Dirichlet series
| L(s) = 1 | + 4·3-s + 3·5-s − 3·9-s − 13·13-s + 12·15-s + 7·17-s + 8·19-s − 11·23-s − 20·25-s − 29·27-s − 3·29-s + 12·31-s − 37-s − 52·39-s + 12·41-s + 9·43-s − 9·45-s + 17·47-s + 28·51-s − 5·53-s + 32·57-s + 33·59-s − 15·61-s − 39·65-s − 5·67-s − 44·69-s − 9·71-s + ⋯ |
| L(s) = 1 | + 2.30·3-s + 1.34·5-s − 9-s − 3.60·13-s + 3.09·15-s + 1.69·17-s + 1.83·19-s − 2.29·23-s − 4·25-s − 5.58·27-s − 0.557·29-s + 2.15·31-s − 0.164·37-s − 8.32·39-s + 1.87·41-s + 1.37·43-s − 1.34·45-s + 2.47·47-s + 3.92·51-s − 0.686·53-s + 4.23·57-s + 4.29·59-s − 1.92·61-s − 4.83·65-s − 0.610·67-s − 5.29·69-s − 1.06·71-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{33} \cdot 7^{22} \cdot 23^{11}\right)^{s/2} \, \Gamma_{\C}(s)^{11} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{33} \cdot 7^{22} \cdot 23^{11}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{11} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(22\) |
| Conductor: | \(2^{33} \cdot 7^{22} \cdot 23^{11}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.69278\times 10^{20}\) |
| Root analytic conductor: | \(8.48487\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((22,\ 2^{33} \cdot 7^{22} \cdot 23^{11} ,\ ( \ : [1/2]^{11} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(43.54059010\) |
| \(L(\frac12)\) | \(\approx\) | \(43.54059010\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) | |
| 23 | \( ( 1 + T )^{11} \) | |
| good | 3 | \( 1 - 4 T + 19 T^{2} - 59 T^{3} + 188 T^{4} - 499 T^{5} + 1258 T^{6} - 965 p T^{7} + 691 p^{2} T^{8} - 1403 p^{2} T^{9} + 23750 T^{10} - 1583 p^{3} T^{11} + 23750 p T^{12} - 1403 p^{4} T^{13} + 691 p^{5} T^{14} - 965 p^{5} T^{15} + 1258 p^{5} T^{16} - 499 p^{6} T^{17} + 188 p^{7} T^{18} - 59 p^{8} T^{19} + 19 p^{9} T^{20} - 4 p^{10} T^{21} + p^{11} T^{22} \) |
| 5 | \( 1 - 3 T + 29 T^{2} - 84 T^{3} + 453 T^{4} - 1201 T^{5} + 4866 T^{6} - 11659 T^{7} + 39206 T^{8} - 84809 T^{9} + 246619 T^{10} - 479249 T^{11} + 246619 p T^{12} - 84809 p^{2} T^{13} + 39206 p^{3} T^{14} - 11659 p^{4} T^{15} + 4866 p^{5} T^{16} - 1201 p^{6} T^{17} + 453 p^{7} T^{18} - 84 p^{8} T^{19} + 29 p^{9} T^{20} - 3 p^{10} T^{21} + p^{11} T^{22} \) | |
| 11 | \( 1 + 48 T^{2} + 25 T^{3} + 1126 T^{4} + 1025 T^{5} + 19763 T^{6} + 19346 T^{7} + 297305 T^{8} + 260732 T^{9} + 3791809 T^{10} + 3051469 T^{11} + 3791809 p T^{12} + 260732 p^{2} T^{13} + 297305 p^{3} T^{14} + 19346 p^{4} T^{15} + 19763 p^{5} T^{16} + 1025 p^{6} T^{17} + 1126 p^{7} T^{18} + 25 p^{8} T^{19} + 48 p^{9} T^{20} + p^{11} T^{22} \) | |
| 13 | \( 1 + p T + 131 T^{2} + 1047 T^{3} + 7363 T^{4} + 44599 T^{5} + 246815 T^{6} + 1239990 T^{7} + 5766203 T^{8} + 1898718 p T^{9} + 99020185 T^{10} + 368940695 T^{11} + 99020185 p T^{12} + 1898718 p^{3} T^{13} + 5766203 p^{3} T^{14} + 1239990 p^{4} T^{15} + 246815 p^{5} T^{16} + 44599 p^{6} T^{17} + 7363 p^{7} T^{18} + 1047 p^{8} T^{19} + 131 p^{9} T^{20} + p^{11} T^{21} + p^{11} T^{22} \) | |
| 17 | \( 1 - 7 T + 105 T^{2} - 616 T^{3} + 5561 T^{4} - 29241 T^{5} + 202867 T^{6} - 954644 T^{7} + 5536486 T^{8} - 23370417 T^{9} + 118292223 T^{10} - 447604735 T^{11} + 118292223 p T^{12} - 23370417 p^{2} T^{13} + 5536486 p^{3} T^{14} - 954644 p^{4} T^{15} + 202867 p^{5} T^{16} - 29241 p^{6} T^{17} + 5561 p^{7} T^{18} - 616 p^{8} T^{19} + 105 p^{9} T^{20} - 7 p^{10} T^{21} + p^{11} T^{22} \) | |
| 19 | \( 1 - 8 T + 164 T^{2} - 1120 T^{3} + 12672 T^{4} - 76201 T^{5} + 617113 T^{6} - 3292388 T^{7} + 21095503 T^{8} - 99612520 T^{9} + 532132974 T^{10} - 115859711 p T^{11} + 532132974 p T^{12} - 99612520 p^{2} T^{13} + 21095503 p^{3} T^{14} - 3292388 p^{4} T^{15} + 617113 p^{5} T^{16} - 76201 p^{6} T^{17} + 12672 p^{7} T^{18} - 1120 p^{8} T^{19} + 164 p^{9} T^{20} - 8 p^{10} T^{21} + p^{11} T^{22} \) | |
| 29 | \( 1 + 3 T + 106 T^{2} + 201 T^{3} + 5953 T^{4} + 11260 T^{5} + 276330 T^{6} + 691657 T^{7} + 10639555 T^{8} + 28787286 T^{9} + 336295963 T^{10} + 894926249 T^{11} + 336295963 p T^{12} + 28787286 p^{2} T^{13} + 10639555 p^{3} T^{14} + 691657 p^{4} T^{15} + 276330 p^{5} T^{16} + 11260 p^{6} T^{17} + 5953 p^{7} T^{18} + 201 p^{8} T^{19} + 106 p^{9} T^{20} + 3 p^{10} T^{21} + p^{11} T^{22} \) | |
| 31 | \( 1 - 12 T + 183 T^{2} - 1409 T^{3} + 441 p T^{4} - 80849 T^{5} + 630262 T^{6} - 3064762 T^{7} + 21562191 T^{8} - 89934897 T^{9} + 633874722 T^{10} - 2564280583 T^{11} + 633874722 p T^{12} - 89934897 p^{2} T^{13} + 21562191 p^{3} T^{14} - 3064762 p^{4} T^{15} + 630262 p^{5} T^{16} - 80849 p^{6} T^{17} + 441 p^{8} T^{18} - 1409 p^{8} T^{19} + 183 p^{9} T^{20} - 12 p^{10} T^{21} + p^{11} T^{22} \) | |
| 37 | \( 1 + T + 275 T^{2} + 257 T^{3} + 35442 T^{4} + 31336 T^{5} + 2868083 T^{6} + 2427104 T^{7} + 166326209 T^{8} + 134581758 T^{9} + 7520840932 T^{10} + 5663586749 T^{11} + 7520840932 p T^{12} + 134581758 p^{2} T^{13} + 166326209 p^{3} T^{14} + 2427104 p^{4} T^{15} + 2868083 p^{5} T^{16} + 31336 p^{6} T^{17} + 35442 p^{7} T^{18} + 257 p^{8} T^{19} + 275 p^{9} T^{20} + p^{10} T^{21} + p^{11} T^{22} \) | |
| 41 | \( 1 - 12 T + 316 T^{2} - 2800 T^{3} + 43100 T^{4} - 305158 T^{5} + 3629584 T^{6} - 21888474 T^{7} + 225335617 T^{8} - 1211297756 T^{9} + 11242056891 T^{10} - 54757133275 T^{11} + 11242056891 p T^{12} - 1211297756 p^{2} T^{13} + 225335617 p^{3} T^{14} - 21888474 p^{4} T^{15} + 3629584 p^{5} T^{16} - 305158 p^{6} T^{17} + 43100 p^{7} T^{18} - 2800 p^{8} T^{19} + 316 p^{9} T^{20} - 12 p^{10} T^{21} + p^{11} T^{22} \) | |
| 43 | \( 1 - 9 T + 194 T^{2} - 1736 T^{3} + 22920 T^{4} - 186397 T^{5} + 1971169 T^{6} - 14445514 T^{7} + 130123698 T^{8} - 866356276 T^{9} + 6888036725 T^{10} - 41482712213 T^{11} + 6888036725 p T^{12} - 866356276 p^{2} T^{13} + 130123698 p^{3} T^{14} - 14445514 p^{4} T^{15} + 1971169 p^{5} T^{16} - 186397 p^{6} T^{17} + 22920 p^{7} T^{18} - 1736 p^{8} T^{19} + 194 p^{9} T^{20} - 9 p^{10} T^{21} + p^{11} T^{22} \) | |
| 47 | \( 1 - 17 T + 485 T^{2} - 6164 T^{3} + 102609 T^{4} - 1054801 T^{5} + 13073806 T^{6} - 113183583 T^{7} + 1139568056 T^{8} - 8473805375 T^{9} + 72077072089 T^{10} - 463159112059 T^{11} + 72077072089 p T^{12} - 8473805375 p^{2} T^{13} + 1139568056 p^{3} T^{14} - 113183583 p^{4} T^{15} + 13073806 p^{5} T^{16} - 1054801 p^{6} T^{17} + 102609 p^{7} T^{18} - 6164 p^{8} T^{19} + 485 p^{9} T^{20} - 17 p^{10} T^{21} + p^{11} T^{22} \) | |
| 53 | \( 1 + 5 T + 188 T^{2} - 272 T^{3} + 12655 T^{4} - 129144 T^{5} + 829383 T^{6} - 12069246 T^{7} + 79883679 T^{8} - 673039935 T^{9} + 6637491519 T^{10} - 32259812565 T^{11} + 6637491519 p T^{12} - 673039935 p^{2} T^{13} + 79883679 p^{3} T^{14} - 12069246 p^{4} T^{15} + 829383 p^{5} T^{16} - 129144 p^{6} T^{17} + 12655 p^{7} T^{18} - 272 p^{8} T^{19} + 188 p^{9} T^{20} + 5 p^{10} T^{21} + p^{11} T^{22} \) | |
| 59 | \( 1 - 33 T + 866 T^{2} - 15816 T^{3} + 253072 T^{4} - 3367815 T^{5} + 40870239 T^{6} - 438167582 T^{7} + 4378601666 T^{8} - 39763123344 T^{9} + 340828535329 T^{10} - 2688775619579 T^{11} + 340828535329 p T^{12} - 39763123344 p^{2} T^{13} + 4378601666 p^{3} T^{14} - 438167582 p^{4} T^{15} + 40870239 p^{5} T^{16} - 3367815 p^{6} T^{17} + 253072 p^{7} T^{18} - 15816 p^{8} T^{19} + 866 p^{9} T^{20} - 33 p^{10} T^{21} + p^{11} T^{22} \) | |
| 61 | \( 1 + 15 T + 521 T^{2} + 6580 T^{3} + 126577 T^{4} + 1377679 T^{5} + 19299189 T^{6} + 183909484 T^{7} + 2077742216 T^{8} + 17443576587 T^{9} + 166648029477 T^{10} + 1228000161081 T^{11} + 166648029477 p T^{12} + 17443576587 p^{2} T^{13} + 2077742216 p^{3} T^{14} + 183909484 p^{4} T^{15} + 19299189 p^{5} T^{16} + 1377679 p^{6} T^{17} + 126577 p^{7} T^{18} + 6580 p^{8} T^{19} + 521 p^{9} T^{20} + 15 p^{10} T^{21} + p^{11} T^{22} \) | |
| 67 | \( 1 + 5 T + 372 T^{2} + 1440 T^{3} + 67539 T^{4} + 222405 T^{5} + 8403029 T^{6} + 25429497 T^{7} + 818884778 T^{8} + 2320861597 T^{9} + 65635789506 T^{10} + 171818406725 T^{11} + 65635789506 p T^{12} + 2320861597 p^{2} T^{13} + 818884778 p^{3} T^{14} + 25429497 p^{4} T^{15} + 8403029 p^{5} T^{16} + 222405 p^{6} T^{17} + 67539 p^{7} T^{18} + 1440 p^{8} T^{19} + 372 p^{9} T^{20} + 5 p^{10} T^{21} + p^{11} T^{22} \) | |
| 71 | \( 1 + 9 T + 514 T^{2} + 4897 T^{3} + 131542 T^{4} + 1255504 T^{5} + 22158599 T^{6} + 201580542 T^{7} + 2710364414 T^{8} + 22619579521 T^{9} + 250652701249 T^{10} + 1861956543819 T^{11} + 250652701249 p T^{12} + 22619579521 p^{2} T^{13} + 2710364414 p^{3} T^{14} + 201580542 p^{4} T^{15} + 22158599 p^{5} T^{16} + 1255504 p^{6} T^{17} + 131542 p^{7} T^{18} + 4897 p^{8} T^{19} + 514 p^{9} T^{20} + 9 p^{10} T^{21} + p^{11} T^{22} \) | |
| 73 | \( 1 - 5 T + 297 T^{2} - 1493 T^{3} + 40705 T^{4} - 179132 T^{5} + 4034363 T^{6} - 15307353 T^{7} + 386150920 T^{8} - 1607407311 T^{9} + 34702563899 T^{10} - 146791527553 T^{11} + 34702563899 p T^{12} - 1607407311 p^{2} T^{13} + 386150920 p^{3} T^{14} - 15307353 p^{4} T^{15} + 4034363 p^{5} T^{16} - 179132 p^{6} T^{17} + 40705 p^{7} T^{18} - 1493 p^{8} T^{19} + 297 p^{9} T^{20} - 5 p^{10} T^{21} + p^{11} T^{22} \) | |
| 79 | \( 1 - 11 T + 231 T^{2} - 680 T^{3} + 21107 T^{4} - 89108 T^{5} + 3012293 T^{6} - 12811505 T^{7} + 219971140 T^{8} - 992034761 T^{9} + 22397120958 T^{10} - 147950507353 T^{11} + 22397120958 p T^{12} - 992034761 p^{2} T^{13} + 219971140 p^{3} T^{14} - 12811505 p^{4} T^{15} + 3012293 p^{5} T^{16} - 89108 p^{6} T^{17} + 21107 p^{7} T^{18} - 680 p^{8} T^{19} + 231 p^{9} T^{20} - 11 p^{10} T^{21} + p^{11} T^{22} \) | |
| 83 | \( 1 - 51 T + 1558 T^{2} - 34315 T^{3} + 615432 T^{4} - 9475668 T^{5} + 130634453 T^{6} - 1640380157 T^{7} + 19007307868 T^{8} - 204085688334 T^{9} + 2046375763905 T^{10} - 19224382138995 T^{11} + 2046375763905 p T^{12} - 204085688334 p^{2} T^{13} + 19007307868 p^{3} T^{14} - 1640380157 p^{4} T^{15} + 130634453 p^{5} T^{16} - 9475668 p^{6} T^{17} + 615432 p^{7} T^{18} - 34315 p^{8} T^{19} + 1558 p^{9} T^{20} - 51 p^{10} T^{21} + p^{11} T^{22} \) | |
| 89 | \( 1 - 26 T + 751 T^{2} - 14371 T^{3} + 259846 T^{4} - 3962930 T^{5} + 56278298 T^{6} - 717885331 T^{7} + 8625068449 T^{8} - 94954248572 T^{9} + 993938720759 T^{10} - 9600690333113 T^{11} + 993938720759 p T^{12} - 94954248572 p^{2} T^{13} + 8625068449 p^{3} T^{14} - 717885331 p^{4} T^{15} + 56278298 p^{5} T^{16} - 3962930 p^{6} T^{17} + 259846 p^{7} T^{18} - 14371 p^{8} T^{19} + 751 p^{9} T^{20} - 26 p^{10} T^{21} + p^{11} T^{22} \) | |
| 97 | \( 1 - 21 T + 920 T^{2} - 15771 T^{3} + 387707 T^{4} - 5595802 T^{5} + 100479312 T^{6} - 1242714589 T^{7} + 17967559877 T^{8} - 192153737056 T^{9} + 2339949051551 T^{10} - 21674949096417 T^{11} + 2339949051551 p T^{12} - 192153737056 p^{2} T^{13} + 17967559877 p^{3} T^{14} - 1242714589 p^{4} T^{15} + 100479312 p^{5} T^{16} - 5595802 p^{6} T^{17} + 387707 p^{7} T^{18} - 15771 p^{8} T^{19} + 920 p^{9} T^{20} - 21 p^{10} T^{21} + p^{11} T^{22} \) | |
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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
−2.45553612983166995923093878250, −2.45135383603758392410312643443, −2.40079168444864018036002141349, −2.22767403483216133771164117021, −2.12379632855668836218440441286, −2.06017234077194066422329964154, −1.91125090111544229574886332345, −1.90318973529274428905143081076, −1.80173559123393126825214868461, −1.78889390748309031813989735731, −1.73953613415890449931285252083, −1.70321144694533958175989599420, −1.52274279284936040328433601381, −1.33785736990279616306194960245, −1.20935028030387140843680477155, −1.06326127653659573741659441556, −0.846269686005170048512764388959, −0.797052825593674203212045996447, −0.76831554304239583647898207331, −0.66460103034884772501532347371, −0.61693432223475350994709300454, −0.49618176398694185537469712155, −0.46246687504421460519397986640, −0.23580906989343918026962216958, −0.15398740646281748432677474325, 0.15398740646281748432677474325, 0.23580906989343918026962216958, 0.46246687504421460519397986640, 0.49618176398694185537469712155, 0.61693432223475350994709300454, 0.66460103034884772501532347371, 0.76831554304239583647898207331, 0.797052825593674203212045996447, 0.846269686005170048512764388959, 1.06326127653659573741659441556, 1.20935028030387140843680477155, 1.33785736990279616306194960245, 1.52274279284936040328433601381, 1.70321144694533958175989599420, 1.73953613415890449931285252083, 1.78889390748309031813989735731, 1.80173559123393126825214868461, 1.90318973529274428905143081076, 1.91125090111544229574886332345, 2.06017234077194066422329964154, 2.12379632855668836218440441286, 2.22767403483216133771164117021, 2.40079168444864018036002141349, 2.45135383603758392410312643443, 2.45553612983166995923093878250
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.