Dirichlet series
| L(s) = 1 | + 2·3-s − 4·4-s + 9-s − 8·12-s − 4·13-s + 15·16-s − 12·17-s + 24·23-s − 31·25-s − 8·27-s + 12·29-s − 4·36-s − 8·39-s + 4·43-s + 30·48-s − 40·49-s − 24·51-s + 16·52-s + 108·53-s − 2·61-s − 40·64-s + 48·68-s + 48·69-s − 62·75-s − 14·79-s − 4·81-s + 24·87-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 2·4-s + 1/3·9-s − 2.30·12-s − 1.10·13-s + 15/4·16-s − 2.91·17-s + 5.00·23-s − 6.19·25-s − 1.53·27-s + 2.22·29-s − 2/3·36-s − 1.28·39-s + 0.609·43-s + 4.33·48-s − 5.71·49-s − 3.36·51-s + 2.21·52-s + 14.8·53-s − 0.256·61-s − 5·64-s + 5.82·68-s + 5.77·69-s − 7.15·75-s − 1.57·79-s − 4/9·81-s + 2.57·87-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{40} \cdot 13^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{40} \cdot 13^{20}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(40\) |
| Conductor: | \(3^{40} \cdot 13^{20}\) |
| Sign: | $1$ |
| Analytic conductor: | \(0.256601\) |
| Root analytic conductor: | \(0.966565\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((40,\ 3^{40} \cdot 13^{20} ,\ ( \ : [1/2]^{20} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.6140185204\) |
| \(L(\frac12)\) | \(\approx\) | \(0.6140185204\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( ( 1 - T + T^{2} + p T^{3} - p^{2} T^{4} + p^{2} T^{5} - p^{3} T^{6} + p^{3} T^{7} + p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| 13 | \( 1 + 4 T + 7 T^{2} + 116 T^{3} + 383 T^{4} - 112 T^{5} + 2738 T^{6} + 7520 T^{7} - 72595 T^{8} - 231820 T^{9} - 216343 T^{10} - 231820 p T^{11} - 72595 p^{2} T^{12} + 7520 p^{3} T^{13} + 2738 p^{4} T^{14} - 112 p^{5} T^{15} + 383 p^{6} T^{16} + 116 p^{7} T^{17} + 7 p^{8} T^{18} + 4 p^{9} T^{19} + p^{10} T^{20} \) | |
| good | 2 | \( 1 + p^{2} T^{2} + T^{4} - p^{4} T^{6} - 31 T^{8} - 55 T^{10} - 7 p^{4} T^{12} + 7 p T^{14} + 307 p T^{16} + 103 p^{3} T^{18} + 161 T^{20} + 103 p^{5} T^{22} + 307 p^{5} T^{24} + 7 p^{7} T^{26} - 7 p^{12} T^{28} - 55 p^{10} T^{30} - 31 p^{12} T^{32} - p^{18} T^{34} + p^{16} T^{36} + p^{20} T^{38} + p^{20} T^{40} \) |
| 5 | \( 1 + 31 T^{2} + 484 T^{4} + 5177 T^{6} + 43181 T^{8} + 297968 T^{10} + 352879 p T^{12} + 9363119 T^{14} + 46666604 T^{16} + 227102657 T^{18} + 1118219951 T^{20} + 227102657 p^{2} T^{22} + 46666604 p^{4} T^{24} + 9363119 p^{6} T^{26} + 352879 p^{9} T^{28} + 297968 p^{10} T^{30} + 43181 p^{12} T^{32} + 5177 p^{14} T^{34} + 484 p^{16} T^{36} + 31 p^{18} T^{38} + p^{20} T^{40} \) | |
| 7 | \( 1 + 40 T^{2} + 766 T^{4} + 9906 T^{6} + 104718 T^{8} + 1007973 T^{10} + 9176841 T^{12} + 79542213 T^{14} + 648455565 T^{16} + 701985460 p T^{18} + 35134876099 T^{20} + 701985460 p^{3} T^{22} + 648455565 p^{4} T^{24} + 79542213 p^{6} T^{26} + 9176841 p^{8} T^{28} + 1007973 p^{10} T^{30} + 104718 p^{12} T^{32} + 9906 p^{14} T^{34} + 766 p^{16} T^{36} + 40 p^{18} T^{38} + p^{20} T^{40} \) | |
| 11 | \( 1 + 61 T^{2} + 1759 T^{4} + 34610 T^{6} + 569738 T^{8} + 8403023 T^{10} + 111458825 T^{12} + 1389384182 T^{14} + 16770404174 T^{16} + 191784152222 T^{18} + 2114206875851 T^{20} + 191784152222 p^{2} T^{22} + 16770404174 p^{4} T^{24} + 1389384182 p^{6} T^{26} + 111458825 p^{8} T^{28} + 8403023 p^{10} T^{30} + 569738 p^{12} T^{32} + 34610 p^{14} T^{34} + 1759 p^{16} T^{36} + 61 p^{18} T^{38} + p^{20} T^{40} \) | |
| 17 | \( ( 1 + 3 T + 52 T^{2} + 114 T^{3} + 1342 T^{4} + 2223 T^{5} + 1342 p T^{6} + 114 p^{2} T^{7} + 52 p^{3} T^{8} + 3 p^{4} T^{9} + p^{5} T^{10} )^{4} \) | |
| 19 | \( ( 1 - 61 T^{2} + 2364 T^{4} - 69849 T^{6} + 1733793 T^{8} - 36234975 T^{10} + 1733793 p^{2} T^{12} - 69849 p^{4} T^{14} + 2364 p^{6} T^{16} - 61 p^{8} T^{18} + p^{10} T^{20} )^{2} \) | |
| 23 | \( ( 1 - 12 T + 29 T^{2} + 234 T^{3} - 1263 T^{4} - 2664 T^{5} + 17913 T^{6} + 156015 T^{7} - 47034 p T^{8} - 1678653 T^{9} + 32543439 T^{10} - 1678653 p T^{11} - 47034 p^{3} T^{12} + 156015 p^{3} T^{13} + 17913 p^{4} T^{14} - 2664 p^{5} T^{15} - 1263 p^{6} T^{16} + 234 p^{7} T^{17} + 29 p^{8} T^{18} - 12 p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 29 | \( ( 1 - 6 T - 76 T^{2} + 720 T^{3} + 2721 T^{4} - 40383 T^{5} - 1692 T^{6} + 1265454 T^{7} - 3176103 T^{8} - 16012737 T^{9} + 145472475 T^{10} - 16012737 p T^{11} - 3176103 p^{2} T^{12} + 1265454 p^{3} T^{13} - 1692 p^{4} T^{14} - 40383 p^{5} T^{15} + 2721 p^{6} T^{16} + 720 p^{7} T^{17} - 76 p^{8} T^{18} - 6 p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 31 | \( 1 + 94 T^{2} + 3496 T^{4} + 60684 T^{6} + 232869 T^{8} - 6200757 T^{10} + 938661372 T^{12} + 54171437946 T^{14} + 206185080135 T^{16} - 63160311473567 T^{18} - 2789343532963817 T^{20} - 63160311473567 p^{2} T^{22} + 206185080135 p^{4} T^{24} + 54171437946 p^{6} T^{26} + 938661372 p^{8} T^{28} - 6200757 p^{10} T^{30} + 232869 p^{12} T^{32} + 60684 p^{14} T^{34} + 3496 p^{16} T^{36} + 94 p^{18} T^{38} + p^{20} T^{40} \) | |
| 37 | \( ( 1 - 139 T^{2} + 9594 T^{4} - 342798 T^{6} + 5828598 T^{8} - 40695039 T^{10} + 5828598 p^{2} T^{12} - 342798 p^{4} T^{14} + 9594 p^{6} T^{16} - 139 p^{8} T^{18} + p^{10} T^{20} )^{2} \) | |
| 41 | \( 1 + 292 T^{2} + 43918 T^{4} + 4601498 T^{6} + 379798502 T^{8} + 26277210809 T^{10} + 1582109241605 T^{12} + 85152971769953 T^{14} + 4179701321374313 T^{16} + 189752134321025000 T^{18} + 8039813413006726667 T^{20} + 189752134321025000 p^{2} T^{22} + 4179701321374313 p^{4} T^{24} + 85152971769953 p^{6} T^{26} + 1582109241605 p^{8} T^{28} + 26277210809 p^{10} T^{30} + 379798502 p^{12} T^{32} + 4601498 p^{14} T^{34} + 43918 p^{16} T^{36} + 292 p^{18} T^{38} + p^{20} T^{40} \) | |
| 43 | \( ( 1 - 2 T - 158 T^{2} + 492 T^{3} + 13593 T^{4} - 46269 T^{5} - 788022 T^{6} + 2313330 T^{7} + 36536745 T^{8} - 44646443 T^{9} - 1561237673 T^{10} - 44646443 p T^{11} + 36536745 p^{2} T^{12} + 2313330 p^{3} T^{13} - 788022 p^{4} T^{14} - 46269 p^{5} T^{15} + 13593 p^{6} T^{16} + 492 p^{7} T^{17} - 158 p^{8} T^{18} - 2 p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 47 | \( 1 + 265 T^{2} + 34225 T^{4} + 3010598 T^{6} + 210824006 T^{8} + 12513305021 T^{10} + 635253973508 T^{12} + 27859495538075 T^{14} + 1100009994766280 T^{16} + 42494666913707087 T^{18} + 1823122682961743357 T^{20} + 42494666913707087 p^{2} T^{22} + 1100009994766280 p^{4} T^{24} + 27859495538075 p^{6} T^{26} + 635253973508 p^{8} T^{28} + 12513305021 p^{10} T^{30} + 210824006 p^{12} T^{32} + 3010598 p^{14} T^{34} + 34225 p^{16} T^{36} + 265 p^{18} T^{38} + p^{20} T^{40} \) | |
| 53 | \( ( 1 - 27 T + 511 T^{2} - 6579 T^{3} + 68041 T^{4} - 545931 T^{5} + 68041 p T^{6} - 6579 p^{2} T^{7} + 511 p^{3} T^{8} - 27 p^{4} T^{9} + p^{5} T^{10} )^{4} \) | |
| 59 | \( 1 + 445 T^{2} + 102148 T^{4} + 16582619 T^{6} + 2153519297 T^{8} + 235789088486 T^{10} + 22391523103049 T^{12} + 1882667448950369 T^{14} + 142010022953796698 T^{16} + 9678367531922928449 T^{18} + \)\(59\!\cdots\!79\)\( T^{20} + 9678367531922928449 p^{2} T^{22} + 142010022953796698 p^{4} T^{24} + 1882667448950369 p^{6} T^{26} + 22391523103049 p^{8} T^{28} + 235789088486 p^{10} T^{30} + 2153519297 p^{12} T^{32} + 16582619 p^{14} T^{34} + 102148 p^{16} T^{36} + 445 p^{18} T^{38} + p^{20} T^{40} \) | |
| 61 | \( ( 1 + T - 125 T^{2} + 656 T^{3} + 6488 T^{4} - 81601 T^{5} + 60518 T^{6} + 4328783 T^{7} - 19244110 T^{8} - 90274741 T^{9} + 1238845421 T^{10} - 90274741 p T^{11} - 19244110 p^{2} T^{12} + 4328783 p^{3} T^{13} + 60518 p^{4} T^{14} - 81601 p^{5} T^{15} + 6488 p^{6} T^{16} + 656 p^{7} T^{17} - 125 p^{8} T^{18} + p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 67 | \( 1 + 400 T^{2} + 85354 T^{4} + 12111222 T^{6} + 1238096910 T^{8} + 91718965425 T^{10} + 4450393864113 T^{12} + 48732104645151 T^{14} - 16365979624555461 T^{16} - 2109184553119824092 T^{18} - \)\(16\!\cdots\!25\)\( T^{20} - 2109184553119824092 p^{2} T^{22} - 16365979624555461 p^{4} T^{24} + 48732104645151 p^{6} T^{26} + 4450393864113 p^{8} T^{28} + 91718965425 p^{10} T^{30} + 1238096910 p^{12} T^{32} + 12111222 p^{14} T^{34} + 85354 p^{16} T^{36} + 400 p^{18} T^{38} + p^{20} T^{40} \) | |
| 71 | \( ( 1 - 613 T^{2} + 174795 T^{4} - 30520538 T^{6} + 3612809909 T^{8} - 302886608319 T^{10} + 3612809909 p^{2} T^{12} - 30520538 p^{4} T^{14} + 174795 p^{6} T^{16} - 613 p^{8} T^{18} + p^{10} T^{20} )^{2} \) | |
| 73 | \( ( 1 - 361 T^{2} + 70830 T^{4} - 9633162 T^{6} + 992875494 T^{8} - 80983596843 T^{10} + 992875494 p^{2} T^{12} - 9633162 p^{4} T^{14} + 70830 p^{6} T^{16} - 361 p^{8} T^{18} + p^{10} T^{20} )^{2} \) | |
| 79 | \( ( 1 + 7 T - 209 T^{2} - 16 T^{3} + 33350 T^{4} - 85333 T^{5} - 2241208 T^{6} + 16415867 T^{7} + 75205370 T^{8} - 565517209 T^{9} + 1637698121 T^{10} - 565517209 p T^{11} + 75205370 p^{2} T^{12} + 16415867 p^{3} T^{13} - 2241208 p^{4} T^{14} - 85333 p^{5} T^{15} + 33350 p^{6} T^{16} - 16 p^{7} T^{17} - 209 p^{8} T^{18} + 7 p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 83 | \( 1 + 250 T^{2} + 25096 T^{4} - 21694 T^{6} - 269014066 T^{8} - 28228984225 T^{10} - 949682247373 T^{12} + 58313788182695 T^{14} + 7938038666343767 T^{16} + 398503536379690220 T^{18} + 18559639389283612403 T^{20} + 398503536379690220 p^{2} T^{22} + 7938038666343767 p^{4} T^{24} + 58313788182695 p^{6} T^{26} - 949682247373 p^{8} T^{28} - 28228984225 p^{10} T^{30} - 269014066 p^{12} T^{32} - 21694 p^{14} T^{34} + 25096 p^{16} T^{36} + 250 p^{18} T^{38} + p^{20} T^{40} \) | |
| 89 | \( ( 1 - 253 T^{2} + 37818 T^{4} - 3626345 T^{6} + 319495067 T^{8} - 26311353663 T^{10} + 319495067 p^{2} T^{12} - 3626345 p^{4} T^{14} + 37818 p^{6} T^{16} - 253 p^{8} T^{18} + p^{10} T^{20} )^{2} \) | |
| 97 | \( 1 + 358 T^{2} + 65539 T^{4} + 6024588 T^{6} - 18344607 T^{8} - 97667063874 T^{10} - 15833253123387 T^{12} - 1284157029694287 T^{14} - 6695236510938888 T^{16} + 132976113035383381 p T^{18} + 196861358090333953 p^{2} T^{20} + 132976113035383381 p^{3} T^{22} - 6695236510938888 p^{4} T^{24} - 1284157029694287 p^{6} T^{26} - 15833253123387 p^{8} T^{28} - 97667063874 p^{10} T^{30} - 18344607 p^{12} T^{32} + 6024588 p^{14} T^{34} + 65539 p^{16} T^{36} + 358 p^{18} T^{38} + p^{20} T^{40} \) | |
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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
−3.67914876013593911457956688986, −3.61058724618862481099785365288, −3.57573180191110194802208780593, −3.38740086623251928302607528661, −3.37160375601497057484405085312, −3.32331454542003528481934627020, −3.27378127595196457363195219638, −2.96992804933466798400569947755, −2.89458481365581741435447923099, −2.73535719804955112267915682791, −2.72063594259214437631451158750, −2.53697279939410730977316605982, −2.45153059066128947207762353384, −2.41753269211083985403222222254, −2.40326758916929363980824957048, −2.38970374191528453158869914150, −2.21594644855176431002979759337, −2.18429954315169107360635852950, −1.72295543123498472015543982165, −1.70541425282673793179447651860, −1.54697697173793777268479535250, −1.34912532480348257495157277875, −1.15285296245287200232375084859, −1.00995761031371622742523934341, −0.49801548062322619528807714181, 0.49801548062322619528807714181, 1.00995761031371622742523934341, 1.15285296245287200232375084859, 1.34912532480348257495157277875, 1.54697697173793777268479535250, 1.70541425282673793179447651860, 1.72295543123498472015543982165, 2.18429954315169107360635852950, 2.21594644855176431002979759337, 2.38970374191528453158869914150, 2.40326758916929363980824957048, 2.41753269211083985403222222254, 2.45153059066128947207762353384, 2.53697279939410730977316605982, 2.72063594259214437631451158750, 2.73535719804955112267915682791, 2.89458481365581741435447923099, 2.96992804933466798400569947755, 3.27378127595196457363195219638, 3.32331454542003528481934627020, 3.37160375601497057484405085312, 3.38740086623251928302607528661, 3.57573180191110194802208780593, 3.61058724618862481099785365288, 3.67914876013593911457956688986
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.