Dirichlet series
| L(s) = 1 | − 8·4-s − 8·9-s + 8·11-s + 36·16-s + 20·19-s + 8·23-s + 30·25-s + 64·36-s − 28·37-s + 32·41-s − 64·44-s − 56·61-s − 120·64-s + 32·67-s − 56·71-s + 88·73-s − 160·76-s + 36·81-s + 8·83-s − 64·92-s − 64·99-s − 240·100-s − 8·101-s + 40·113-s + 31·121-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 4·4-s − 8/3·9-s + 2.41·11-s + 9·16-s + 4.58·19-s + 1.66·23-s + 6·25-s + 32/3·36-s − 4.60·37-s + 4.99·41-s − 9.64·44-s − 7.17·61-s − 15·64-s + 3.90·67-s − 6.64·71-s + 10.2·73-s − 18.3·76-s + 4·81-s + 0.878·83-s − 6.67·92-s − 6.43·99-s − 24·100-s − 0.796·101-s + 3.76·113-s + 2.81·121-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 7^{32} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 7^{32} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(2^{16} \cdot 3^{16} \cdot 7^{32} \cdot 11^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.91087\times 10^{22}\) |
| Root analytic conductor: | \(5.08169\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 2^{16} \cdot 3^{16} \cdot 7^{32} \cdot 11^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(2.007509891\) |
| \(L(\frac12)\) | \(\approx\) | \(2.007509891\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + T^{2} )^{8} \) |
| 3 | \( ( 1 + T^{2} )^{8} \) | |
| 7 | \( 1 \) | |
| 11 | \( 1 - 8 T + 3 p T^{2} - 56 T^{3} - 206 T^{4} + 120 p T^{5} - 107 p T^{6} - 15528 T^{7} + 86618 T^{8} - 15528 p T^{9} - 107 p^{3} T^{10} + 120 p^{4} T^{11} - 206 p^{4} T^{12} - 56 p^{5} T^{13} + 3 p^{7} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| good | 5 | \( 1 - 6 p T^{2} + 471 T^{4} - 986 p T^{6} + 37781 T^{8} - 44492 p T^{10} + 1052266 T^{12} - 4383184 T^{14} + 19642634 T^{16} - 4383184 p^{2} T^{18} + 1052266 p^{4} T^{20} - 44492 p^{7} T^{22} + 37781 p^{8} T^{24} - 986 p^{11} T^{26} + 471 p^{12} T^{28} - 6 p^{15} T^{30} + p^{16} T^{32} \) |
| 13 | \( ( 1 + 40 T^{2} - 8 T^{3} + 576 T^{4} + 648 T^{5} + 3672 T^{6} + 30816 T^{7} + 19294 T^{8} + 30816 p T^{9} + 3672 p^{2} T^{10} + 648 p^{3} T^{11} + 576 p^{4} T^{12} - 8 p^{5} T^{13} + 40 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 17 | \( ( 1 + 83 T^{2} + 4 p T^{3} + 3413 T^{4} + 4044 T^{5} + 95086 T^{6} + 108260 T^{7} + 1908750 T^{8} + 108260 p T^{9} + 95086 p^{2} T^{10} + 4044 p^{3} T^{11} + 3413 p^{4} T^{12} + 4 p^{6} T^{13} + 83 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 19 | \( ( 1 - 10 T + 145 T^{2} - 1030 T^{3} + 8870 T^{4} - 49582 T^{5} + 317127 T^{6} - 1444786 T^{7} + 7373490 T^{8} - 1444786 p T^{9} + 317127 p^{2} T^{10} - 49582 p^{3} T^{11} + 8870 p^{4} T^{12} - 1030 p^{5} T^{13} + 145 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 23 | \( ( 1 - 4 T + 57 T^{2} - 124 T^{3} + 2103 T^{4} - 4828 T^{5} + 65020 T^{6} - 100872 T^{7} + 1438254 T^{8} - 100872 p T^{9} + 65020 p^{2} T^{10} - 4828 p^{3} T^{11} + 2103 p^{4} T^{12} - 124 p^{5} T^{13} + 57 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 29 | \( 1 - 220 T^{2} + 25766 T^{4} - 2101752 T^{6} + 132425201 T^{8} - 6784185928 T^{10} + 290786240918 T^{12} - 10599475438788 T^{14} + 331431007100868 T^{16} - 10599475438788 p^{2} T^{18} + 290786240918 p^{4} T^{20} - 6784185928 p^{6} T^{22} + 132425201 p^{8} T^{24} - 2101752 p^{10} T^{26} + 25766 p^{12} T^{28} - 220 p^{14} T^{30} + p^{16} T^{32} \) | |
| 31 | \( 1 - 386 T^{2} + 71901 T^{4} - 8595466 T^{6} + 739336166 T^{8} - 48582533718 T^{10} + 2525237875987 T^{12} - 105975133331486 T^{14} + 3630081582174546 T^{16} - 105975133331486 p^{2} T^{18} + 2525237875987 p^{4} T^{20} - 48582533718 p^{6} T^{22} + 739336166 p^{8} T^{24} - 8595466 p^{10} T^{26} + 71901 p^{12} T^{28} - 386 p^{14} T^{30} + p^{16} T^{32} \) | |
| 37 | \( ( 1 + 14 T + 229 T^{2} + 2126 T^{3} + 20986 T^{4} + 154618 T^{5} + 1182051 T^{6} + 7476394 T^{7} + 49108426 T^{8} + 7476394 p T^{9} + 1182051 p^{2} T^{10} + 154618 p^{3} T^{11} + 20986 p^{4} T^{12} + 2126 p^{5} T^{13} + 229 p^{6} T^{14} + 14 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 41 | \( ( 1 - 16 T + 330 T^{2} - 3640 T^{3} + 42745 T^{4} - 362832 T^{5} + 3123194 T^{6} - 21761352 T^{7} + 152229812 T^{8} - 21761352 p T^{9} + 3123194 p^{2} T^{10} - 362832 p^{3} T^{11} + 42745 p^{4} T^{12} - 3640 p^{5} T^{13} + 330 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 43 | \( 1 - 370 T^{2} + 69205 T^{4} - 8714850 T^{6} + 829915302 T^{8} - 63533346462 T^{10} + 4048663635627 T^{12} - 219017168933710 T^{14} + 10156745084830162 T^{16} - 219017168933710 p^{2} T^{18} + 4048663635627 p^{4} T^{20} - 63533346462 p^{6} T^{22} + 829915302 p^{8} T^{24} - 8714850 p^{10} T^{26} + 69205 p^{12} T^{28} - 370 p^{14} T^{30} + p^{16} T^{32} \) | |
| 47 | \( 1 - 62 T^{2} + 2031 T^{4} - 368922 T^{6} + 20631365 T^{8} - 625319636 T^{10} + 67952187226 T^{12} - 3345437111480 T^{14} + 84733582253162 T^{16} - 3345437111480 p^{2} T^{18} + 67952187226 p^{4} T^{20} - 625319636 p^{6} T^{22} + 20631365 p^{8} T^{24} - 368922 p^{10} T^{26} + 2031 p^{12} T^{28} - 62 p^{14} T^{30} + p^{16} T^{32} \) | |
| 53 | \( ( 1 + 189 T^{2} - 116 T^{3} + 18282 T^{4} - 21964 T^{5} + 1293871 T^{6} - 2238960 T^{7} + 74773842 T^{8} - 2238960 p T^{9} + 1293871 p^{2} T^{10} - 21964 p^{3} T^{11} + 18282 p^{4} T^{12} - 116 p^{5} T^{13} + 189 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 59 | \( 1 - 412 T^{2} + 84726 T^{4} - 11812648 T^{6} + 1278873825 T^{8} - 115825341832 T^{10} + 9135071092582 T^{12} - 639128365018580 T^{14} + 39901635887620324 T^{16} - 639128365018580 p^{2} T^{18} + 9135071092582 p^{4} T^{20} - 115825341832 p^{6} T^{22} + 1278873825 p^{8} T^{24} - 11812648 p^{10} T^{26} + 84726 p^{12} T^{28} - 412 p^{14} T^{30} + p^{16} T^{32} \) | |
| 61 | \( ( 1 + 28 T + 652 T^{2} + 10788 T^{3} + 153837 T^{4} + 1841484 T^{5} + 19466748 T^{6} + 180831988 T^{7} + 1498320772 T^{8} + 180831988 p T^{9} + 19466748 p^{2} T^{10} + 1841484 p^{3} T^{11} + 153837 p^{4} T^{12} + 10788 p^{5} T^{13} + 652 p^{6} T^{14} + 28 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 67 | \( ( 1 - 16 T + 442 T^{2} - 5580 T^{3} + 88929 T^{4} - 911040 T^{5} + 10786038 T^{6} - 91408468 T^{7} + 872545804 T^{8} - 91408468 p T^{9} + 10786038 p^{2} T^{10} - 911040 p^{3} T^{11} + 88929 p^{4} T^{12} - 5580 p^{5} T^{13} + 442 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 71 | \( ( 1 + 28 T + 665 T^{2} + 9556 T^{3} + 124346 T^{4} + 1164764 T^{5} + 10825387 T^{6} + 79684020 T^{7} + 712940082 T^{8} + 79684020 p T^{9} + 10825387 p^{2} T^{10} + 1164764 p^{3} T^{11} + 124346 p^{4} T^{12} + 9556 p^{5} T^{13} + 665 p^{6} T^{14} + 28 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 73 | \( ( 1 - 44 T + 1204 T^{2} - 23988 T^{3} + 389092 T^{4} - 5297980 T^{5} + 62491788 T^{6} - 643636388 T^{7} + 5856468854 T^{8} - 643636388 p T^{9} + 62491788 p^{2} T^{10} - 5297980 p^{3} T^{11} + 389092 p^{4} T^{12} - 23988 p^{5} T^{13} + 1204 p^{6} T^{14} - 44 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 79 | \( 1 - 538 T^{2} + 147807 T^{4} - 27647974 T^{6} + 3965128581 T^{8} - 467618207428 T^{10} + 47603630292250 T^{12} - 4323347326389920 T^{14} + 357043246909249546 T^{16} - 4323347326389920 p^{2} T^{18} + 47603630292250 p^{4} T^{20} - 467618207428 p^{6} T^{22} + 3965128581 p^{8} T^{24} - 27647974 p^{10} T^{26} + 147807 p^{12} T^{28} - 538 p^{14} T^{30} + p^{16} T^{32} \) | |
| 83 | \( ( 1 - 4 T + 523 T^{2} - 1544 T^{3} + 123705 T^{4} - 263372 T^{5} + 17791866 T^{6} - 28394068 T^{7} + 1752220430 T^{8} - 28394068 p T^{9} + 17791866 p^{2} T^{10} - 263372 p^{3} T^{11} + 123705 p^{4} T^{12} - 1544 p^{5} T^{13} + 523 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 89 | \( 1 - 640 T^{2} + 210552 T^{4} - 47499904 T^{6} + 8295415452 T^{8} - 1194559990144 T^{10} + 146915919459400 T^{12} - 15750756401654144 T^{14} + 1489545499043303110 T^{16} - 15750756401654144 p^{2} T^{18} + 146915919459400 p^{4} T^{20} - 1194559990144 p^{6} T^{22} + 8295415452 p^{8} T^{24} - 47499904 p^{10} T^{26} + 210552 p^{12} T^{28} - 640 p^{14} T^{30} + p^{16} T^{32} \) | |
| 97 | \( 1 - 608 T^{2} + 187588 T^{4} - 40328000 T^{6} + 6818823050 T^{8} - 967972100768 T^{10} + 120672937256208 T^{12} - 13517338511335328 T^{14} + 1373778662576096019 T^{16} - 13517338511335328 p^{2} T^{18} + 120672937256208 p^{4} T^{20} - 967972100768 p^{6} T^{22} + 6818823050 p^{8} T^{24} - 40328000 p^{10} T^{26} + 187588 p^{12} T^{28} - 608 p^{14} T^{30} + p^{16} T^{32} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.07240023985030544876612032473, −1.96719987783428621593002963155, −1.95596242184774366048249181858, −1.90583004164163440106523263931, −1.85528230593989528276360160258, −1.74263102916599994517658895277, −1.67238664917644168751968503859, −1.49957265606966964124899083731, −1.35851482365329429954883293635, −1.27348831162938794860104873539, −1.22744128553198481828209561447, −1.20179162467525464734821760701, −1.19002645595443562233348759593, −1.06687952232510860562252417739, −0.972569963400701920287643377100, −0.940995186793132486034605140703, −0.904018424139354618623068360577, −0.807350578047623732827039029240, −0.74239451148666151060875932193, −0.68657432964413323449201382952, −0.55487955351336026486162839218, −0.50984542782332521979403800101, −0.29983057353607263992314778028, −0.13451720269080045353572959893, −0.090660373876815401402317262248, 0.090660373876815401402317262248, 0.13451720269080045353572959893, 0.29983057353607263992314778028, 0.50984542782332521979403800101, 0.55487955351336026486162839218, 0.68657432964413323449201382952, 0.74239451148666151060875932193, 0.807350578047623732827039029240, 0.904018424139354618623068360577, 0.940995186793132486034605140703, 0.972569963400701920287643377100, 1.06687952232510860562252417739, 1.19002645595443562233348759593, 1.20179162467525464734821760701, 1.22744128553198481828209561447, 1.27348831162938794860104873539, 1.35851482365329429954883293635, 1.49957265606966964124899083731, 1.67238664917644168751968503859, 1.74263102916599994517658895277, 1.85528230593989528276360160258, 1.90583004164163440106523263931, 1.95596242184774366048249181858, 1.96719987783428621593002963155, 2.07240023985030544876612032473
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.