Dirichlet series
| L(s) = 1 | + 4·4-s + 12·5-s − 4·9-s + 8·11-s + 6·16-s + 48·20-s + 16·23-s + 52·25-s + 12·31-s − 16·36-s − 16·37-s + 32·44-s − 48·45-s − 24·47-s − 28·53-s + 96·55-s − 60·59-s + 12·67-s + 8·71-s + 72·80-s + 18·81-s − 96·89-s + 64·92-s − 32·99-s + 208·100-s + 36·103-s − 48·113-s + ⋯ |
| L(s) = 1 | + 2·4-s + 5.36·5-s − 4/3·9-s + 2.41·11-s + 3/2·16-s + 10.7·20-s + 3.33·23-s + 52/5·25-s + 2.15·31-s − 8/3·36-s − 2.63·37-s + 4.82·44-s − 7.15·45-s − 3.50·47-s − 3.84·53-s + 12.9·55-s − 7.81·59-s + 1.46·67-s + 0.949·71-s + 8.04·80-s + 2·81-s − 10.1·89-s + 6.67·92-s − 3.21·99-s + 20.7·100-s + 3.54·103-s − 4.51·113-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{32} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{32} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(2^{16} \cdot 7^{32} \cdot 11^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(9.08518\times 10^{14}\) |
| Root analytic conductor: | \(2.93391\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 2^{16} \cdot 7^{32} \cdot 11^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(19.30544167\) |
| \(L(\frac12)\) | \(\approx\) | \(19.30544167\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - T^{2} + T^{4} )^{4} \) |
| 7 | \( 1 \) | |
| 11 | \( 1 - 8 T + 4 p T^{2} - 240 T^{3} + 1082 T^{4} - 4552 T^{5} + 17200 T^{6} - 62088 T^{7} + 220771 T^{8} - 62088 p T^{9} + 17200 p^{2} T^{10} - 4552 p^{3} T^{11} + 1082 p^{4} T^{12} - 240 p^{5} T^{13} + 4 p^{7} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| good | 3 | \( ( 1 - 2 T - 2 T^{3} + 10 T^{4} - 2 p T^{5} - 2 p^{3} T^{7} + p^{4} T^{8} )^{2}( 1 + 2 T + 2 p T^{2} + 14 T^{3} + 19 T^{4} + 14 p T^{5} + 2 p^{3} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 5 | \( ( 1 - 6 T + 28 T^{2} - 96 T^{3} + 56 p T^{4} - 792 T^{5} + 2128 T^{6} - 1098 p T^{7} + 2603 p T^{8} - 1098 p^{2} T^{9} + 2128 p^{2} T^{10} - 792 p^{3} T^{11} + 56 p^{5} T^{12} - 96 p^{5} T^{13} + 28 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 13 | \( ( 1 + 60 T^{2} + 1810 T^{4} + 37104 T^{6} + 559467 T^{8} + 37104 p^{2} T^{10} + 1810 p^{4} T^{12} + 60 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 17 | \( 1 - 44 T^{2} + 876 T^{4} - 20488 T^{6} + 372650 T^{8} - 2186196 T^{10} - 4370480 T^{12} + 757707148 T^{14} - 25935551901 T^{16} + 757707148 p^{2} T^{18} - 4370480 p^{4} T^{20} - 2186196 p^{6} T^{22} + 372650 p^{8} T^{24} - 20488 p^{10} T^{26} + 876 p^{12} T^{28} - 44 p^{14} T^{30} + p^{16} T^{32} \) | |
| 19 | \( 1 - 72 T^{2} + 2924 T^{4} - 67896 T^{6} + 796942 T^{8} + 4900284 T^{10} - 422590480 T^{12} + 10241692560 T^{14} - 201732257981 T^{16} + 10241692560 p^{2} T^{18} - 422590480 p^{4} T^{20} + 4900284 p^{6} T^{22} + 796942 p^{8} T^{24} - 67896 p^{10} T^{26} + 2924 p^{12} T^{28} - 72 p^{14} T^{30} + p^{16} T^{32} \) | |
| 23 | \( ( 1 - 8 T - 22 T^{2} + 348 T^{3} + 16 p T^{4} - 10318 T^{5} + 15976 T^{6} + 94008 T^{7} - 488945 T^{8} + 94008 p T^{9} + 15976 p^{2} T^{10} - 10318 p^{3} T^{11} + 16 p^{5} T^{12} + 348 p^{5} T^{13} - 22 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 29 | \( ( 1 - 196 T^{2} + 17650 T^{4} - 953200 T^{6} + 33744235 T^{8} - 953200 p^{2} T^{10} + 17650 p^{4} T^{12} - 196 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 31 | \( ( 1 - 6 T + 60 T^{2} - 288 T^{3} + 1654 T^{4} - 2958 T^{5} - 45048 T^{6} + 281862 T^{7} - 2543793 T^{8} + 281862 p T^{9} - 45048 p^{2} T^{10} - 2958 p^{3} T^{11} + 1654 p^{4} T^{12} - 288 p^{5} T^{13} + 60 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 37 | \( ( 1 + 8 T - 96 T^{2} - 500 T^{3} + 9380 T^{4} + 29730 T^{5} - 489260 T^{6} - 281056 T^{7} + 23065323 T^{8} - 281056 p T^{9} - 489260 p^{2} T^{10} + 29730 p^{3} T^{11} + 9380 p^{4} T^{12} - 500 p^{5} T^{13} - 96 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 41 | \( ( 1 + 128 T^{2} + 10456 T^{4} + 593396 T^{6} + 27172090 T^{8} + 593396 p^{2} T^{10} + 10456 p^{4} T^{12} + 128 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 43 | \( ( 1 - 100 T^{2} + 8356 T^{4} - 517756 T^{6} + 23891542 T^{8} - 517756 p^{2} T^{10} + 8356 p^{4} T^{12} - 100 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 47 | \( ( 1 + 12 T + 220 T^{2} + 2064 T^{3} + 23770 T^{4} + 187500 T^{5} + 1740112 T^{6} + 11915748 T^{7} + 94622851 T^{8} + 11915748 p T^{9} + 1740112 p^{2} T^{10} + 187500 p^{3} T^{11} + 23770 p^{4} T^{12} + 2064 p^{5} T^{13} + 220 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 53 | \( ( 1 + 14 T - 40 T^{2} - 504 T^{3} + 10724 T^{4} + 37408 T^{5} - 784280 T^{6} - 1391586 T^{7} + 37358647 T^{8} - 1391586 p T^{9} - 784280 p^{2} T^{10} + 37408 p^{3} T^{11} + 10724 p^{4} T^{12} - 504 p^{5} T^{13} - 40 p^{6} T^{14} + 14 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 59 | \( ( 1 + 30 T + 538 T^{2} + 7140 T^{3} + 78637 T^{4} + 744282 T^{5} + 6314518 T^{6} + 50427036 T^{7} + 391144336 T^{8} + 50427036 p T^{9} + 6314518 p^{2} T^{10} + 744282 p^{3} T^{11} + 78637 p^{4} T^{12} + 7140 p^{5} T^{13} + 538 p^{6} T^{14} + 30 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 61 | \( 1 - 188 T^{2} + 14118 T^{4} - 225784 T^{6} - 33981775 T^{8} + 2162610456 T^{10} + 41215697590 T^{12} - 13509345277892 T^{14} + 1095095924907012 T^{16} - 13509345277892 p^{2} T^{18} + 41215697590 p^{4} T^{20} + 2162610456 p^{6} T^{22} - 33981775 p^{8} T^{24} - 225784 p^{10} T^{26} + 14118 p^{12} T^{28} - 188 p^{14} T^{30} + p^{16} T^{32} \) | |
| 67 | \( ( 1 - 6 T - 178 T^{2} + 1032 T^{3} + 18733 T^{4} - 1254 p T^{5} - 1507870 T^{6} + 2345184 T^{7} + 110694208 T^{8} + 2345184 p T^{9} - 1507870 p^{2} T^{10} - 1254 p^{4} T^{11} + 18733 p^{4} T^{12} + 1032 p^{5} T^{13} - 178 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 71 | \( ( 1 - 2 T + 110 T^{2} - 764 T^{3} + 8800 T^{4} - 764 p T^{5} + 110 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{4} \) | |
| 73 | \( 1 - 380 T^{2} + 72672 T^{4} - 9659248 T^{6} + 1042287158 T^{8} - 101074214712 T^{10} + 9223660177504 T^{12} - 784363064092172 T^{14} + 60491708327485827 T^{16} - 784363064092172 p^{2} T^{18} + 9223660177504 p^{4} T^{20} - 101074214712 p^{6} T^{22} + 1042287158 p^{8} T^{24} - 9659248 p^{10} T^{26} + 72672 p^{12} T^{28} - 380 p^{14} T^{30} + p^{16} T^{32} \) | |
| 79 | \( 1 + 232 T^{2} + 22494 T^{4} + 1979384 T^{6} + 156416777 T^{8} + 1173839700 T^{10} - 1024342395362 T^{12} - 135072893469332 T^{14} - 12452187847957308 T^{16} - 135072893469332 p^{2} T^{18} - 1024342395362 p^{4} T^{20} + 1173839700 p^{6} T^{22} + 156416777 p^{8} T^{24} + 1979384 p^{10} T^{26} + 22494 p^{12} T^{28} + 232 p^{14} T^{30} + p^{16} T^{32} \) | |
| 83 | \( ( 1 + 404 T^{2} + 73300 T^{4} + 8358284 T^{6} + 744742582 T^{8} + 8358284 p^{2} T^{10} + 73300 p^{4} T^{12} + 404 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 89 | \( ( 1 + 48 T + 1280 T^{2} + 24576 T^{3} + 373030 T^{4} + 4735908 T^{5} + 52673264 T^{6} + 535621992 T^{7} + 5154774259 T^{8} + 535621992 p T^{9} + 52673264 p^{2} T^{10} + 4735908 p^{3} T^{11} + 373030 p^{4} T^{12} + 24576 p^{5} T^{13} + 1280 p^{6} T^{14} + 48 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 97 | \( ( 1 - 132 T^{2} + 34546 T^{4} - 2728224 T^{6} + 439608027 T^{8} - 2728224 p^{2} T^{10} + 34546 p^{4} T^{12} - 132 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
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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
−2.58274977632026422659757661831, −2.54495810305820921863869443528, −2.44258993291243989754293276084, −2.42003462193485422101926746770, −2.25521139168646423939545131902, −2.03893296702203040542380038200, −2.00656896824602390526253713071, −1.90021321318706503358214563604, −1.81479321317165929786873535885, −1.79581537253154584115322457438, −1.60722021518839903881329624619, −1.56795644452009039168990427575, −1.54815776409799270004565621062, −1.54685544601342892446318750917, −1.53774227637290971458706038967, −1.49895927287988687398743638645, −1.39409054957845690923201087040, −1.37461387009119838532791142367, −1.17934812278454468052750856203, −0.958291665713892427355227610811, −0.904286900774247887509463967626, −0.53140600647998185395252741392, −0.52614176845887850635022525092, −0.17299783568427873261975074277, −0.16811917301608920703580924438, 0.16811917301608920703580924438, 0.17299783568427873261975074277, 0.52614176845887850635022525092, 0.53140600647998185395252741392, 0.904286900774247887509463967626, 0.958291665713892427355227610811, 1.17934812278454468052750856203, 1.37461387009119838532791142367, 1.39409054957845690923201087040, 1.49895927287988687398743638645, 1.53774227637290971458706038967, 1.54685544601342892446318750917, 1.54815776409799270004565621062, 1.56795644452009039168990427575, 1.60722021518839903881329624619, 1.79581537253154584115322457438, 1.81479321317165929786873535885, 1.90021321318706503358214563604, 2.00656896824602390526253713071, 2.03893296702203040542380038200, 2.25521139168646423939545131902, 2.42003462193485422101926746770, 2.44258993291243989754293276084, 2.54495810305820921863869443528, 2.58274977632026422659757661831
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.