Dirichlet series
| L(s) = 1 | + 3-s − 5·5-s + 10·7-s − 7·9-s − 3·11-s + 13-s − 5·15-s + 6·17-s + 15·19-s + 10·21-s − 6·23-s − 25-s − 9·27-s − 31·29-s + 14·31-s − 3·33-s − 50·35-s + 37-s + 39-s + 22·41-s + 37·43-s + 35·45-s − 2·47-s + 34·49-s + 6·51-s − 29·53-s + 15·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 2.23·5-s + 3.77·7-s − 7/3·9-s − 0.904·11-s + 0.277·13-s − 1.29·15-s + 1.45·17-s + 3.44·19-s + 2.18·21-s − 1.25·23-s − 1/5·25-s − 1.73·27-s − 5.75·29-s + 2.51·31-s − 0.522·33-s − 8.45·35-s + 0.164·37-s + 0.160·39-s + 3.43·41-s + 5.64·43-s + 5.21·45-s − 0.291·47-s + 34/7·49-s + 0.840·51-s − 3.98·53-s + 2.02·55-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{66} \cdot 83^{11}\right)^{s/2} \, \Gamma_{\C}(s)^{11} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{66} \cdot 83^{11}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{11} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(22\) |
| Conductor: | \(2^{66} \cdot 83^{11}\) |
| Sign: | $1$ |
| Analytic conductor: | \(7.99626\times 10^{17}\) |
| Root analytic conductor: | \(6.51279\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((22,\ 2^{66} \cdot 83^{11} ,\ ( \ : [1/2]^{11} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(17.97522113\) |
| \(L(\frac12)\) | \(\approx\) | \(17.97522113\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 83 | \( ( 1 - T )^{11} \) | |
| good | 3 | \( 1 - T + 8 T^{2} - 2 p T^{3} + 34 T^{4} - 26 T^{5} + 128 T^{6} - 118 T^{7} + 152 p T^{8} - 391 T^{9} + 1391 T^{10} - 1156 T^{11} + 1391 p T^{12} - 391 p^{2} T^{13} + 152 p^{4} T^{14} - 118 p^{4} T^{15} + 128 p^{5} T^{16} - 26 p^{6} T^{17} + 34 p^{7} T^{18} - 2 p^{9} T^{19} + 8 p^{9} T^{20} - p^{10} T^{21} + p^{11} T^{22} \) |
| 5 | \( 1 + p T + 26 T^{2} + 92 T^{3} + 74 p T^{4} + 1143 T^{5} + 3749 T^{6} + 10116 T^{7} + 28886 T^{8} + 14044 p T^{9} + 178588 T^{10} + 390208 T^{11} + 178588 p T^{12} + 14044 p^{3} T^{13} + 28886 p^{3} T^{14} + 10116 p^{4} T^{15} + 3749 p^{5} T^{16} + 1143 p^{6} T^{17} + 74 p^{8} T^{18} + 92 p^{8} T^{19} + 26 p^{9} T^{20} + p^{11} T^{21} + p^{11} T^{22} \) | |
| 7 | \( 1 - 10 T + 66 T^{2} - 332 T^{3} + 1471 T^{4} - 821 p T^{5} + 20758 T^{6} - 69576 T^{7} + 221351 T^{8} - 663666 T^{9} + 38764 p^{2} T^{10} - 5146342 T^{11} + 38764 p^{3} T^{12} - 663666 p^{2} T^{13} + 221351 p^{3} T^{14} - 69576 p^{4} T^{15} + 20758 p^{5} T^{16} - 821 p^{7} T^{17} + 1471 p^{7} T^{18} - 332 p^{8} T^{19} + 66 p^{9} T^{20} - 10 p^{10} T^{21} + p^{11} T^{22} \) | |
| 11 | \( 1 + 3 T + 4 p T^{2} + 178 T^{3} + 1260 T^{4} + 4880 T^{5} + 27252 T^{6} + 93200 T^{7} + 450688 T^{8} + 1413007 T^{9} + 540411 p T^{10} + 17323736 T^{11} + 540411 p^{2} T^{12} + 1413007 p^{2} T^{13} + 450688 p^{3} T^{14} + 93200 p^{4} T^{15} + 27252 p^{5} T^{16} + 4880 p^{6} T^{17} + 1260 p^{7} T^{18} + 178 p^{8} T^{19} + 4 p^{10} T^{20} + 3 p^{10} T^{21} + p^{11} T^{22} \) | |
| 13 | \( 1 - T + 54 T^{2} - 132 T^{3} + 1746 T^{4} - 443 p T^{5} + 40525 T^{6} - 160388 T^{7} + 58022 p T^{8} - 3132600 T^{9} + 11639912 T^{10} - 46661872 T^{11} + 11639912 p T^{12} - 3132600 p^{2} T^{13} + 58022 p^{4} T^{14} - 160388 p^{4} T^{15} + 40525 p^{5} T^{16} - 443 p^{7} T^{17} + 1746 p^{7} T^{18} - 132 p^{8} T^{19} + 54 p^{9} T^{20} - p^{10} T^{21} + p^{11} T^{22} \) | |
| 17 | \( 1 - 6 T + 112 T^{2} - 558 T^{3} + 5585 T^{4} - 24097 T^{5} + 171016 T^{6} - 675408 T^{7} + 3845475 T^{8} - 14612074 T^{9} + 72496868 T^{10} - 267030852 T^{11} + 72496868 p T^{12} - 14612074 p^{2} T^{13} + 3845475 p^{3} T^{14} - 675408 p^{4} T^{15} + 171016 p^{5} T^{16} - 24097 p^{6} T^{17} + 5585 p^{7} T^{18} - 558 p^{8} T^{19} + 112 p^{9} T^{20} - 6 p^{10} T^{21} + p^{11} T^{22} \) | |
| 19 | \( 1 - 15 T + 216 T^{2} - 1908 T^{3} + 15970 T^{4} - 98997 T^{5} + 589117 T^{6} - 2699020 T^{7} + 12386672 T^{8} - 43911684 T^{9} + 183594586 T^{10} - 34467264 p T^{11} + 183594586 p T^{12} - 43911684 p^{2} T^{13} + 12386672 p^{3} T^{14} - 2699020 p^{4} T^{15} + 589117 p^{5} T^{16} - 98997 p^{6} T^{17} + 15970 p^{7} T^{18} - 1908 p^{8} T^{19} + 216 p^{9} T^{20} - 15 p^{10} T^{21} + p^{11} T^{22} \) | |
| 23 | \( 1 + 6 T + 170 T^{2} + 717 T^{3} + 12292 T^{4} + 35743 T^{5} + 536209 T^{6} + 1059550 T^{7} + 17316344 T^{8} + 24895707 T^{9} + 466078938 T^{10} + 569908810 T^{11} + 466078938 p T^{12} + 24895707 p^{2} T^{13} + 17316344 p^{3} T^{14} + 1059550 p^{4} T^{15} + 536209 p^{5} T^{16} + 35743 p^{6} T^{17} + 12292 p^{7} T^{18} + 717 p^{8} T^{19} + 170 p^{9} T^{20} + 6 p^{10} T^{21} + p^{11} T^{22} \) | |
| 29 | \( 1 + 31 T + 608 T^{2} + 8812 T^{3} + 105176 T^{4} + 1072448 T^{5} + 9647626 T^{6} + 77627766 T^{7} + 566383644 T^{8} + 3768716175 T^{9} + 23014331721 T^{10} + 129151568552 T^{11} + 23014331721 p T^{12} + 3768716175 p^{2} T^{13} + 566383644 p^{3} T^{14} + 77627766 p^{4} T^{15} + 9647626 p^{5} T^{16} + 1072448 p^{6} T^{17} + 105176 p^{7} T^{18} + 8812 p^{8} T^{19} + 608 p^{9} T^{20} + 31 p^{10} T^{21} + p^{11} T^{22} \) | |
| 31 | \( 1 - 14 T + 246 T^{2} - 74 p T^{3} + 24301 T^{4} - 174149 T^{5} + 1424122 T^{6} - 8450210 T^{7} + 59378453 T^{8} - 310065850 T^{9} + 2023398610 T^{10} - 9929498050 T^{11} + 2023398610 p T^{12} - 310065850 p^{2} T^{13} + 59378453 p^{3} T^{14} - 8450210 p^{4} T^{15} + 1424122 p^{5} T^{16} - 174149 p^{6} T^{17} + 24301 p^{7} T^{18} - 74 p^{9} T^{19} + 246 p^{9} T^{20} - 14 p^{10} T^{21} + p^{11} T^{22} \) | |
| 37 | \( 1 - T + 156 T^{2} - 284 T^{3} + 12400 T^{4} - 28156 T^{5} + 684334 T^{6} - 1587434 T^{7} + 30531940 T^{8} - 62466849 T^{9} + 1218214221 T^{10} - 2220683208 T^{11} + 1218214221 p T^{12} - 62466849 p^{2} T^{13} + 30531940 p^{3} T^{14} - 1587434 p^{4} T^{15} + 684334 p^{5} T^{16} - 28156 p^{6} T^{17} + 12400 p^{7} T^{18} - 284 p^{8} T^{19} + 156 p^{9} T^{20} - p^{10} T^{21} + p^{11} T^{22} \) | |
| 41 | \( 1 - 22 T + 386 T^{2} - 4589 T^{3} + 48874 T^{4} - 430575 T^{5} + 3588121 T^{6} - 26564038 T^{7} + 190968042 T^{8} - 1267126531 T^{9} + 8399912440 T^{10} - 53243631658 T^{11} + 8399912440 p T^{12} - 1267126531 p^{2} T^{13} + 190968042 p^{3} T^{14} - 26564038 p^{4} T^{15} + 3588121 p^{5} T^{16} - 430575 p^{6} T^{17} + 48874 p^{7} T^{18} - 4589 p^{8} T^{19} + 386 p^{9} T^{20} - 22 p^{10} T^{21} + p^{11} T^{22} \) | |
| 43 | \( 1 - 37 T + 862 T^{2} - 14338 T^{3} + 191934 T^{4} - 2124993 T^{5} + 20281283 T^{6} - 169126472 T^{7} + 1271498886 T^{8} - 8796133490 T^{9} + 58412362816 T^{10} - 381776207148 T^{11} + 58412362816 p T^{12} - 8796133490 p^{2} T^{13} + 1271498886 p^{3} T^{14} - 169126472 p^{4} T^{15} + 20281283 p^{5} T^{16} - 2124993 p^{6} T^{17} + 191934 p^{7} T^{18} - 14338 p^{8} T^{19} + 862 p^{9} T^{20} - 37 p^{10} T^{21} + p^{11} T^{22} \) | |
| 47 | \( 1 + 2 T + 273 T^{2} + 362 T^{3} + 38859 T^{4} + 38698 T^{5} + 3807451 T^{6} + 2996408 T^{7} + 281672154 T^{8} + 184416020 T^{9} + 16471154586 T^{10} + 9521542844 T^{11} + 16471154586 p T^{12} + 184416020 p^{2} T^{13} + 281672154 p^{3} T^{14} + 2996408 p^{4} T^{15} + 3807451 p^{5} T^{16} + 38698 p^{6} T^{17} + 38859 p^{7} T^{18} + 362 p^{8} T^{19} + 273 p^{9} T^{20} + 2 p^{10} T^{21} + p^{11} T^{22} \) | |
| 53 | \( 1 + 29 T + 714 T^{2} + 12026 T^{3} + 179810 T^{4} + 2209945 T^{5} + 24839329 T^{6} + 244738216 T^{7} + 2258713698 T^{8} + 18930887970 T^{9} + 151585078612 T^{10} + 1124207929468 T^{11} + 151585078612 p T^{12} + 18930887970 p^{2} T^{13} + 2258713698 p^{3} T^{14} + 244738216 p^{4} T^{15} + 24839329 p^{5} T^{16} + 2209945 p^{6} T^{17} + 179810 p^{7} T^{18} + 12026 p^{8} T^{19} + 714 p^{9} T^{20} + 29 p^{10} T^{21} + p^{11} T^{22} \) | |
| 59 | \( 1 + T + 378 T^{2} + 1388 T^{3} + 66242 T^{4} + 407224 T^{5} + 7802890 T^{6} + 58262242 T^{7} + 735298734 T^{8} + 5249841149 T^{9} + 56212645265 T^{10} + 348922081976 T^{11} + 56212645265 p T^{12} + 5249841149 p^{2} T^{13} + 735298734 p^{3} T^{14} + 58262242 p^{4} T^{15} + 7802890 p^{5} T^{16} + 407224 p^{6} T^{17} + 66242 p^{7} T^{18} + 1388 p^{8} T^{19} + 378 p^{9} T^{20} + p^{10} T^{21} + p^{11} T^{22} \) | |
| 61 | \( 1 + 15 T + 504 T^{2} + 5768 T^{3} + 112900 T^{4} + 1054924 T^{5} + 15574722 T^{6} + 123757914 T^{7} + 1526560100 T^{8} + 10631728119 T^{9} + 115500415517 T^{10} + 720548296688 T^{11} + 115500415517 p T^{12} + 10631728119 p^{2} T^{13} + 1526560100 p^{3} T^{14} + 123757914 p^{4} T^{15} + 15574722 p^{5} T^{16} + 1054924 p^{6} T^{17} + 112900 p^{7} T^{18} + 5768 p^{8} T^{19} + 504 p^{9} T^{20} + 15 p^{10} T^{21} + p^{11} T^{22} \) | |
| 67 | \( 1 - 45 T + 1280 T^{2} - 27628 T^{3} + 496488 T^{4} - 7662697 T^{5} + 104590695 T^{6} - 1279764736 T^{7} + 14204363314 T^{8} - 143796374674 T^{9} + 1335554426528 T^{10} - 11396229567656 T^{11} + 1335554426528 p T^{12} - 143796374674 p^{2} T^{13} + 14204363314 p^{3} T^{14} - 1279764736 p^{4} T^{15} + 104590695 p^{5} T^{16} - 7662697 p^{6} T^{17} + 496488 p^{7} T^{18} - 27628 p^{8} T^{19} + 1280 p^{9} T^{20} - 45 p^{10} T^{21} + p^{11} T^{22} \) | |
| 71 | \( 1 + 541 T^{2} - 432 T^{3} + 141351 T^{4} - 194624 T^{5} + 23786563 T^{6} - 40466112 T^{7} + 2883218106 T^{8} - 5129176000 T^{9} + 264829078962 T^{10} - 437729080096 T^{11} + 264829078962 p T^{12} - 5129176000 p^{2} T^{13} + 2883218106 p^{3} T^{14} - 40466112 p^{4} T^{15} + 23786563 p^{5} T^{16} - 194624 p^{6} T^{17} + 141351 p^{7} T^{18} - 432 p^{8} T^{19} + 541 p^{9} T^{20} + p^{11} T^{22} \) | |
| 73 | \( 1 + 4 T + 397 T^{2} + 2350 T^{3} + 82961 T^{4} + 574236 T^{5} + 12369477 T^{6} + 86072344 T^{7} + 1429995346 T^{8} + 9281669920 T^{9} + 130739174794 T^{10} + 767947935732 T^{11} + 130739174794 p T^{12} + 9281669920 p^{2} T^{13} + 1429995346 p^{3} T^{14} + 86072344 p^{4} T^{15} + 12369477 p^{5} T^{16} + 574236 p^{6} T^{17} + 82961 p^{7} T^{18} + 2350 p^{8} T^{19} + 397 p^{9} T^{20} + 4 p^{10} T^{21} + p^{11} T^{22} \) | |
| 79 | \( 1 - 10 T + 417 T^{2} - 3154 T^{3} + 80935 T^{4} - 520810 T^{5} + 11060631 T^{6} - 65596968 T^{7} + 1216140454 T^{8} - 6549568076 T^{9} + 110178653494 T^{10} - 545537513996 T^{11} + 110178653494 p T^{12} - 6549568076 p^{2} T^{13} + 1216140454 p^{3} T^{14} - 65596968 p^{4} T^{15} + 11060631 p^{5} T^{16} - 520810 p^{6} T^{17} + 80935 p^{7} T^{18} - 3154 p^{8} T^{19} + 417 p^{9} T^{20} - 10 p^{10} T^{21} + p^{11} T^{22} \) | |
| 89 | \( 1 + 423 T^{2} + 312 T^{3} + 92539 T^{4} + 54016 T^{5} + 13874429 T^{6} - 3659616 T^{7} + 1618530554 T^{8} - 1962901760 T^{9} + 159849890390 T^{10} - 255313733040 T^{11} + 159849890390 p T^{12} - 1962901760 p^{2} T^{13} + 1618530554 p^{3} T^{14} - 3659616 p^{4} T^{15} + 13874429 p^{5} T^{16} + 54016 p^{6} T^{17} + 92539 p^{7} T^{18} + 312 p^{8} T^{19} + 423 p^{9} T^{20} + p^{11} T^{22} \) | |
| 97 | \( 1 + 18 T + 887 T^{2} + 12348 T^{3} + 345907 T^{4} + 3930090 T^{5} + 81893093 T^{6} + 787607120 T^{7} + 13551034890 T^{8} + 113191040388 T^{9} + 1688228249382 T^{10} + 12434429339368 T^{11} + 1688228249382 p T^{12} + 113191040388 p^{2} T^{13} + 13551034890 p^{3} T^{14} + 787607120 p^{4} T^{15} + 81893093 p^{5} T^{16} + 3930090 p^{6} T^{17} + 345907 p^{7} T^{18} + 12348 p^{8} T^{19} + 887 p^{9} T^{20} + 18 p^{10} T^{21} + p^{11} T^{22} \) | |
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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
−2.57492575706951274487881901394, −2.57380681820074691190389863231, −2.52845518024999443602248104570, −2.24427377687637600751400603888, −2.22434535418272217723240573251, −2.16256637132308285696250714463, −2.13851648757093660184603404845, −2.02356617983755171362972324383, −1.84361729721159942500948974689, −1.80099728192678521419399286340, −1.74903611699325058944845181995, −1.65619000633189583100033510055, −1.47585548637476200908206648231, −1.38864244801790573097032259583, −1.37013075365319531907343666082, −1.25665555534761804939259484114, −1.05304305319646978120256679099, −0.937614210246776329969505166871, −0.916665770773439725557272321345, −0.806675063899257739382091501278, −0.59409011274126506829526468487, −0.48338513316367990371699408096, −0.32274874664282204413808336217, −0.30169614390409789935182510338, −0.21298952271226919579573875676, 0.21298952271226919579573875676, 0.30169614390409789935182510338, 0.32274874664282204413808336217, 0.48338513316367990371699408096, 0.59409011274126506829526468487, 0.806675063899257739382091501278, 0.916665770773439725557272321345, 0.937614210246776329969505166871, 1.05304305319646978120256679099, 1.25665555534761804939259484114, 1.37013075365319531907343666082, 1.38864244801790573097032259583, 1.47585548637476200908206648231, 1.65619000633189583100033510055, 1.74903611699325058944845181995, 1.80099728192678521419399286340, 1.84361729721159942500948974689, 2.02356617983755171362972324383, 2.13851648757093660184603404845, 2.16256637132308285696250714463, 2.22434535418272217723240573251, 2.24427377687637600751400603888, 2.52845518024999443602248104570, 2.57380681820074691190389863231, 2.57492575706951274487881901394
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.