Dirichlet series
| L(s) = 1 | + 5·4-s + 22·9-s − 20·11-s + 8·16-s + 2·19-s − 2·29-s + 26·31-s + 110·36-s − 4·41-s − 100·44-s + 69·49-s + 16·59-s + 50·61-s − 7·64-s − 52·71-s + 10·76-s + 26·79-s + 231·81-s − 26·89-s − 440·99-s − 54·101-s + 22·109-s − 10·116-s + 210·121-s + 130·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 5/2·4-s + 22/3·9-s − 6.03·11-s + 2·16-s + 0.458·19-s − 0.371·29-s + 4.66·31-s + 55/3·36-s − 0.624·41-s − 15.0·44-s + 69/7·49-s + 2.08·59-s + 6.40·61-s − 7/8·64-s − 6.17·71-s + 1.14·76-s + 2.92·79-s + 77/3·81-s − 2.75·89-s − 44.2·99-s − 5.37·101-s + 2.10·109-s − 0.928·116-s + 19.0·121-s + 11.6·124-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
| \(L(1)\) | \(\approx\) | \(986.0003538\) |
| \(L(\frac12)\) | \(\approx\) | \(986.0003538\) |
| \(L(\frac{3}{2})\) | 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 | \( 1 - 5 T^{2} + 17 T^{4} - 19 p T^{6} + 21 p T^{8} + 7 p T^{10} - 189 T^{12} + 75 p^{2} T^{14} + 179 p T^{16} - 717 p^{2} T^{18} + 7801 T^{20} - 717 p^{4} T^{22} + 179 p^{5} T^{24} + 75 p^{8} T^{26} - 189 p^{8} T^{28} + 7 p^{11} T^{30} + 21 p^{13} T^{32} - 19 p^{15} T^{34} + 17 p^{16} T^{36} - 5 p^{18} T^{38} + p^{20} T^{40} \) |
| 3 | \( 1 - 22 T^{2} + 253 T^{4} - 692 p T^{6} + 13642 T^{8} - 75652 T^{10} + 365080 T^{12} - 1562402 T^{14} + 5997413 T^{16} - 6934897 p T^{18} + 65486833 T^{20} - 6934897 p^{3} T^{22} + 5997413 p^{4} T^{24} - 1562402 p^{6} T^{26} + 365080 p^{8} T^{28} - 75652 p^{10} T^{30} + 13642 p^{12} T^{32} - 692 p^{15} T^{34} + 253 p^{16} T^{36} - 22 p^{18} T^{38} + p^{20} T^{40} \) | |
| 7 | \( 1 - 69 T^{2} + 2301 T^{4} - 49859 T^{6} + 796543 T^{8} - 10084224 T^{10} + 106124335 T^{12} - 963675676 T^{14} + 7819457662 T^{16} - 1197662320 p^{2} T^{18} + 418904384397 T^{20} - 1197662320 p^{4} T^{22} + 7819457662 p^{4} T^{24} - 963675676 p^{6} T^{26} + 106124335 p^{8} T^{28} - 10084224 p^{10} T^{30} + 796543 p^{12} T^{32} - 49859 p^{14} T^{34} + 2301 p^{16} T^{36} - 69 p^{18} T^{38} + p^{20} T^{40} \) | |
| 13 | \( 1 - 107 T^{2} + 5924 T^{4} - 226937 T^{6} + 519613 p T^{8} - 166196137 T^{10} + 3510850038 T^{12} - 65274447171 T^{14} + 83489995346 p T^{16} - 16299123379631 T^{18} + 222205167926459 T^{20} - 16299123379631 p^{2} T^{22} + 83489995346 p^{5} T^{24} - 65274447171 p^{6} T^{26} + 3510850038 p^{8} T^{28} - 166196137 p^{10} T^{30} + 519613 p^{13} T^{32} - 226937 p^{14} T^{34} + 5924 p^{16} T^{36} - 107 p^{18} T^{38} + p^{20} T^{40} \) | |
| 17 | \( 1 - 156 T^{2} + 11806 T^{4} - 584023 T^{6} + 1266637 p T^{8} - 640541936 T^{10} + 16233063580 T^{12} - 364188107421 T^{14} + 436804393688 p T^{16} - 139930941973568 T^{18} + 2459526902891385 T^{20} - 139930941973568 p^{2} T^{22} + 436804393688 p^{5} T^{24} - 364188107421 p^{6} T^{26} + 16233063580 p^{8} T^{28} - 640541936 p^{10} T^{30} + 1266637 p^{13} T^{32} - 584023 p^{14} T^{34} + 11806 p^{16} T^{36} - 156 p^{18} T^{38} + p^{20} T^{40} \) | |
| 19 | \( ( 1 - T + 93 T^{2} - 79 T^{3} + 3552 T^{4} - 4217 T^{5} + 3571 p T^{6} - 188492 T^{7} + 556606 T^{8} - 5753093 T^{9} + 1603089 T^{10} - 5753093 p T^{11} + 556606 p^{2} T^{12} - 188492 p^{3} T^{13} + 3571 p^{5} T^{14} - 4217 p^{5} T^{15} + 3552 p^{6} T^{16} - 79 p^{7} T^{17} + 93 p^{8} T^{18} - p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 23 | \( 1 - 195 T^{2} + 20672 T^{4} - 1540510 T^{6} + 89454648 T^{8} - 4271138370 T^{10} + 173192827005 T^{12} - 6088501006570 T^{14} + 188057383978622 T^{16} - 5146548355148560 T^{18} + 125370403985921919 T^{20} - 5146548355148560 p^{2} T^{22} + 188057383978622 p^{4} T^{24} - 6088501006570 p^{6} T^{26} + 173192827005 p^{8} T^{28} - 4271138370 p^{10} T^{30} + 89454648 p^{12} T^{32} - 1540510 p^{14} T^{34} + 20672 p^{16} T^{36} - 195 p^{18} T^{38} + p^{20} T^{40} \) | |
| 29 | \( ( 1 + T + 144 T^{2} - 14 T^{3} + 10144 T^{4} - 10089 T^{5} + 504232 T^{6} - 664494 T^{7} + 20370266 T^{8} - 24038668 T^{9} + 665474627 T^{10} - 24038668 p T^{11} + 20370266 p^{2} T^{12} - 664494 p^{3} T^{13} + 504232 p^{4} T^{14} - 10089 p^{5} T^{15} + 10144 p^{6} T^{16} - 14 p^{7} T^{17} + 144 p^{8} T^{18} + p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 31 | \( ( 1 - 13 T + 294 T^{2} - 2737 T^{3} + 35509 T^{4} - 257137 T^{5} + 2470245 T^{6} - 14677737 T^{7} + 116212281 T^{8} - 592153645 T^{9} + 4087260953 T^{10} - 592153645 p T^{11} + 116212281 p^{2} T^{12} - 14677737 p^{3} T^{13} + 2470245 p^{4} T^{14} - 257137 p^{5} T^{15} + 35509 p^{6} T^{16} - 2737 p^{7} T^{17} + 294 p^{8} T^{18} - 13 p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 37 | \( 1 - 330 T^{2} + 56067 T^{4} - 6478950 T^{6} + 15436279 p T^{8} - 40983174060 T^{10} + 2497414086430 T^{12} - 133048620310395 T^{14} + 6317229969916727 T^{16} - 270431261506759135 T^{18} + 10498349557335411589 T^{20} - 270431261506759135 p^{2} T^{22} + 6317229969916727 p^{4} T^{24} - 133048620310395 p^{6} T^{26} + 2497414086430 p^{8} T^{28} - 40983174060 p^{10} T^{30} + 15436279 p^{13} T^{32} - 6478950 p^{14} T^{34} + 56067 p^{16} T^{36} - 330 p^{18} T^{38} + p^{20} T^{40} \) | |
| 41 | \( ( 1 + 2 T + 210 T^{2} + 431 T^{3} + 24478 T^{4} + 49602 T^{5} + 1963123 T^{6} + 3804993 T^{7} + 117560042 T^{8} + 210099013 T^{9} + 5452977215 T^{10} + 210099013 p T^{11} + 117560042 p^{2} T^{12} + 3804993 p^{3} T^{13} + 1963123 p^{4} T^{14} + 49602 p^{5} T^{15} + 24478 p^{6} T^{16} + 431 p^{7} T^{17} + 210 p^{8} T^{18} + 2 p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 43 | \( 1 - 217 T^{2} + 26143 T^{4} - 2176216 T^{6} + 139336411 T^{8} - 7236500378 T^{10} + 315134618177 T^{12} - 11510976082068 T^{14} + 346627854600506 T^{16} - 8745990218949021 T^{18} + 267527614065362469 T^{20} - 8745990218949021 p^{2} T^{22} + 346627854600506 p^{4} T^{24} - 11510976082068 p^{6} T^{26} + 315134618177 p^{8} T^{28} - 7236500378 p^{10} T^{30} + 139336411 p^{12} T^{32} - 2176216 p^{14} T^{34} + 26143 p^{16} T^{36} - 217 p^{18} T^{38} + p^{20} T^{40} \) | |
| 47 | \( 1 - 505 T^{2} + 127215 T^{4} - 21158805 T^{6} + 2600431270 T^{8} - 251211712030 T^{10} + 19880933206855 T^{12} - 1332582137396330 T^{14} + 78049895212372060 T^{16} - 4117333633060247180 T^{18} + \)\(20\!\cdots\!23\)\( T^{20} - 4117333633060247180 p^{2} T^{22} + 78049895212372060 p^{4} T^{24} - 1332582137396330 p^{6} T^{26} + 19880933206855 p^{8} T^{28} - 251211712030 p^{10} T^{30} + 2600431270 p^{12} T^{32} - 21158805 p^{14} T^{34} + 127215 p^{16} T^{36} - 505 p^{18} T^{38} + p^{20} T^{40} \) | |
| 53 | \( 1 - 619 T^{2} + 185716 T^{4} - 36124362 T^{6} + 5154472554 T^{8} - 580188957774 T^{10} + 54157938049125 T^{12} - 4343692082603259 T^{14} + 306265454073011286 T^{16} - 19210618534331542792 T^{18} + \)\(10\!\cdots\!85\)\( T^{20} - 19210618534331542792 p^{2} T^{22} + 306265454073011286 p^{4} T^{24} - 4343692082603259 p^{6} T^{26} + 54157938049125 p^{8} T^{28} - 580188957774 p^{10} T^{30} + 5154472554 p^{12} T^{32} - 36124362 p^{14} T^{34} + 185716 p^{16} T^{36} - 619 p^{18} T^{38} + p^{20} T^{40} \) | |
| 59 | \( ( 1 - 8 T + 362 T^{2} - 2839 T^{3} + 69276 T^{4} - 489400 T^{5} + 8713692 T^{6} - 55188538 T^{7} + 788130044 T^{8} - 4418104176 T^{9} + 53479384451 T^{10} - 4418104176 p T^{11} + 788130044 p^{2} T^{12} - 55188538 p^{3} T^{13} + 8713692 p^{4} T^{14} - 489400 p^{5} T^{15} + 69276 p^{6} T^{16} - 2839 p^{7} T^{17} + 362 p^{8} T^{18} - 8 p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 61 | \( ( 1 - 25 T + 558 T^{2} - 8200 T^{3} + 108783 T^{4} - 1177655 T^{5} + 11582650 T^{6} - 101899745 T^{7} + 831201247 T^{8} - 6621252455 T^{9} + 50963457531 T^{10} - 6621252455 p T^{11} + 831201247 p^{2} T^{12} - 101899745 p^{3} T^{13} + 11582650 p^{4} T^{14} - 1177655 p^{5} T^{15} + 108783 p^{6} T^{16} - 8200 p^{7} T^{17} + 558 p^{8} T^{18} - 25 p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 67 | \( 1 - 535 T^{2} + 143460 T^{4} - 25403540 T^{6} + 3331637020 T^{8} - 346671029101 T^{10} + 30248254519585 T^{12} - 2335698228574745 T^{14} + 167948438406621585 T^{16} - 11648261666820596270 T^{18} + \)\(78\!\cdots\!17\)\( T^{20} - 11648261666820596270 p^{2} T^{22} + 167948438406621585 p^{4} T^{24} - 2335698228574745 p^{6} T^{26} + 30248254519585 p^{8} T^{28} - 346671029101 p^{10} T^{30} + 3331637020 p^{12} T^{32} - 25403540 p^{14} T^{34} + 143460 p^{16} T^{36} - 535 p^{18} T^{38} + p^{20} T^{40} \) | |
| 71 | \( ( 1 + 26 T + 818 T^{2} + 14337 T^{3} + 266806 T^{4} + 3591360 T^{5} + 49669128 T^{6} + 543718754 T^{7} + 6047784394 T^{8} + 55252608642 T^{9} + 511030357589 T^{10} + 55252608642 p T^{11} + 6047784394 p^{2} T^{12} + 543718754 p^{3} T^{13} + 49669128 p^{4} T^{14} + 3591360 p^{5} T^{15} + 266806 p^{6} T^{16} + 14337 p^{7} T^{17} + 818 p^{8} T^{18} + 26 p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 73 | \( 1 - 824 T^{2} + 349162 T^{4} - 100183696 T^{6} + 21705883179 T^{8} - 3760312089093 T^{10} + 538999742690418 T^{12} - 65332061245569367 T^{14} + 6791610068357078021 T^{16} - \)\(61\!\cdots\!81\)\( T^{18} + \)\(47\!\cdots\!53\)\( T^{20} - \)\(61\!\cdots\!81\)\( p^{2} T^{22} + 6791610068357078021 p^{4} T^{24} - 65332061245569367 p^{6} T^{26} + 538999742690418 p^{8} T^{28} - 3760312089093 p^{10} T^{30} + 21705883179 p^{12} T^{32} - 100183696 p^{14} T^{34} + 349162 p^{16} T^{36} - 824 p^{18} T^{38} + p^{20} T^{40} \) | |
| 79 | \( ( 1 - 13 T + 440 T^{2} - 5315 T^{3} + 103728 T^{4} - 1115410 T^{5} + 16597847 T^{6} - 157833483 T^{7} + 1952986697 T^{8} - 16469781464 T^{9} + 175597273193 T^{10} - 16469781464 p T^{11} + 1952986697 p^{2} T^{12} - 157833483 p^{3} T^{13} + 16597847 p^{4} T^{14} - 1115410 p^{5} T^{15} + 103728 p^{6} T^{16} - 5315 p^{7} T^{17} + 440 p^{8} T^{18} - 13 p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 83 | \( 1 - 611 T^{2} + 202032 T^{4} - 47385954 T^{6} + 8747971404 T^{8} - 1345990157782 T^{10} + 178835767670753 T^{12} - 20999478754028418 T^{14} + 2212902255909314986 T^{16} - \)\(21\!\cdots\!64\)\( T^{18} + \)\(18\!\cdots\!03\)\( T^{20} - \)\(21\!\cdots\!64\)\( p^{2} T^{22} + 2212902255909314986 p^{4} T^{24} - 20999478754028418 p^{6} T^{26} + 178835767670753 p^{8} T^{28} - 1345990157782 p^{10} T^{30} + 8747971404 p^{12} T^{32} - 47385954 p^{14} T^{34} + 202032 p^{16} T^{36} - 611 p^{18} T^{38} + p^{20} T^{40} \) | |
| 89 | \( ( 1 + 13 T + 585 T^{2} + 6326 T^{3} + 161639 T^{4} + 1505784 T^{5} + 28760441 T^{6} + 235591488 T^{7} + 3732793936 T^{8} + 27171822187 T^{9} + 374535022251 T^{10} + 27171822187 p T^{11} + 3732793936 p^{2} T^{12} + 235591488 p^{3} T^{13} + 28760441 p^{4} T^{14} + 1505784 p^{5} T^{15} + 161639 p^{6} T^{16} + 6326 p^{7} T^{17} + 585 p^{8} T^{18} + 13 p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 97 | \( 1 - 826 T^{2} + 350911 T^{4} - 100654626 T^{6} + 21773494909 T^{8} - 3774915480369 T^{10} + 546393095142740 T^{12} - 68273438698372551 T^{14} + 7615132804778173446 T^{16} - \)\(78\!\cdots\!55\)\( T^{18} + \)\(77\!\cdots\!97\)\( T^{20} - \)\(78\!\cdots\!55\)\( p^{2} T^{22} + 7615132804778173446 p^{4} T^{24} - 68273438698372551 p^{6} T^{26} + 546393095142740 p^{8} T^{28} - 3774915480369 p^{10} T^{30} + 21773494909 p^{12} T^{32} - 100654626 p^{14} T^{34} + 350911 p^{16} T^{36} - 826 p^{18} T^{38} + p^{20} T^{40} \) | |
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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
−2.00021060123234925123712055660, −1.99529170574072356839059655544, −1.90646460785858801112797543504, −1.86035506741931585170440418412, −1.85534380561689454886307679509, −1.76324737709603378067498309751, −1.72460815517086239740235804305, −1.69850187934252642925203772383, −1.67615024460498389301035151995, −1.52474706758031045752641853518, −1.43298005897251210360355259560, −1.27810673159284638757765303689, −1.23355644924681498546464694935, −1.22648507845260830845117905424, −1.12632883303286095318325487353, −0.926306238625420625035117691642, −0.912251860344067475336222340901, −0.833134196311723978454999954287, −0.75912169187566056620325458828, −0.70545732372611657549434205237, −0.64884487927436933639223310734, −0.48154123050252622172506178257, −0.44135115906205021760568659991, −0.37272392060730960413019111827, −0.37193592861621900519404255817, 0.37193592861621900519404255817, 0.37272392060730960413019111827, 0.44135115906205021760568659991, 0.48154123050252622172506178257, 0.64884487927436933639223310734, 0.70545732372611657549434205237, 0.75912169187566056620325458828, 0.833134196311723978454999954287, 0.912251860344067475336222340901, 0.926306238625420625035117691642, 1.12632883303286095318325487353, 1.22648507845260830845117905424, 1.23355644924681498546464694935, 1.27810673159284638757765303689, 1.43298005897251210360355259560, 1.52474706758031045752641853518, 1.67615024460498389301035151995, 1.69850187934252642925203772383, 1.72460815517086239740235804305, 1.76324737709603378067498309751, 1.85534380561689454886307679509, 1.86035506741931585170440418412, 1.90646460785858801112797543504, 1.99529170574072356839059655544, 2.00021060123234925123712055660
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.