Dirichlet series
| L(s) = 1 | + 9·4-s + 22·9-s + 20·11-s + 36·16-s − 10·19-s + 14·29-s − 6·31-s + 198·36-s − 8·41-s + 180·44-s + 65·49-s − 8·59-s − 30·61-s + 85·64-s + 44·71-s − 90·76-s + 18·79-s + 215·81-s + 86·89-s + 440·99-s + 38·101-s − 14·109-s + 126·116-s + 210·121-s − 54·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 9/2·4-s + 22/3·9-s + 6.03·11-s + 9·16-s − 2.29·19-s + 2.59·29-s − 1.07·31-s + 33·36-s − 1.24·41-s + 27.1·44-s + 65/7·49-s − 1.04·59-s − 3.84·61-s + 85/8·64-s + 5.22·71-s − 10.3·76-s + 2.02·79-s + 23.8·81-s + 9.11·89-s + 44.2·99-s + 3.78·101-s − 1.34·109-s + 11.6·116-s + 19.0·121-s − 4.84·124-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{60} \cdot 11^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{60} \cdot 11^{20}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(40\) |
| Conductor: | \(5^{60} \cdot 11^{20}\) |
| Sign: | $1$ |
| Analytic conductor: | \(6.48031\times 10^{20}\) |
| Root analytic conductor: | \(3.31352\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((40,\ 5^{60} \cdot 11^{20} ,\ ( \ : [1/2]^{20} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(7957.600275\) |
| \(L(\frac12)\) | \(\approx\) | \(7957.600275\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 5 | \( 1 \) |
| 11 | \( ( 1 - T )^{20} \) | |
| good | 2 | \( 1 - 9 T^{2} + 45 T^{4} - 83 p T^{6} + 257 p T^{8} - 725 p T^{10} + 3791 T^{12} - 2319 p^{2} T^{14} + 10725 p T^{16} - 11761 p^{2} T^{18} + 97313 T^{20} - 11761 p^{4} T^{22} + 10725 p^{5} T^{24} - 2319 p^{8} T^{26} + 3791 p^{8} T^{28} - 725 p^{11} T^{30} + 257 p^{13} T^{32} - 83 p^{15} T^{34} + 45 p^{16} T^{36} - 9 p^{18} T^{38} + p^{20} T^{40} \) |
| 3 | \( 1 - 22 T^{2} + 269 T^{4} - 2300 T^{6} + 15274 T^{8} - 83116 T^{10} + 385016 T^{12} - 1561250 T^{14} + 5682541 T^{16} - 18948467 T^{18} + 58782793 T^{20} - 18948467 p^{2} T^{22} + 5682541 p^{4} T^{24} - 1561250 p^{6} T^{26} + 385016 p^{8} T^{28} - 83116 p^{10} T^{30} + 15274 p^{12} T^{32} - 2300 p^{14} T^{34} + 269 p^{16} T^{36} - 22 p^{18} T^{38} + p^{20} T^{40} \) | |
| 7 | \( 1 - 65 T^{2} + 2105 T^{4} - 45595 T^{6} + 746295 T^{8} - 9863680 T^{10} + 15682565 p T^{12} - 1061093820 T^{14} + 9140525070 T^{16} - 10245550020 p T^{18} + 520515176773 T^{20} - 10245550020 p^{3} T^{22} + 9140525070 p^{4} T^{24} - 1061093820 p^{6} T^{26} + 15682565 p^{9} T^{28} - 9863680 p^{10} T^{30} + 746295 p^{12} T^{32} - 45595 p^{14} T^{34} + 2105 p^{16} T^{36} - 65 p^{18} T^{38} + p^{20} T^{40} \) | |
| 13 | \( 1 - 163 T^{2} + 12800 T^{4} - 646469 T^{6} + 23681649 T^{8} - 673821521 T^{10} + 15601591186 T^{12} - 304656535059 T^{14} + 5166884383790 T^{16} - 77934223229723 T^{18} + 1062535025938883 T^{20} - 77934223229723 p^{2} T^{22} + 5166884383790 p^{4} T^{24} - 304656535059 p^{6} T^{26} + 15601591186 p^{8} T^{28} - 673821521 p^{10} T^{30} + 23681649 p^{12} T^{32} - 646469 p^{14} T^{34} + 12800 p^{16} T^{36} - 163 p^{18} T^{38} + p^{20} T^{40} \) | |
| 17 | \( 1 - 120 T^{2} + 7086 T^{4} - 270095 T^{6} + 7326801 T^{8} - 147565340 T^{10} + 2246274684 T^{12} - 26205942325 T^{14} + 248749912664 T^{16} - 2385112054180 T^{18} + 32168144752873 T^{20} - 2385112054180 p^{2} T^{22} + 248749912664 p^{4} T^{24} - 26205942325 p^{6} T^{26} + 2246274684 p^{8} T^{28} - 147565340 p^{10} T^{30} + 7326801 p^{12} T^{32} - 270095 p^{14} T^{34} + 7086 p^{16} T^{36} - 120 p^{18} T^{38} + p^{20} T^{40} \) | |
| 19 | \( ( 1 + 5 T + 105 T^{2} + 525 T^{3} + 5920 T^{4} + 28015 T^{5} + 225955 T^{6} + 994050 T^{7} + 6362470 T^{8} + 25369905 T^{9} + 137436573 T^{10} + 25369905 p T^{11} + 6362470 p^{2} T^{12} + 994050 p^{3} T^{13} + 225955 p^{4} T^{14} + 28015 p^{5} T^{15} + 5920 p^{6} T^{16} + 525 p^{7} T^{17} + 105 p^{8} T^{18} + 5 p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 23 | \( 1 - 187 T^{2} + 19464 T^{4} - 1434150 T^{6} + 82628344 T^{8} - 3921444306 T^{10} + 6879808827 p T^{12} - 5540754340210 T^{14} + 170635811611086 T^{16} - 4661162001247192 T^{18} + 113476075004432703 T^{20} - 4661162001247192 p^{2} T^{22} + 170635811611086 p^{4} T^{24} - 5540754340210 p^{6} T^{26} + 6879808827 p^{9} T^{28} - 3921444306 p^{10} T^{30} + 82628344 p^{12} T^{32} - 1434150 p^{14} T^{34} + 19464 p^{16} T^{36} - 187 p^{18} T^{38} + p^{20} T^{40} \) | |
| 29 | \( ( 1 - 7 T + 180 T^{2} - 1106 T^{3} + 15864 T^{4} - 2981 p T^{5} + 923156 T^{6} - 4516596 T^{7} + 39661130 T^{8} - 173846642 T^{9} + 1307718203 T^{10} - 173846642 p T^{11} + 39661130 p^{2} T^{12} - 4516596 p^{3} T^{13} + 923156 p^{4} T^{14} - 2981 p^{6} T^{15} + 15864 p^{6} T^{16} - 1106 p^{7} T^{17} + 180 p^{8} T^{18} - 7 p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 31 | \( ( 1 + 3 T + 110 T^{2} + 487 T^{3} + 8625 T^{4} + 40051 T^{5} + 467169 T^{6} + 2185387 T^{7} + 20721261 T^{8} + 87968563 T^{9} + 708031821 T^{10} + 87968563 p T^{11} + 20721261 p^{2} T^{12} + 2185387 p^{3} T^{13} + 467169 p^{4} T^{14} + 40051 p^{5} T^{15} + 8625 p^{6} T^{16} + 487 p^{7} T^{17} + 110 p^{8} T^{18} + 3 p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 37 | \( 1 - 442 T^{2} + 98155 T^{4} - 14539006 T^{6} + 1609800259 T^{8} - 141591480284 T^{10} + 10265419661926 T^{12} - 628306868539131 T^{14} + 32980296667675135 T^{16} - 1499593039404577687 T^{18} + 59395567578449950533 T^{20} - 1499593039404577687 p^{2} T^{22} + 32980296667675135 p^{4} T^{24} - 628306868539131 p^{6} T^{26} + 10265419661926 p^{8} T^{28} - 141591480284 p^{10} T^{30} + 1609800259 p^{12} T^{32} - 14539006 p^{14} T^{34} + 98155 p^{16} T^{36} - 442 p^{18} T^{38} + p^{20} T^{40} \) | |
| 41 | \( ( 1 + 4 T + 74 T^{2} + 85 T^{3} + 5316 T^{4} + 19428 T^{5} + 288303 T^{6} + 851097 T^{7} + 11868974 T^{8} + 58175357 T^{9} + 594095867 T^{10} + 58175357 p T^{11} + 11868974 p^{2} T^{12} + 851097 p^{3} T^{13} + 288303 p^{4} T^{14} + 19428 p^{5} T^{15} + 5316 p^{6} T^{16} + 85 p^{7} T^{17} + 74 p^{8} T^{18} + 4 p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 43 | \( 1 - 445 T^{2} + 2285 p T^{4} - 14160820 T^{6} + 1480797575 T^{8} - 118465135870 T^{10} + 7474200142525 T^{12} - 380487435846820 T^{14} + 16164920634359870 T^{16} - 620243784167340445 T^{18} + 24875343780396148173 T^{20} - 620243784167340445 p^{2} T^{22} + 16164920634359870 p^{4} T^{24} - 380487435846820 p^{6} T^{26} + 7474200142525 p^{8} T^{28} - 118465135870 p^{10} T^{30} + 1480797575 p^{12} T^{32} - 14160820 p^{14} T^{34} + 2285 p^{17} T^{36} - 445 p^{18} T^{38} + p^{20} T^{40} \) | |
| 47 | \( 1 - 209 T^{2} + 25351 T^{4} - 2254733 T^{6} + 166900758 T^{8} - 10945385086 T^{10} + 659646268807 T^{12} - 37208136303698 T^{14} + 1982142667975804 T^{16} - 100079176849481524 T^{18} + 4815194824723847203 T^{20} - 100079176849481524 p^{2} T^{22} + 1982142667975804 p^{4} T^{24} - 37208136303698 p^{6} T^{26} + 659646268807 p^{8} T^{28} - 10945385086 p^{10} T^{30} + 166900758 p^{12} T^{32} - 2254733 p^{14} T^{34} + 25351 p^{16} T^{36} - 209 p^{18} T^{38} + p^{20} T^{40} \) | |
| 53 | \( 1 - 387 T^{2} + 82548 T^{4} - 12725994 T^{6} + 1565108802 T^{8} - 161286858678 T^{10} + 14340269825613 T^{12} - 1120202896676499 T^{14} + 77769553886249302 T^{16} - 4833700639774932752 T^{18} + \)\(27\!\cdots\!13\)\( T^{20} - 4833700639774932752 p^{2} T^{22} + 77769553886249302 p^{4} T^{24} - 1120202896676499 p^{6} T^{26} + 14340269825613 p^{8} T^{28} - 161286858678 p^{10} T^{30} + 1565108802 p^{12} T^{32} - 12725994 p^{14} T^{34} + 82548 p^{16} T^{36} - 387 p^{18} T^{38} + p^{20} T^{40} \) | |
| 59 | \( ( 1 + 4 T + 302 T^{2} + 1205 T^{3} + 47180 T^{4} + 173852 T^{5} + 4956060 T^{6} + 277650 p T^{7} + 394376888 T^{8} + 1170258984 T^{9} + 25499969083 T^{10} + 1170258984 p T^{11} + 394376888 p^{2} T^{12} + 277650 p^{4} T^{13} + 4956060 p^{4} T^{14} + 173852 p^{5} T^{15} + 47180 p^{6} T^{16} + 1205 p^{7} T^{17} + 302 p^{8} T^{18} + 4 p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 61 | \( ( 1 + 15 T + 564 T^{2} + 6806 T^{3} + 141779 T^{4} + 1430533 T^{5} + 21412784 T^{6} + 183870463 T^{7} + 2177563907 T^{8} + 15975349537 T^{9} + 156740644207 T^{10} + 15975349537 p T^{11} + 2177563907 p^{2} T^{12} + 183870463 p^{3} T^{13} + 21412784 p^{4} T^{14} + 1430533 p^{5} T^{15} + 141779 p^{6} T^{16} + 6806 p^{7} T^{17} + 564 p^{8} T^{18} + 15 p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 67 | \( 1 - 999 T^{2} + 487340 T^{4} - 154484196 T^{6} + 35721989084 T^{8} - 6410880872245 T^{10} + 927323013846401 T^{12} - 110786093946157681 T^{14} + 11108328821669980865 T^{16} - \)\(94\!\cdots\!14\)\( T^{18} + \)\(68\!\cdots\!13\)\( T^{20} - \)\(94\!\cdots\!14\)\( p^{2} T^{22} + 11108328821669980865 p^{4} T^{24} - 110786093946157681 p^{6} T^{26} + 927323013846401 p^{8} T^{28} - 6410880872245 p^{10} T^{30} + 35721989084 p^{12} T^{32} - 154484196 p^{14} T^{34} + 487340 p^{16} T^{36} - 999 p^{18} T^{38} + p^{20} T^{40} \) | |
| 71 | \( ( 1 - 22 T + 494 T^{2} - 6411 T^{3} + 82806 T^{4} - 745128 T^{5} + 7359560 T^{6} - 53715734 T^{7} + 514068426 T^{8} - 3667854834 T^{9} + 36568580129 T^{10} - 3667854834 p T^{11} + 514068426 p^{2} T^{12} - 53715734 p^{3} T^{13} + 7359560 p^{4} T^{14} - 745128 p^{5} T^{15} + 82806 p^{6} T^{16} - 6411 p^{7} T^{17} + 494 p^{8} T^{18} - 22 p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 73 | \( 1 - 928 T^{2} + 435310 T^{4} - 136344044 T^{6} + 31837265899 T^{8} - 5872668871321 T^{10} + 885828029406766 T^{12} - 111694855449025279 T^{14} + 11942267406321249645 T^{16} - \)\(10\!\cdots\!93\)\( T^{18} + \)\(85\!\cdots\!73\)\( T^{20} - \)\(10\!\cdots\!93\)\( p^{2} T^{22} + 11942267406321249645 p^{4} T^{24} - 111694855449025279 p^{6} T^{26} + 885828029406766 p^{8} T^{28} - 5872668871321 p^{10} T^{30} + 31837265899 p^{12} T^{32} - 136344044 p^{14} T^{34} + 435310 p^{16} T^{36} - 928 p^{18} T^{38} + p^{20} T^{40} \) | |
| 79 | \( ( 1 - 9 T + 426 T^{2} - 3173 T^{3} + 73518 T^{4} - 390886 T^{5} + 5653917 T^{6} - 8695893 T^{7} + 50608099 T^{8} + 2340768596 T^{9} - 17121703687 T^{10} + 2340768596 p T^{11} + 50608099 p^{2} T^{12} - 8695893 p^{3} T^{13} + 5653917 p^{4} T^{14} - 390886 p^{5} T^{15} + 73518 p^{6} T^{16} - 3173 p^{7} T^{17} + 426 p^{8} T^{18} - 9 p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 83 | \( 1 - 1195 T^{2} + 703220 T^{4} - 270630130 T^{6} + 76343041800 T^{8} - 16776350991650 T^{10} + 2979877940099225 T^{12} - 438170493171157930 T^{14} + 54182751576444150130 T^{16} - \)\(56\!\cdots\!20\)\( T^{18} + \)\(51\!\cdots\!23\)\( T^{20} - \)\(56\!\cdots\!20\)\( p^{2} T^{22} + 54182751576444150130 p^{4} T^{24} - 438170493171157930 p^{6} T^{26} + 2979877940099225 p^{8} T^{28} - 16776350991650 p^{10} T^{30} + 76343041800 p^{12} T^{32} - 270630130 p^{14} T^{34} + 703220 p^{16} T^{36} - 1195 p^{18} T^{38} + p^{20} T^{40} \) | |
| 89 | \( ( 1 - 43 T + 1313 T^{2} - 29866 T^{3} + 572807 T^{4} - 9408252 T^{5} + 137512633 T^{6} - 1795376896 T^{7} + 21304289952 T^{8} - 229609662833 T^{9} + 2267505014343 T^{10} - 229609662833 p T^{11} + 21304289952 p^{2} T^{12} - 1795376896 p^{3} T^{13} + 137512633 p^{4} T^{14} - 9408252 p^{5} T^{15} + 572807 p^{6} T^{16} - 29866 p^{7} T^{17} + 1313 p^{8} T^{18} - 43 p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 97 | \( 1 - 842 T^{2} + 337095 T^{4} - 86817706 T^{6} + 16548824829 T^{8} - 2557662706249 T^{10} + 341121237025636 T^{12} - 40729885823297591 T^{14} + 4448237411316781790 T^{16} - \)\(45\!\cdots\!47\)\( T^{18} + \)\(44\!\cdots\!53\)\( T^{20} - \)\(45\!\cdots\!47\)\( p^{2} T^{22} + 4448237411316781790 p^{4} T^{24} - 40729885823297591 p^{6} T^{26} + 341121237025636 p^{8} T^{28} - 2557662706249 p^{10} T^{30} + 16548824829 p^{12} T^{32} - 86817706 p^{14} T^{34} + 337095 p^{16} T^{36} - 842 p^{18} T^{38} + p^{20} T^{40} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{40} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.08312002845122447189537623474, −2.07500776893038330588217801948, −1.96363396133806323612623636643, −1.83699318612265971308935665519, −1.78094858099167051584133223149, −1.75808871280501675970341261426, −1.66481629707194025547336581553, −1.60086798312543430707935806884, −1.59274197733345509563261385540, −1.54754073041249080739639403097, −1.50078860812508858262969484021, −1.45459620297247129448149061617, −1.28101906087845264894993672094, −1.15723760341953614287101109870, −1.10744601362703471159411220377, −1.02492445154516623988981056963, −0.971787214360421212524700567363, −0.910910261742552237775741593680, −0.895145342462767177578857777332, −0.877770068167603994978797440835, −0.73468271889801983012390484363, −0.66153826605776664628568456474, −0.53418785274202648597652347045, −0.41231699472417868265307984006, −0.32358644026898687692554926965, 0.32358644026898687692554926965, 0.41231699472417868265307984006, 0.53418785274202648597652347045, 0.66153826605776664628568456474, 0.73468271889801983012390484363, 0.877770068167603994978797440835, 0.895145342462767177578857777332, 0.910910261742552237775741593680, 0.971787214360421212524700567363, 1.02492445154516623988981056963, 1.10744601362703471159411220377, 1.15723760341953614287101109870, 1.28101906087845264894993672094, 1.45459620297247129448149061617, 1.50078860812508858262969484021, 1.54754073041249080739639403097, 1.59274197733345509563261385540, 1.60086798312543430707935806884, 1.66481629707194025547336581553, 1.75808871280501675970341261426, 1.78094858099167051584133223149, 1.83699318612265971308935665519, 1.96363396133806323612623636643, 2.07500776893038330588217801948, 2.08312002845122447189537623474
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.