Dirichlet series
| L(s) = 1 | + 4·7-s − 12·17-s + 4·23-s − 14·25-s + 136·31-s − 8·41-s − 188·47-s + 8·49-s − 248·71-s − 124·73-s + 114·81-s + 100·97-s + 516·103-s − 284·113-s − 48·119-s + 1.31e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 16·161-s + 163-s + 167-s + 173-s + ⋯ |
| L(s) = 1 | + 4/7·7-s − 0.705·17-s + 4/23·23-s − 0.559·25-s + 4.38·31-s − 0.195·41-s − 4·47-s + 8/49·49-s − 3.49·71-s − 1.69·73-s + 1.40·81-s + 1.03·97-s + 5.00·103-s − 2.51·113-s − 0.403·119-s + 10.8·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.0993·161-s + 0.00613·163-s + 0.00598·167-s + 0.00578·173-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{100} \cdot 5^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{100} \cdot 5^{20}\right)^{s/2} \, \Gamma_{\C}(s+1)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(40\) |
| Conductor: | \(2^{100} \cdot 5^{20}\) |
| Sign: | $1$ |
| Analytic conductor: | \(6.15319\times 10^{12}\) |
| Root analytic conductor: | \(2.08798\) |
| Motivic weight: | \(2\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((40,\ 2^{100} \cdot 5^{20} ,\ ( \ : [1]^{20} ),\ 1 )\) |
Particular Values
| \(L(\frac{3}{2})\) | \(\approx\) | \(3.188286757\) |
| \(L(\frac12)\) | \(\approx\) | \(3.188286757\) |
| \(L(2)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + 14 T^{2} + 41 p T^{4} - 2808 T^{6} + 2034 p^{2} T^{8} + 8084 p^{4} T^{10} + 2034 p^{6} T^{12} - 2808 p^{8} T^{14} + 41 p^{13} T^{16} + 14 p^{16} T^{18} + p^{20} T^{20} \) | |
| good | 3 | \( 1 - 38 p T^{4} - 1859 T^{8} + 1114504 T^{12} - 14869022 T^{16} - 50382700 p^{4} T^{20} - 14869022 p^{8} T^{24} + 1114504 p^{16} T^{28} - 1859 p^{24} T^{32} - 38 p^{33} T^{36} + p^{40} T^{40} \) |
| 7 | \( ( 1 - 2 T + 2 T^{2} - 610 T^{3} + 271 p T^{4} + 18320 T^{5} + 145616 T^{6} - 1137328 T^{7} - 8045434 T^{8} + 18540420 T^{9} + 564357756 T^{10} + 18540420 p^{2} T^{11} - 8045434 p^{4} T^{12} - 1137328 p^{6} T^{13} + 145616 p^{8} T^{14} + 18320 p^{10} T^{15} + 271 p^{13} T^{16} - 610 p^{14} T^{17} + 2 p^{16} T^{18} - 2 p^{18} T^{19} + p^{20} T^{20} )^{2} \) | |
| 11 | \( ( 1 - 658 T^{2} + 224781 T^{4} - 52340600 T^{6} + 9120139298 T^{8} - 1240167831404 T^{10} + 9120139298 p^{4} T^{12} - 52340600 p^{8} T^{14} + 224781 p^{12} T^{16} - 658 p^{16} T^{18} + p^{20} T^{20} )^{2} \) | |
| 13 | \( 1 - 68714 T^{4} + 3087675853 T^{8} - 85473147028472 T^{12} + 2398905635263377746 T^{16} - \)\(60\!\cdots\!28\)\( T^{20} + 2398905635263377746 p^{8} T^{24} - 85473147028472 p^{16} T^{28} + 3087675853 p^{24} T^{32} - 68714 p^{32} T^{36} + p^{40} T^{40} \) | |
| 17 | \( ( 1 + 6 T + 18 T^{2} + 12134 T^{3} + 9389 p T^{4} - 1173048 T^{5} + 63705656 T^{6} + 1574926728 T^{7} - 7035754814 T^{8} + 54207870724 T^{9} + 10081821662572 T^{10} + 54207870724 p^{2} T^{11} - 7035754814 p^{4} T^{12} + 1574926728 p^{6} T^{13} + 63705656 p^{8} T^{14} - 1173048 p^{10} T^{15} + 9389 p^{13} T^{16} + 12134 p^{14} T^{17} + 18 p^{16} T^{18} + 6 p^{18} T^{19} + p^{20} T^{20} )^{2} \) | |
| 19 | \( ( 1 + 2406 T^{2} + 2930557 T^{4} + 2315128904 T^{6} + 1303230895506 T^{8} + 543406660187172 T^{10} + 1303230895506 p^{4} T^{12} + 2315128904 p^{8} T^{14} + 2930557 p^{12} T^{16} + 2406 p^{16} T^{18} + p^{20} T^{20} )^{2} \) | |
| 23 | \( ( 1 - 2 T + 2 T^{2} + 2494 T^{3} + 280233 T^{4} - 2455824 T^{5} + 324400 p T^{6} + 589949040 T^{7} - 128728992634 T^{8} - 513870857180 T^{9} + 3681193905276 T^{10} - 513870857180 p^{2} T^{11} - 128728992634 p^{4} T^{12} + 589949040 p^{6} T^{13} + 324400 p^{9} T^{14} - 2455824 p^{10} T^{15} + 280233 p^{12} T^{16} + 2494 p^{14} T^{17} + 2 p^{16} T^{18} - 2 p^{18} T^{19} + p^{20} T^{20} )^{2} \) | |
| 29 | \( ( 1 + 5862 T^{2} + 16207421 T^{4} + 28469762120 T^{6} + 35887152044178 T^{8} + 34347575552131876 T^{10} + 35887152044178 p^{4} T^{12} + 28469762120 p^{8} T^{14} + 16207421 p^{12} T^{16} + 5862 p^{16} T^{18} + p^{20} T^{20} )^{2} \) | |
| 31 | \( ( 1 - 34 T + 119 p T^{2} - 73552 T^{3} + 5377158 T^{4} - 77314812 T^{5} + 5377158 p^{2} T^{6} - 73552 p^{4} T^{7} + 119 p^{7} T^{8} - 34 p^{8} T^{9} + p^{10} T^{10} )^{4} \) | |
| 37 | \( 1 + 4173782 T^{4} + 12570586938765 T^{8} + 36325113880823571720 T^{12} + \)\(81\!\cdots\!38\)\( T^{16} + \)\(15\!\cdots\!68\)\( T^{20} + \)\(81\!\cdots\!38\)\( p^{8} T^{24} + 36325113880823571720 p^{16} T^{28} + 12570586938765 p^{24} T^{32} + 4173782 p^{32} T^{36} + p^{40} T^{40} \) | |
| 41 | \( ( 1 + 2 T + 3929 T^{2} - 43776 T^{3} + 9954182 T^{4} - 83797156 T^{5} + 9954182 p^{2} T^{6} - 43776 p^{4} T^{7} + 3929 p^{6} T^{8} + 2 p^{8} T^{9} + p^{10} T^{10} )^{4} \) | |
| 43 | \( 1 - 3802546 T^{4} + 26831738947197 T^{8} - 91894552069346003832 T^{12} + \)\(50\!\cdots\!70\)\( T^{16} - \)\(13\!\cdots\!80\)\( T^{20} + \)\(50\!\cdots\!70\)\( p^{8} T^{24} - 91894552069346003832 p^{16} T^{28} + 26831738947197 p^{24} T^{32} - 3802546 p^{32} T^{36} + p^{40} T^{40} \) | |
| 47 | \( ( 1 + 2 p T + 2 p^{2} T^{2} + 248926 T^{3} + 12120201 T^{4} + 392566800 T^{5} + 14336307920 T^{6} + 634261663440 T^{7} + 14862450450758 T^{8} + 118302473355652 T^{9} + 6637757880275964 T^{10} + 118302473355652 p^{2} T^{11} + 14862450450758 p^{4} T^{12} + 634261663440 p^{6} T^{13} + 14336307920 p^{8} T^{14} + 392566800 p^{10} T^{15} + 12120201 p^{12} T^{16} + 248926 p^{14} T^{17} + 2 p^{18} T^{18} + 2 p^{19} T^{19} + p^{20} T^{20} )^{2} \) | |
| 53 | \( 1 - 15039146 T^{4} + 133684220395021 T^{8} + 39218946710640122632 T^{12} - \)\(96\!\cdots\!02\)\( T^{16} + \)\(12\!\cdots\!28\)\( T^{20} - \)\(96\!\cdots\!02\)\( p^{8} T^{24} + 39218946710640122632 p^{16} T^{28} + 133684220395021 p^{24} T^{32} - 15039146 p^{32} T^{36} + p^{40} T^{40} \) | |
| 59 | \( ( 1 + 20358 T^{2} + 205085277 T^{4} + 1346166325448 T^{6} + 6536734169700626 T^{8} + 25203640704687976420 T^{10} + 6536734169700626 p^{4} T^{12} + 1346166325448 p^{8} T^{14} + 205085277 p^{12} T^{16} + 20358 p^{16} T^{18} + p^{20} T^{20} )^{2} \) | |
| 61 | \( ( 1 - 18802 T^{2} + 189117357 T^{4} - 1333577301752 T^{6} + 7106541440264866 T^{8} - 29659798453891124140 T^{10} + 7106541440264866 p^{4} T^{12} - 1333577301752 p^{8} T^{14} + 189117357 p^{12} T^{16} - 18802 p^{16} T^{18} + p^{20} T^{20} )^{2} \) | |
| 67 | \( 1 + 6517774 T^{4} + 162116992238781 T^{8} - \)\(33\!\cdots\!08\)\( T^{12} + \)\(23\!\cdots\!98\)\( T^{16} + \)\(28\!\cdots\!68\)\( T^{20} + \)\(23\!\cdots\!98\)\( p^{8} T^{24} - \)\(33\!\cdots\!08\)\( p^{16} T^{28} + 162116992238781 p^{24} T^{32} + 6517774 p^{32} T^{36} + p^{40} T^{40} \) | |
| 71 | \( ( 1 + 62 T + 12585 T^{2} + 1087696 T^{3} + 91333814 T^{4} + 7391992324 T^{5} + 91333814 p^{2} T^{6} + 1087696 p^{4} T^{7} + 12585 p^{6} T^{8} + 62 p^{8} T^{9} + p^{10} T^{10} )^{4} \) | |
| 73 | \( ( 1 + 62 T + 1922 T^{2} + 293406 T^{3} + 52885149 T^{4} + 1992907656 T^{5} + 64958558712 T^{6} + 9681205197576 T^{7} + 2398655765914626 T^{8} + 112307032540117748 T^{9} + 3743840733961712140 T^{10} + 112307032540117748 p^{2} T^{11} + 2398655765914626 p^{4} T^{12} + 9681205197576 p^{6} T^{13} + 64958558712 p^{8} T^{14} + 1992907656 p^{10} T^{15} + 52885149 p^{12} T^{16} + 293406 p^{14} T^{17} + 1922 p^{16} T^{18} + 62 p^{18} T^{19} + p^{20} T^{20} )^{2} \) | |
| 79 | \( ( 1 - 46346 T^{2} + 1023688173 T^{4} - 14286438751864 T^{6} + 140168501571945426 T^{8} - \)\(10\!\cdots\!20\)\( T^{10} + 140168501571945426 p^{4} T^{12} - 14286438751864 p^{8} T^{14} + 1023688173 p^{12} T^{16} - 46346 p^{16} T^{18} + p^{20} T^{20} )^{2} \) | |
| 83 | \( 1 - 209971186 T^{4} + 203835292037679 p T^{8} - \)\(34\!\cdots\!52\)\( T^{12} - \)\(43\!\cdots\!90\)\( T^{16} + \)\(36\!\cdots\!20\)\( T^{20} - \)\(43\!\cdots\!90\)\( p^{8} T^{24} - \)\(34\!\cdots\!52\)\( p^{16} T^{28} + 203835292037679 p^{25} T^{32} - 209971186 p^{32} T^{36} + p^{40} T^{40} \) | |
| 89 | \( ( 1 - 46986 T^{2} + 1163729293 T^{4} - 215914071096 p T^{6} + 230705121354497106 T^{8} - \)\(20\!\cdots\!60\)\( T^{10} + 230705121354497106 p^{4} T^{12} - 215914071096 p^{9} T^{14} + 1163729293 p^{12} T^{16} - 46986 p^{16} T^{18} + p^{20} T^{20} )^{2} \) | |
| 97 | \( ( 1 - 50 T + 1250 T^{2} - 914226 T^{3} + 90832893 T^{4} + 7094951240 T^{5} - 50384088712 T^{6} + 11147294809672 T^{7} - 3740546910279294 T^{8} + 653351137546335316 T^{9} - 8437320896266469556 T^{10} + 653351137546335316 p^{2} T^{11} - 3740546910279294 p^{4} T^{12} + 11147294809672 p^{6} T^{13} - 50384088712 p^{8} T^{14} + 7094951240 p^{10} T^{15} + 90832893 p^{12} T^{16} - 914226 p^{14} T^{17} + 1250 p^{16} T^{18} - 50 p^{18} T^{19} + p^{20} T^{20} )^{2} \) | |
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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
−3.02878896273069215571353086641, −2.92930207216142546411213077661, −2.91538082145007012691973105096, −2.69039405599357588105941227665, −2.65370403351739257986545832013, −2.63129950264265721849820806214, −2.42302226682149180536455567434, −2.40411754626911255726855711376, −2.36247294702625153469859971215, −2.24651062898452295193772436094, −2.00297783892652583735740341010, −1.95496074922009698580550575625, −1.78897653077391960579341822836, −1.74278947917878184215171573022, −1.71192332277655219207281132029, −1.61223040739077806774761123099, −1.46873799985940028876846820852, −1.39407383235160307719312885845, −1.05368837344791461721012531743, −0.903583637409107382885236592608, −0.872981040050345257121315903951, −0.76425047923520404724294204667, −0.58762804741124329717946617472, −0.28428959623764864941835423069, −0.12879466634240432482580080998, 0.12879466634240432482580080998, 0.28428959623764864941835423069, 0.58762804741124329717946617472, 0.76425047923520404724294204667, 0.872981040050345257121315903951, 0.903583637409107382885236592608, 1.05368837344791461721012531743, 1.39407383235160307719312885845, 1.46873799985940028876846820852, 1.61223040739077806774761123099, 1.71192332277655219207281132029, 1.74278947917878184215171573022, 1.78897653077391960579341822836, 1.95496074922009698580550575625, 2.00297783892652583735740341010, 2.24651062898452295193772436094, 2.36247294702625153469859971215, 2.40411754626911255726855711376, 2.42302226682149180536455567434, 2.63129950264265721849820806214, 2.65370403351739257986545832013, 2.69039405599357588105941227665, 2.91538082145007012691973105096, 2.92930207216142546411213077661, 3.02878896273069215571353086641
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.