 3 T + 504 T^{2} - 1533 T^{3} + 119624 T^{4} - 367229 T^{5} + 17792142 T^{6} - 53981851 T^{7} + 1858021409 T^{8} - 5363837721 T^{9} + 144074909069 T^{10} - 375090942553 T^{11} + 144074909069 p T^{12} - 5363837721 p^{2} T^{13} + 1858021409 p^{3} T^{14} - 53981851 p^{4} T^{15} + 17792142 p^{5} T^{16} - 367229 p^{6} T^{17} + 119624 p^{7} T^{18} - 1533 p^{8} T^{19} + 504 p^{9} T^{20} - 3 p^{10} T^{21} + p^{11} T^{22} \)
67 \( 1 + 24 T + 746 T^{2} + 11525 T^{3} + 205735 T^{4} + 2338279 T^{5} + 30851917 T^{6} + 278464946 T^{7} + 3068071825 T^{8} + 23594529666 T^{9} + 236961820908 T^{10} + 1668482725813 T^{11} + 236961820908 p T^{12} + 23594529666 p^{2} T^{13} + 3068071825 p^{3} T^{14} + 278464946 p^{4} T^{15} + 30851917 p^{5} T^{16} + 2338279 p^{6} T^{17} + 205735 p^{7} T^{18} + 11525 p^{8} T^{19} + 746 p^{9} T^{20} + 24 p^{10} T^{21} + p^{11} T^{22} \)
71 \( 1 - 7 T + 423 T^{2} - 2938 T^{3} + 94891 T^{4} - 634016 T^{5} + 14512884 T^{6} - 91820873 T^{7} + 1664655219 T^{8} - 9737118634 T^{9} + 148962046506 T^{10} - 787502321349 T^{11} + 148962046506 p T^{12} - 9737118634 p^{2} T^{13} + 1664655219 p^{3} T^{14} - 91820873 p^{4} T^{15} + 14512884 p^{5} T^{16} - 634016 p^{6} T^{17} + 94891 p^{7} T^{18} - 2938 p^{8} T^{19} + 423 p^{9} T^{20} - 7 p^{10} T^{21} + p^{11} T^{22} \)
73 \( 1 + 42 T + 1222 T^{2} + 25622 T^{3} + 447305 T^{4} + 6597184 T^{5} + 86476323 T^{6} + 1015901577 T^{7} + 10992487226 T^{8} + 109895882058 T^{9} + 1031664342334 T^{10} + 124233446049 p T^{11} + 1031664342334 p T^{12} + 109895882058 p^{2} T^{13} + 10992487226 p^{3} T^{14} + 1015901577 p^{4} T^{15} + 86476323 p^{5} T^{16} + 6597184 p^{6} T^{17} + 447305 p^{7} T^{18} + 25622 p^{8} T^{19} + 1222 p^{9} T^{20} + 42 p^{10} T^{21} + p^{11} T^{22} \)
79 \( 1 + 11 T + 690 T^{2} + 6490 T^{3} + 218178 T^{4} + 1770975 T^{5} + 42469286 T^{6} + 300331876 T^{7} + 5775405374 T^{8} + 35916525340 T^{9} + 588476371774 T^{10} + 3234511194975 T^{11} + 588476371774 p T^{12} + 35916525340 p^{2} T^{13} + 5775405374 p^{3} T^{14} + 300331876 p^{4} T^{15} + 42469286 p^{5} T^{16} + 1770975 p^{6} T^{17} + 218178 p^{7} T^{18} + 6490 p^{8} T^{19} + 690 p^{9} T^{20} + 11 p^{10} T^{21} + p^{11} T^{22} \)
83 \( 1 + 34 T + 1164 T^{2} + 24674 T^{3} + 499533 T^{4} + 7832010 T^{5} + 117485561 T^{6} + 1476053272 T^{7} + 17943932359 T^{8} + 189853527918 T^{9} + 1966784410629 T^{10} + 18060985019151 T^{11} + 1966784410629 p T^{12} + 189853527918 p^{2} T^{13} + 17943932359 p^{3} T^{14} + 1476053272 p^{4} T^{15} + 117485561 p^{5} T^{16} + 7832010 p^{6} T^{17} + 499533 p^{7} T^{18} + 24674 p^{8} T^{19} + 1164 p^{9} T^{20} + 34 p^{10} T^{21} + p^{11} T^{22} \)
89 \( 1 - 29 T + 1006 T^{2} - 20891 T^{3} + 434731 T^{4} - 7113247 T^{5} + 111586710 T^{6} - 1505638719 T^{7} + 19231282025 T^{8} - 219069317588 T^{9} + 2354328494051 T^{10} - 22873997682689 T^{11} + 2354328494051 p T^{12} - 219069317588 p^{2} T^{13} + 19231282025 p^{3} T^{14} - 1505638719 p^{4} T^{15} + 111586710 p^{5} T^{16} - 7113247 p^{6} T^{17} + 434731 p^{7} T^{18} - 20891 p^{8} T^{19} + 1006 p^{9} T^{20} - 29 p^{10} T^{21} + p^{11} T^{22} \)
97 \( 1 + 45 T + 1376 T^{2} + 29931 T^{3} + 545438 T^{4} + 8375059 T^{5} + 116449942 T^{6} + 1465766429 T^{7} + 17473897571 T^{8} + 194927586130 T^{9} + 2087374168964 T^{10} + 20998963042063 T^{11} + 2087374168964 p T^{12} + 194927586130 p^{2} T^{13} + 17473897571 p^{3} T^{14} + 1465766429 p^{4} T^{15} + 116449942 p^{5} T^{16} + 8375059 p^{6} T^{17} + 545438 p^{7} T^{18} + 29931 p^{8} T^{19} + 1376 p^{9} T^{20} + 45 p^{10} T^{21} + p^{11} T^{22} \)
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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

−3.73437634926029018992666736670, −3.65275218246625698282978143269, −3.44052520774153059532441303016, −3.28410550009955990852021099286, −3.20628576425907611425832402693, −3.12125811403305475252376225291, −3.09427569684751321414514729698, −2.82864028327393829810635523595, −2.78789312824455701287437423431, −2.72463097361773273368120129041, −2.57100234094703296651475495025, −2.27722901705640007760320447445, −2.21320520023075297658826176159, −2.13095291340323130698797509112, −2.04079537719894268774447475633, −1.99950814378009503297542023261, −1.87109474575817373958350049942, −1.67350482400661752720018192840, −1.53763001638617841916906055884, −1.43923357828260642503621591934, −1.31881167459163369689341966559, −1.31145948635138763594681064837, −1.13216430346626463757447886548, −0.999133069216803901406858833985, −0.970846795256844661266912820344, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.970846795256844661266912820344, 0.999133069216803901406858833985, 1.13216430346626463757447886548, 1.31145948635138763594681064837, 1.31881167459163369689341966559, 1.43923357828260642503621591934, 1.53763001638617841916906055884, 1.67350482400661752720018192840, 1.87109474575817373958350049942, 1.99950814378009503297542023261, 2.04079537719894268774447475633, 2.13095291340323130698797509112, 2.21320520023075297658826176159, 2.27722901705640007760320447445, 2.57100234094703296651475495025, 2.72463097361773273368120129041, 2.78789312824455701287437423431, 2.82864028327393829810635523595, 3.09427569684751321414514729698, 3.12125811403305475252376225291, 3.20628576425907611425832402693, 3.28410550009955990852021099286, 3.44052520774153059532441303016, 3.65275218246625698282978143269, 3.73437634926029018992666736670

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

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