Dirichlet series
| L(s) = 1 | − 9·4-s + 19·9-s + 8·11-s + 45·16-s − 12·19-s + 11·25-s − 30·29-s − 12·31-s − 171·36-s + 54·41-s − 72·44-s + 73·49-s − 40·59-s − 38·61-s − 165·64-s − 20·71-s + 108·76-s + 28·79-s + 195·81-s − 86·89-s + 152·99-s − 99·100-s − 14·101-s − 18·109-s + 270·116-s − 60·121-s + 108·124-s + ⋯ |
| L(s) = 1 | − 9/2·4-s + 19/3·9-s + 2.41·11-s + 45/4·16-s − 2.75·19-s + 11/5·25-s − 5.57·29-s − 2.15·31-s − 28.5·36-s + 8.43·41-s − 10.8·44-s + 73/7·49-s − 5.20·59-s − 4.86·61-s − 20.6·64-s − 2.37·71-s + 12.3·76-s + 3.15·79-s + 65/3·81-s − 9.11·89-s + 15.2·99-s − 9.89·100-s − 1.39·101-s − 1.72·109-s + 25.0·116-s − 5.45·121-s + 9.69·124-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 5^{18} \cdot 13^{36}\right)^{s/2} \, \Gamma_{\C}(s)^{18} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 5^{18} \cdot 13^{36}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{18} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(36\) |
| Conductor: | \(2^{18} \cdot 5^{18} \cdot 13^{36}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.20266\times 10^{20}\) |
| Root analytic conductor: | \(3.67351\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((36,\ 2^{18} \cdot 5^{18} \cdot 13^{36} ,\ ( \ : [1/2]^{18} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(6.019163332\) |
| \(L(\frac12)\) | \(\approx\) | \(6.019163332\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + T^{2} )^{9} \) |
| 5 | \( 1 - 11 T^{2} - 12 T^{3} + 88 T^{4} + 144 T^{5} - 16 p^{2} T^{6} - 1028 T^{7} + 678 T^{8} + 6336 T^{9} + 678 p T^{10} - 1028 p^{2} T^{11} - 16 p^{5} T^{12} + 144 p^{4} T^{13} + 88 p^{5} T^{14} - 12 p^{6} T^{15} - 11 p^{7} T^{16} + p^{9} T^{18} \) | |
| 13 | \( 1 \) | |
| good | 3 | \( 1 - 19 T^{2} + 166 T^{4} - 886 T^{6} + 3239 T^{8} - 2879 p T^{10} + 17516 T^{12} - 27689 T^{14} + 34912 T^{16} - 56237 T^{18} + 34912 p^{2} T^{20} - 27689 p^{4} T^{22} + 17516 p^{6} T^{24} - 2879 p^{9} T^{26} + 3239 p^{10} T^{28} - 886 p^{12} T^{30} + 166 p^{14} T^{32} - 19 p^{16} T^{34} + p^{18} T^{36} \) |
| 7 | \( 1 - 73 T^{2} + 2563 T^{4} - 57135 T^{6} + 900394 T^{8} - 1513146 p T^{10} + 96322374 T^{12} - 704872417 T^{14} + 4516318140 T^{16} - 29647825570 T^{18} + 4516318140 p^{2} T^{20} - 704872417 p^{4} T^{22} + 96322374 p^{6} T^{24} - 1513146 p^{9} T^{26} + 900394 p^{10} T^{28} - 57135 p^{12} T^{30} + 2563 p^{14} T^{32} - 73 p^{16} T^{34} + p^{18} T^{36} \) | |
| 11 | \( ( 1 - 4 T + 54 T^{2} - 226 T^{3} + 1563 T^{4} - 6174 T^{5} + 31234 T^{6} - 108642 T^{7} + 460356 T^{8} - 1382188 T^{9} + 460356 p T^{10} - 108642 p^{2} T^{11} + 31234 p^{3} T^{12} - 6174 p^{4} T^{13} + 1563 p^{5} T^{14} - 226 p^{6} T^{15} + 54 p^{7} T^{16} - 4 p^{8} T^{17} + p^{9} T^{18} )^{2} \) | |
| 17 | \( 1 - 140 T^{2} + 10446 T^{4} - 542132 T^{6} + 21721465 T^{8} - 709645220 T^{10} + 19533329208 T^{12} - 462527257290 T^{14} + 9545961762368 T^{16} - 172983069913188 T^{18} + 9545961762368 p^{2} T^{20} - 462527257290 p^{4} T^{22} + 19533329208 p^{6} T^{24} - 709645220 p^{8} T^{26} + 21721465 p^{10} T^{28} - 542132 p^{12} T^{30} + 10446 p^{14} T^{32} - 140 p^{16} T^{34} + p^{18} T^{36} \) | |
| 19 | \( ( 1 + 6 T + 100 T^{2} + 610 T^{3} + 5309 T^{4} + 29618 T^{5} + 191988 T^{6} + 920766 T^{7} + 5003266 T^{8} + 20407128 T^{9} + 5003266 p T^{10} + 920766 p^{2} T^{11} + 191988 p^{3} T^{12} + 29618 p^{4} T^{13} + 5309 p^{5} T^{14} + 610 p^{6} T^{15} + 100 p^{7} T^{16} + 6 p^{8} T^{17} + p^{9} T^{18} )^{2} \) | |
| 23 | \( 1 - 189 T^{2} + 18547 T^{4} - 1260871 T^{6} + 66450970 T^{8} - 2875772502 T^{10} + 105602315862 T^{12} - 3354862741769 T^{14} + 93276063568428 T^{16} - 2283856292179514 T^{18} + 93276063568428 p^{2} T^{20} - 3354862741769 p^{4} T^{22} + 105602315862 p^{6} T^{24} - 2875772502 p^{8} T^{26} + 66450970 p^{10} T^{28} - 1260871 p^{12} T^{30} + 18547 p^{14} T^{32} - 189 p^{16} T^{34} + p^{18} T^{36} \) | |
| 29 | \( ( 1 + 15 T + 239 T^{2} + 2305 T^{3} + 23198 T^{4} + 175696 T^{5} + 1370430 T^{6} + 8630975 T^{7} + 55644200 T^{8} + 296660234 T^{9} + 55644200 p T^{10} + 8630975 p^{2} T^{11} + 1370430 p^{3} T^{12} + 175696 p^{4} T^{13} + 23198 p^{5} T^{14} + 2305 p^{6} T^{15} + 239 p^{7} T^{16} + 15 p^{8} T^{17} + p^{9} T^{18} )^{2} \) | |
| 31 | \( ( 1 + 6 T + 127 T^{2} + 436 T^{3} + 8464 T^{4} + 23872 T^{5} + 432792 T^{6} + 1011532 T^{7} + 16920774 T^{8} + 34745716 T^{9} + 16920774 p T^{10} + 1011532 p^{2} T^{11} + 432792 p^{3} T^{12} + 23872 p^{4} T^{13} + 8464 p^{5} T^{14} + 436 p^{6} T^{15} + 127 p^{7} T^{16} + 6 p^{8} T^{17} + p^{9} T^{18} )^{2} \) | |
| 37 | \( 1 - 326 T^{2} + 53737 T^{4} - 5985392 T^{6} + 508093676 T^{8} - 35092908872 T^{10} + 2050036741932 T^{12} - 103691263111184 T^{14} + 4604927567685814 T^{16} - 180924599670008420 T^{18} + 4604927567685814 p^{2} T^{20} - 103691263111184 p^{4} T^{22} + 2050036741932 p^{6} T^{24} - 35092908872 p^{8} T^{26} + 508093676 p^{10} T^{28} - 5985392 p^{12} T^{30} + 53737 p^{14} T^{32} - 326 p^{16} T^{34} + p^{18} T^{36} \) | |
| 41 | \( ( 1 - 27 T + 582 T^{2} - 8750 T^{3} + 113479 T^{4} - 1218237 T^{5} + 11671064 T^{6} - 97454357 T^{7} + 737337724 T^{8} - 4948932329 T^{9} + 737337724 p T^{10} - 97454357 p^{2} T^{11} + 11671064 p^{3} T^{12} - 1218237 p^{4} T^{13} + 113479 p^{5} T^{14} - 8750 p^{6} T^{15} + 582 p^{7} T^{16} - 27 p^{8} T^{17} + p^{9} T^{18} )^{2} \) | |
| 43 | \( 1 - 287 T^{2} + 43142 T^{4} - 4573446 T^{6} + 386135651 T^{8} - 27572203969 T^{10} + 1720293277144 T^{12} - 95577110144845 T^{14} + 4778336185255988 T^{16} - 215967522595737853 T^{18} + 4778336185255988 p^{2} T^{20} - 95577110144845 p^{4} T^{22} + 1720293277144 p^{6} T^{24} - 27572203969 p^{8} T^{26} + 386135651 p^{10} T^{28} - 4573446 p^{12} T^{30} + 43142 p^{14} T^{32} - 287 p^{16} T^{34} + p^{18} T^{36} \) | |
| 47 | \( 1 - 473 T^{2} + 104383 T^{4} - 14402055 T^{6} + 1417579994 T^{8} - 109177509830 T^{10} + 7129159929262 T^{12} - 419342159324281 T^{14} + 22714253660536472 T^{16} - 1122547435551063170 T^{18} + 22714253660536472 p^{2} T^{20} - 419342159324281 p^{4} T^{22} + 7129159929262 p^{6} T^{24} - 109177509830 p^{8} T^{26} + 1417579994 p^{10} T^{28} - 14402055 p^{12} T^{30} + 104383 p^{14} T^{32} - 473 p^{16} T^{34} + p^{18} T^{36} \) | |
| 53 | \( 1 - 358 T^{2} + 68665 T^{4} - 9268656 T^{6} + 978790780 T^{8} - 85687154248 T^{10} + 6454221349468 T^{12} - 430140795687632 T^{14} + 25911787804458438 T^{16} - 1430791011247333924 T^{18} + 25911787804458438 p^{2} T^{20} - 430140795687632 p^{4} T^{22} + 6454221349468 p^{6} T^{24} - 85687154248 p^{8} T^{26} + 978790780 p^{10} T^{28} - 9268656 p^{12} T^{30} + 68665 p^{14} T^{32} - 358 p^{16} T^{34} + p^{18} T^{36} \) | |
| 59 | \( ( 1 + 20 T + 348 T^{2} + 3034 T^{3} + 23675 T^{4} + 60808 T^{5} + 276670 T^{6} - 657672 T^{7} + 44991594 T^{8} + 259362732 T^{9} + 44991594 p T^{10} - 657672 p^{2} T^{11} + 276670 p^{3} T^{12} + 60808 p^{4} T^{13} + 23675 p^{5} T^{14} + 3034 p^{6} T^{15} + 348 p^{7} T^{16} + 20 p^{8} T^{17} + p^{9} T^{18} )^{2} \) | |
| 61 | \( ( 1 + 19 T + 425 T^{2} + 5589 T^{3} + 76490 T^{4} + 812130 T^{5} + 8657310 T^{6} + 77872615 T^{7} + 11445042 p T^{8} + 5450104502 T^{9} + 11445042 p^{2} T^{10} + 77872615 p^{2} T^{11} + 8657310 p^{3} T^{12} + 812130 p^{4} T^{13} + 76490 p^{5} T^{14} + 5589 p^{6} T^{15} + 425 p^{7} T^{16} + 19 p^{8} T^{17} + p^{9} T^{18} )^{2} \) | |
| 67 | \( 1 - 523 T^{2} + 2322 p T^{4} - 32711982 T^{6} + 5356850087 T^{8} - 715731820749 T^{10} + 80306544218460 T^{12} - 7696020504340553 T^{14} + 636781611642255824 T^{16} - 45743256436046064717 T^{18} + 636781611642255824 p^{2} T^{20} - 7696020504340553 p^{4} T^{22} + 80306544218460 p^{6} T^{24} - 715731820749 p^{8} T^{26} + 5356850087 p^{10} T^{28} - 32711982 p^{12} T^{30} + 2322 p^{15} T^{32} - 523 p^{16} T^{34} + p^{18} T^{36} \) | |
| 71 | \( ( 1 + 10 T + 327 T^{2} + 1896 T^{3} + 45400 T^{4} + 156120 T^{5} + 4644512 T^{6} + 13734472 T^{7} + 431152038 T^{8} + 1231171612 T^{9} + 431152038 p T^{10} + 13734472 p^{2} T^{11} + 4644512 p^{3} T^{12} + 156120 p^{4} T^{13} + 45400 p^{5} T^{14} + 1896 p^{6} T^{15} + 327 p^{7} T^{16} + 10 p^{8} T^{17} + p^{9} T^{18} )^{2} \) | |
| 73 | \( 1 - 780 T^{2} + 294470 T^{4} - 71161600 T^{6} + 12302934697 T^{8} - 1617258851460 T^{10} + 168942302080216 T^{12} - 14691108678158610 T^{14} + 1132286834792695128 T^{16} - 83161513108798344316 T^{18} + 1132286834792695128 p^{2} T^{20} - 14691108678158610 p^{4} T^{22} + 168942302080216 p^{6} T^{24} - 1617258851460 p^{8} T^{26} + 12302934697 p^{10} T^{28} - 71161600 p^{12} T^{30} + 294470 p^{14} T^{32} - 780 p^{16} T^{34} + p^{18} T^{36} \) | |
| 79 | \( ( 1 - 14 T + 547 T^{2} - 4976 T^{3} + 114240 T^{4} - 625832 T^{5} + 12515544 T^{6} - 29864720 T^{7} + 939925930 T^{8} - 723600532 T^{9} + 939925930 p T^{10} - 29864720 p^{2} T^{11} + 12515544 p^{3} T^{12} - 625832 p^{4} T^{13} + 114240 p^{5} T^{14} - 4976 p^{6} T^{15} + 547 p^{7} T^{16} - 14 p^{8} T^{17} + p^{9} T^{18} )^{2} \) | |
| 83 | \( 1 - 679 T^{2} + 227550 T^{4} - 50883710 T^{6} + 8645808499 T^{8} - 1200451087873 T^{10} + 142532728970696 T^{12} - 14928187696817437 T^{14} + 1410598721603975972 T^{16} - \)\(12\!\cdots\!65\)\( T^{18} + 1410598721603975972 p^{2} T^{20} - 14928187696817437 p^{4} T^{22} + 142532728970696 p^{6} T^{24} - 1200451087873 p^{8} T^{26} + 8645808499 p^{10} T^{28} - 50883710 p^{12} T^{30} + 227550 p^{14} T^{32} - 679 p^{16} T^{34} + p^{18} T^{36} \) | |
| 89 | \( ( 1 + 43 T + 1178 T^{2} + 24086 T^{3} + 408999 T^{4} + 5989185 T^{5} + 78187192 T^{6} + 919048921 T^{7} + 9864857592 T^{8} + 97010730453 T^{9} + 9864857592 p T^{10} + 919048921 p^{2} T^{11} + 78187192 p^{3} T^{12} + 5989185 p^{4} T^{13} + 408999 p^{5} T^{14} + 24086 p^{6} T^{15} + 1178 p^{7} T^{16} + 43 p^{8} T^{17} + p^{9} T^{18} )^{2} \) | |
| 97 | \( 1 - 1192 T^{2} + 708894 T^{4} - 278185232 T^{6} + 80450784097 T^{8} - 18163365255476 T^{10} + 3311749242376296 T^{12} - 497979146340746546 T^{14} + 62547423445663191864 T^{16} - \)\(66\!\cdots\!84\)\( T^{18} + 62547423445663191864 p^{2} T^{20} - 497979146340746546 p^{4} T^{22} + 3311749242376296 p^{6} T^{24} - 18163365255476 p^{8} T^{26} + 80450784097 p^{10} T^{28} - 278185232 p^{12} T^{30} + 708894 p^{14} T^{32} - 1192 p^{16} T^{34} + p^{18} T^{36} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{36} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.19354149987486205671914629237, −2.08735739526647091101813023790, −1.93484346128057124482966499810, −1.81843109692839822937030479486, −1.76948934922001609071441447403, −1.69588430174620927476555953968, −1.68770337498658150328832292909, −1.56492364036353746374183372966, −1.51464898330310025867480988673, −1.50255259839526569912528954135, −1.39834054928661050692309143943, −1.30624407024964647346227829307, −1.18679277058021278720196352453, −1.13418953932730081021989468177, −1.13354236811951322805229040602, −1.09288178806792848404226358609, −1.07742269273263193630485407048, −0.942969854750463373466764066242, −0.71058893013848685569287001854, −0.69935852107848483148856577292, −0.52384560214269815444622559028, −0.51684312798472444812812462662, −0.28687897157419935748800358349, −0.18908076105386363095694728043, −0.15483989264099229879947520316, 0.15483989264099229879947520316, 0.18908076105386363095694728043, 0.28687897157419935748800358349, 0.51684312798472444812812462662, 0.52384560214269815444622559028, 0.69935852107848483148856577292, 0.71058893013848685569287001854, 0.942969854750463373466764066242, 1.07742269273263193630485407048, 1.09288178806792848404226358609, 1.13354236811951322805229040602, 1.13418953932730081021989468177, 1.18679277058021278720196352453, 1.30624407024964647346227829307, 1.39834054928661050692309143943, 1.50255259839526569912528954135, 1.51464898330310025867480988673, 1.56492364036353746374183372966, 1.68770337498658150328832292909, 1.69588430174620927476555953968, 1.76948934922001609071441447403, 1.81843109692839822937030479486, 1.93484346128057124482966499810, 2.08735739526647091101813023790, 2.19354149987486205671914629237
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.