Dirichlet series
| L(s) = 1 | + 11·2-s − 11·3-s + 66·4-s + 4·5-s − 121·6-s + 3·7-s + 286·8-s + 66·9-s + 44·10-s + 9·11-s − 726·12-s + 6·13-s + 33·14-s − 44·15-s + 1.00e3·16-s − 11·17-s + 726·18-s − 19-s + 264·20-s − 33·21-s + 99·22-s + 10·23-s − 3.14e3·24-s − 12·25-s + 66·26-s − 286·27-s + 198·28-s + ⋯ |
| L(s) = 1 | + 7.77·2-s − 6.35·3-s + 33·4-s + 1.78·5-s − 49.3·6-s + 1.13·7-s + 101.·8-s + 22·9-s + 13.9·10-s + 2.71·11-s − 209.·12-s + 1.66·13-s + 8.81·14-s − 11.3·15-s + 250.·16-s − 2.66·17-s + 171.·18-s − 0.229·19-s + 59.0·20-s − 7.20·21-s + 21.1·22-s + 2.08·23-s − 642.·24-s − 2.39·25-s + 12.9·26-s − 55.0·27-s + 37.4·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{11} \cdot 3^{11} \cdot 17^{11} \cdot 59^{11}\right)^{s/2} \, \Gamma_{\C}(s)^{11} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{11} \cdot 3^{11} \cdot 17^{11} \cdot 59^{11}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{11} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(22\) |
| Conductor: | \(2^{11} \cdot 3^{11} \cdot 17^{11} \cdot 59^{11}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.15516\times 10^{18}\) |
| Root analytic conductor: | \(6.93209\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((22,\ 2^{11} \cdot 3^{11} \cdot 17^{11} \cdot 59^{11} ,\ ( \ : [1/2]^{11} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(23641.42032\) |
| \(L(\frac12)\) | \(\approx\) | \(23641.42032\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - T )^{11} \) |
| 3 | \( ( 1 + T )^{11} \) | |
| 17 | \( ( 1 + T )^{11} \) | |
| 59 | \( ( 1 + T )^{11} \) | |
| good | 5 | \( 1 - 4 T + 28 T^{2} - 83 T^{3} + 72 p T^{4} - 843 T^{5} + 2841 T^{6} - 5387 T^{7} + 16212 T^{8} - 25937 T^{9} + 79238 T^{10} - 24164 p T^{11} + 79238 p T^{12} - 25937 p^{2} T^{13} + 16212 p^{3} T^{14} - 5387 p^{4} T^{15} + 2841 p^{5} T^{16} - 843 p^{6} T^{17} + 72 p^{8} T^{18} - 83 p^{8} T^{19} + 28 p^{9} T^{20} - 4 p^{10} T^{21} + p^{11} T^{22} \) |
| 7 | \( 1 - 3 T + 34 T^{2} - 92 T^{3} + 690 T^{4} - 1635 T^{5} + 9612 T^{6} - 20703 T^{7} + 103651 T^{8} - 200742 T^{9} + 889845 T^{10} - 1567634 T^{11} + 889845 p T^{12} - 200742 p^{2} T^{13} + 103651 p^{3} T^{14} - 20703 p^{4} T^{15} + 9612 p^{5} T^{16} - 1635 p^{6} T^{17} + 690 p^{7} T^{18} - 92 p^{8} T^{19} + 34 p^{9} T^{20} - 3 p^{10} T^{21} + p^{11} T^{22} \) | |
| 11 | \( 1 - 9 T + 95 T^{2} - 548 T^{3} + 3577 T^{4} - 16218 T^{5} + 83482 T^{6} - 322251 T^{7} + 1422693 T^{8} - 4864829 T^{9} + 19170922 T^{10} - 59179954 T^{11} + 19170922 p T^{12} - 4864829 p^{2} T^{13} + 1422693 p^{3} T^{14} - 322251 p^{4} T^{15} + 83482 p^{5} T^{16} - 16218 p^{6} T^{17} + 3577 p^{7} T^{18} - 548 p^{8} T^{19} + 95 p^{9} T^{20} - 9 p^{10} T^{21} + p^{11} T^{22} \) | |
| 13 | \( 1 - 6 T + 74 T^{2} - 290 T^{3} + 2111 T^{4} - 6170 T^{5} + 36064 T^{6} - 88300 T^{7} + 446947 T^{8} - 925648 T^{9} + 4430087 T^{10} - 9224868 T^{11} + 4430087 p T^{12} - 925648 p^{2} T^{13} + 446947 p^{3} T^{14} - 88300 p^{4} T^{15} + 36064 p^{5} T^{16} - 6170 p^{6} T^{17} + 2111 p^{7} T^{18} - 290 p^{8} T^{19} + 74 p^{9} T^{20} - 6 p^{10} T^{21} + p^{11} T^{22} \) | |
| 19 | \( 1 + T + 128 T^{2} - 29 T^{3} + 7321 T^{4} - 12492 T^{5} + 249745 T^{6} - 894519 T^{7} + 5895112 T^{8} - 33805701 T^{9} + 113851209 T^{10} - 797637736 T^{11} + 113851209 p T^{12} - 33805701 p^{2} T^{13} + 5895112 p^{3} T^{14} - 894519 p^{4} T^{15} + 249745 p^{5} T^{16} - 12492 p^{6} T^{17} + 7321 p^{7} T^{18} - 29 p^{8} T^{19} + 128 p^{9} T^{20} + p^{10} T^{21} + p^{11} T^{22} \) | |
| 23 | \( 1 - 10 T + 195 T^{2} - 1516 T^{3} + 17310 T^{4} - 113347 T^{5} + 972036 T^{6} - 5545469 T^{7} + 38924230 T^{8} - 195399715 T^{9} + 1168720699 T^{10} - 224336782 p T^{11} + 1168720699 p T^{12} - 195399715 p^{2} T^{13} + 38924230 p^{3} T^{14} - 5545469 p^{4} T^{15} + 972036 p^{5} T^{16} - 113347 p^{6} T^{17} + 17310 p^{7} T^{18} - 1516 p^{8} T^{19} + 195 p^{9} T^{20} - 10 p^{10} T^{21} + p^{11} T^{22} \) | |
| 29 | \( 1 - 14 T + 238 T^{2} - 2399 T^{3} + 25358 T^{4} - 206327 T^{5} + 1689265 T^{6} - 11698249 T^{7} + 80416878 T^{8} - 488919195 T^{9} + 2938404708 T^{10} - 15920222560 T^{11} + 2938404708 p T^{12} - 488919195 p^{2} T^{13} + 80416878 p^{3} T^{14} - 11698249 p^{4} T^{15} + 1689265 p^{5} T^{16} - 206327 p^{6} T^{17} + 25358 p^{7} T^{18} - 2399 p^{8} T^{19} + 238 p^{9} T^{20} - 14 p^{10} T^{21} + p^{11} T^{22} \) | |
| 31 | \( 1 - 17 T + 247 T^{2} - 2812 T^{3} + 29042 T^{4} - 260389 T^{5} + 2160344 T^{6} - 16316368 T^{7} + 115440467 T^{8} - 753625354 T^{9} + 4643318967 T^{10} - 26611855720 T^{11} + 4643318967 p T^{12} - 753625354 p^{2} T^{13} + 115440467 p^{3} T^{14} - 16316368 p^{4} T^{15} + 2160344 p^{5} T^{16} - 260389 p^{6} T^{17} + 29042 p^{7} T^{18} - 2812 p^{8} T^{19} + 247 p^{9} T^{20} - 17 p^{10} T^{21} + p^{11} T^{22} \) | |
| 37 | \( 1 - 30 T + 516 T^{2} - 6239 T^{3} + 58186 T^{4} - 436507 T^{5} + 2715113 T^{6} - 14343215 T^{7} + 67275402 T^{8} - 301517259 T^{9} + 1443858134 T^{10} - 8033227004 T^{11} + 1443858134 p T^{12} - 301517259 p^{2} T^{13} + 67275402 p^{3} T^{14} - 14343215 p^{4} T^{15} + 2715113 p^{5} T^{16} - 436507 p^{6} T^{17} + 58186 p^{7} T^{18} - 6239 p^{8} T^{19} + 516 p^{9} T^{20} - 30 p^{10} T^{21} + p^{11} T^{22} \) | |
| 41 | \( 1 - 10 T + 365 T^{2} - 2954 T^{3} + 60470 T^{4} - 408733 T^{5} + 6148614 T^{6} - 35486467 T^{7} + 436134586 T^{8} - 2187405683 T^{9} + 23131300421 T^{10} - 102024602272 T^{11} + 23131300421 p T^{12} - 2187405683 p^{2} T^{13} + 436134586 p^{3} T^{14} - 35486467 p^{4} T^{15} + 6148614 p^{5} T^{16} - 408733 p^{6} T^{17} + 60470 p^{7} T^{18} - 2954 p^{8} T^{19} + 365 p^{9} T^{20} - 10 p^{10} T^{21} + p^{11} T^{22} \) | |
| 43 | \( 1 - 11 T + 281 T^{2} - 3140 T^{3} + 43317 T^{4} - 432094 T^{5} + 4527972 T^{6} - 38909167 T^{7} + 340428471 T^{8} - 2551415567 T^{9} + 19127035900 T^{10} - 126007952442 T^{11} + 19127035900 p T^{12} - 2551415567 p^{2} T^{13} + 340428471 p^{3} T^{14} - 38909167 p^{4} T^{15} + 4527972 p^{5} T^{16} - 432094 p^{6} T^{17} + 43317 p^{7} T^{18} - 3140 p^{8} T^{19} + 281 p^{9} T^{20} - 11 p^{10} T^{21} + p^{11} T^{22} \) | |
| 47 | \( 1 + 6 T + 5 p T^{2} + 1237 T^{3} + 27328 T^{4} + 144510 T^{5} + 2253074 T^{6} + 12507376 T^{7} + 149276661 T^{8} + 836868396 T^{9} + 8262758745 T^{10} + 44092833302 T^{11} + 8262758745 p T^{12} + 836868396 p^{2} T^{13} + 149276661 p^{3} T^{14} + 12507376 p^{4} T^{15} + 2253074 p^{5} T^{16} + 144510 p^{6} T^{17} + 27328 p^{7} T^{18} + 1237 p^{8} T^{19} + 5 p^{10} T^{20} + 6 p^{10} T^{21} + p^{11} T^{22} \) | |
| 53 | \( 1 - 10 T + 500 T^{2} - 4484 T^{3} + 117486 T^{4} - 937195 T^{5} + 17086977 T^{6} - 120430875 T^{7} + 1707151194 T^{8} - 10546849619 T^{9} + 122980029886 T^{10} - 658040847186 T^{11} + 122980029886 p T^{12} - 10546849619 p^{2} T^{13} + 1707151194 p^{3} T^{14} - 120430875 p^{4} T^{15} + 17086977 p^{5} T^{16} - 937195 p^{6} T^{17} + 117486 p^{7} T^{18} - 4484 p^{8} T^{19} + 500 p^{9} T^{20} - 10 p^{10} T^{21} + p^{11} T^{22} \) | |
| 61 | \( 1 - 13 T + 563 T^{2} - 110 p T^{3} + 150081 T^{4} - 1616270 T^{5} + 24905612 T^{6} - 239441415 T^{7} + 2850655351 T^{8} - 24166137901 T^{9} + 235892367236 T^{10} - 1736218384774 T^{11} + 235892367236 p T^{12} - 24166137901 p^{2} T^{13} + 2850655351 p^{3} T^{14} - 239441415 p^{4} T^{15} + 24905612 p^{5} T^{16} - 1616270 p^{6} T^{17} + 150081 p^{7} T^{18} - 110 p^{9} T^{19} + 563 p^{9} T^{20} - 13 p^{10} T^{21} + p^{11} T^{22} \) | |
| 67 | \( 1 - 26 T + 589 T^{2} - 9799 T^{3} + 148746 T^{4} - 1931036 T^{5} + 23257443 T^{6} - 253616403 T^{7} + 2593504252 T^{8} - 24594268782 T^{9} + 220624587087 T^{10} - 1855391455220 T^{11} + 220624587087 p T^{12} - 24594268782 p^{2} T^{13} + 2593504252 p^{3} T^{14} - 253616403 p^{4} T^{15} + 23257443 p^{5} T^{16} - 1931036 p^{6} T^{17} + 148746 p^{7} T^{18} - 9799 p^{8} T^{19} + 589 p^{9} T^{20} - 26 p^{10} T^{21} + p^{11} T^{22} \) | |
| 71 | \( 1 - 14 T + 589 T^{2} - 6467 T^{3} + 154714 T^{4} - 1395290 T^{5} + 24883062 T^{6} - 190493176 T^{7} + 2830896977 T^{8} - 18945968312 T^{9} + 248604974441 T^{10} - 1490384924298 T^{11} + 248604974441 p T^{12} - 18945968312 p^{2} T^{13} + 2830896977 p^{3} T^{14} - 190493176 p^{4} T^{15} + 24883062 p^{5} T^{16} - 1395290 p^{6} T^{17} + 154714 p^{7} T^{18} - 6467 p^{8} T^{19} + 589 p^{9} T^{20} - 14 p^{10} T^{21} + p^{11} T^{22} \) | |
| 73 | \( 1 - 20 T + 281 T^{2} - 4322 T^{3} + 51068 T^{4} - 531865 T^{5} + 5992894 T^{6} - 60219675 T^{7} + 565373456 T^{8} - 5604724289 T^{9} + 51028787635 T^{10} - 428234401720 T^{11} + 51028787635 p T^{12} - 5604724289 p^{2} T^{13} + 565373456 p^{3} T^{14} - 60219675 p^{4} T^{15} + 5992894 p^{5} T^{16} - 531865 p^{6} T^{17} + 51068 p^{7} T^{18} - 4322 p^{8} T^{19} + 281 p^{9} T^{20} - 20 p^{10} T^{21} + p^{11} T^{22} \) | |
| 79 | \( 1 - 15 T + 571 T^{2} - 6507 T^{3} + 138192 T^{4} - 1162572 T^{5} + 18355298 T^{6} - 102770505 T^{7} + 1474103091 T^{8} - 3917430053 T^{9} + 87211762843 T^{10} - 67971873592 T^{11} + 87211762843 p T^{12} - 3917430053 p^{2} T^{13} + 1474103091 p^{3} T^{14} - 102770505 p^{4} T^{15} + 18355298 p^{5} T^{16} - 1162572 p^{6} T^{17} + 138192 p^{7} T^{18} - 6507 p^{8} T^{19} + 571 p^{9} T^{20} - 15 p^{10} T^{21} + p^{11} T^{22} \) | |
| 83 | \( 1 - 2 T + 305 T^{2} + 404 T^{3} + 47807 T^{4} + 172945 T^{5} + 6235712 T^{6} + 25348229 T^{7} + 713062419 T^{8} + 2727590341 T^{9} + 68597783678 T^{10} + 246823772998 T^{11} + 68597783678 p T^{12} + 2727590341 p^{2} T^{13} + 713062419 p^{3} T^{14} + 25348229 p^{4} T^{15} + 6235712 p^{5} T^{16} + 172945 p^{6} T^{17} + 47807 p^{7} T^{18} + 404 p^{8} T^{19} + 305 p^{9} T^{20} - 2 p^{10} T^{21} + p^{11} T^{22} \) | |
| 89 | \( 1 - T + 467 T^{2} + 587 T^{3} + 104971 T^{4} + 426358 T^{5} + 15306383 T^{6} + 114752722 T^{7} + 1669796932 T^{8} + 18343921590 T^{9} + 155470497383 T^{10} + 1963819049122 T^{11} + 155470497383 p T^{12} + 18343921590 p^{2} T^{13} + 1669796932 p^{3} T^{14} + 114752722 p^{4} T^{15} + 15306383 p^{5} T^{16} + 426358 p^{6} T^{17} + 104971 p^{7} T^{18} + 587 p^{8} T^{19} + 467 p^{9} T^{20} - p^{10} T^{21} + p^{11} T^{22} \) | |
| 97 | \( 1 - 33 T + 1061 T^{2} - 23268 T^{3} + 469045 T^{4} - 7907015 T^{5} + 123492206 T^{6} - 1709966267 T^{7} + 22157957367 T^{8} - 260674072892 T^{9} + 2884883742560 T^{10} - 29273429105914 T^{11} + 2884883742560 p T^{12} - 260674072892 p^{2} T^{13} + 22157957367 p^{3} T^{14} - 1709966267 p^{4} T^{15} + 123492206 p^{5} T^{16} - 7907015 p^{6} T^{17} + 469045 p^{7} T^{18} - 23268 p^{8} T^{19} + 1061 p^{9} T^{20} - 33 p^{10} T^{21} + p^{11} T^{22} \) | |
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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
−2.75070915590016032023586819177, −2.61241176035099434387262271426, −2.45455601266496512566813941194, −2.20222983245879077922312157105, −2.06024302936800896301402439425, −2.05195083562061125482126463201, −2.03106197751387998016086735521, −2.01069980279419607931013602299, −1.88785560161936967126275458425, −1.85609906875105967730868431364, −1.79993681441941106128517571576, −1.75400090188036055137575818394, −1.74940356495530072567829678744, −1.61755516512236721951446419328, −1.28009256159153936207391914742, −1.08863142844628626707575259056, −1.07084154415259211426971947153, −1.01546051318070919505274825744, −0.866260996621311559799543001104, −0.854593226449390796254641889613, −0.810234281344195161223751030555, −0.76009521285221804483052448223, −0.62130052108315092691545390764, −0.54757526782698107701746789297, −0.34900632650713423448396281525, 0.34900632650713423448396281525, 0.54757526782698107701746789297, 0.62130052108315092691545390764, 0.76009521285221804483052448223, 0.810234281344195161223751030555, 0.854593226449390796254641889613, 0.866260996621311559799543001104, 1.01546051318070919505274825744, 1.07084154415259211426971947153, 1.08863142844628626707575259056, 1.28009256159153936207391914742, 1.61755516512236721951446419328, 1.74940356495530072567829678744, 1.75400090188036055137575818394, 1.79993681441941106128517571576, 1.85609906875105967730868431364, 1.88785560161936967126275458425, 2.01069980279419607931013602299, 2.03106197751387998016086735521, 2.05195083562061125482126463201, 2.06024302936800896301402439425, 2.20222983245879077922312157105, 2.45455601266496512566813941194, 2.61241176035099434387262271426, 2.75070915590016032023586819177
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.