Dirichlet series
| L(s) = 1 | − 56·7-s + 612·19-s + 490·25-s − 1.12e3·31-s − 1.19e3·37-s − 328·43-s + 1.96e3·49-s − 1.63e3·61-s − 308·67-s + 4.06e3·73-s + 2.17e3·79-s − 1.38e3·103-s − 1.07e3·109-s − 3.20e3·121-s + 127-s + 131-s − 3.42e4·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 9.00e3·169-s + 173-s − 2.74e4·175-s + ⋯ |
| L(s) = 1 | − 3.02·7-s + 7.38·19-s + 3.91·25-s − 6.53·31-s − 5.31·37-s − 1.16·43-s + 40/7·49-s − 3.42·61-s − 0.561·67-s + 6.52·73-s + 3.09·79-s − 1.32·103-s − 0.945·109-s − 2.40·121-s + 0.000698·127-s + 0.000666·131-s − 22.3·133-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 4.09·169-s + 0.000439·173-s − 11.8·175-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 3^{32} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 3^{32} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(2^{64} \cdot 3^{32} \cdot 7^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.45033\times 10^{28}\) |
| Root analytic conductor: | \(7.71193\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 2^{64} \cdot 3^{32} \cdot 7^{16} ,\ ( \ : [3/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(\approx\) | \(16.94260354\) |
| \(L(\frac12)\) | \(\approx\) | \(16.94260354\) |
| \(L(\frac{5}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) | |
| 7 | \( ( 1 + 4 p T + 4 p^{2} T^{2} - 4 p^{4} T^{3} - 5359 p^{2} T^{4} - 4 p^{7} T^{5} + 4 p^{8} T^{6} + 4 p^{10} T^{7} + p^{12} T^{8} )^{2} \) | |
| good | 5 | \( 1 - 98 p T^{2} + 4047 p^{2} T^{4} - 13467734 T^{6} + 2004738389 T^{8} - 76183349844 p T^{10} + 60059957637514 T^{12} - 6902983281820528 T^{14} + 756156244566152106 T^{16} - 6902983281820528 p^{6} T^{18} + 60059957637514 p^{12} T^{20} - 76183349844 p^{19} T^{22} + 2004738389 p^{24} T^{24} - 13467734 p^{30} T^{26} + 4047 p^{38} T^{28} - 98 p^{43} T^{30} + p^{48} T^{32} \) |
| 11 | \( 1 + 3202 T^{2} + 1797447 T^{4} - 5608813218 T^{6} - 4531249774539 T^{8} + 15353827063546164 T^{10} + 27034229520225124106 T^{12} + \)\(48\!\cdots\!88\)\( T^{14} - \)\(31\!\cdots\!02\)\( T^{16} + \)\(48\!\cdots\!88\)\( p^{6} T^{18} + 27034229520225124106 p^{12} T^{20} + 15353827063546164 p^{18} T^{22} - 4531249774539 p^{24} T^{24} - 5608813218 p^{30} T^{26} + 1797447 p^{36} T^{28} + 3202 p^{42} T^{30} + p^{48} T^{32} \) | |
| 13 | \( ( 1 - 4502 T^{2} + 10097461 T^{4} - 24107307194 T^{6} + 60089639013832 T^{8} - 24107307194 p^{6} T^{10} + 10097461 p^{12} T^{12} - 4502 p^{18} T^{14} + p^{24} T^{16} )^{2} \) | |
| 17 | \( 1 - 16948 T^{2} + 84203208 T^{4} - 399083701352 T^{6} + 6508712666429186 T^{8} - 42615253832024962764 T^{10} + \)\(11\!\cdots\!04\)\( T^{12} - \)\(92\!\cdots\!84\)\( T^{14} + \)\(77\!\cdots\!75\)\( T^{16} - \)\(92\!\cdots\!84\)\( p^{6} T^{18} + \)\(11\!\cdots\!04\)\( p^{12} T^{20} - 42615253832024962764 p^{18} T^{22} + 6508712666429186 p^{24} T^{24} - 399083701352 p^{30} T^{26} + 84203208 p^{36} T^{28} - 16948 p^{42} T^{30} + p^{48} T^{32} \) | |
| 19 | \( ( 1 - 306 T + 57757 T^{2} - 8122770 T^{3} + 990235135 T^{4} - 107235980892 T^{5} + 10750308055972 T^{6} - 990048157019604 T^{7} + 85251569497003126 T^{8} - 990048157019604 p^{3} T^{9} + 10750308055972 p^{6} T^{10} - 107235980892 p^{9} T^{11} + 990235135 p^{12} T^{12} - 8122770 p^{15} T^{13} + 57757 p^{18} T^{14} - 306 p^{21} T^{15} + p^{24} T^{16} )^{2} \) | |
| 23 | \( 1 + 54964 T^{2} + 1326151704 T^{4} + 25415665738344 T^{6} + 524305919802157986 T^{8} + \)\(89\!\cdots\!72\)\( T^{10} + \)\(12\!\cdots\!96\)\( T^{12} + \)\(17\!\cdots\!32\)\( T^{14} + \)\(23\!\cdots\!87\)\( T^{16} + \)\(17\!\cdots\!32\)\( p^{6} T^{18} + \)\(12\!\cdots\!96\)\( p^{12} T^{20} + \)\(89\!\cdots\!72\)\( p^{18} T^{22} + 524305919802157986 p^{24} T^{24} + 25415665738344 p^{30} T^{26} + 1326151704 p^{36} T^{28} + 54964 p^{42} T^{30} + p^{48} T^{32} \) | |
| 29 | \( ( 1 - 102034 T^{2} + 6213617737 T^{4} - 245008733294114 T^{6} + 7064972052959541764 T^{8} - 245008733294114 p^{6} T^{10} + 6213617737 p^{12} T^{12} - 102034 p^{18} T^{14} + p^{24} T^{16} )^{2} \) | |
| 31 | \( ( 1 + 564 T + 243388 T^{2} + 77468784 T^{3} + 21469133071 T^{4} + 5073861272220 T^{5} + 1094289736794676 T^{6} + 211880592093523344 T^{7} + 38213365513366414528 T^{8} + 211880592093523344 p^{3} T^{9} + 1094289736794676 p^{6} T^{10} + 5073861272220 p^{9} T^{11} + 21469133071 p^{12} T^{12} + 77468784 p^{15} T^{13} + 243388 p^{18} T^{14} + 564 p^{21} T^{15} + p^{24} T^{16} )^{2} \) | |
| 37 | \( ( 1 + 598 T + 47601 T^{2} - 11729418 T^{3} + 10699876539 T^{4} + 3891656263464 T^{5} + 114219217143080 T^{6} + 66547288562775176 T^{7} + 48227467333244423670 T^{8} + 66547288562775176 p^{3} T^{9} + 114219217143080 p^{6} T^{10} + 3891656263464 p^{9} T^{11} + 10699876539 p^{12} T^{12} - 11729418 p^{15} T^{13} + 47601 p^{18} T^{14} + 598 p^{21} T^{15} + p^{24} T^{16} )^{2} \) | |
| 41 | \( ( 1 + 148396 T^{2} + 19955386648 T^{4} + 1828971842691940 T^{6} + \)\(14\!\cdots\!50\)\( T^{8} + 1828971842691940 p^{6} T^{10} + 19955386648 p^{12} T^{12} + 148396 p^{18} T^{14} + p^{24} T^{16} )^{2} \) | |
| 43 | \( ( 1 + 82 T + 122239 T^{2} + 47448038 T^{3} + 6939760430 T^{4} + 47448038 p^{3} T^{5} + 122239 p^{6} T^{6} + 82 p^{9} T^{7} + p^{12} T^{8} )^{4} \) | |
| 47 | \( 1 - 358312 T^{2} + 90535322508 T^{4} - 16795335458693168 T^{6} + \)\(25\!\cdots\!46\)\( T^{8} - \)\(71\!\cdots\!68\)\( p T^{10} + \)\(39\!\cdots\!64\)\( T^{12} - \)\(42\!\cdots\!76\)\( T^{14} + \)\(45\!\cdots\!75\)\( T^{16} - \)\(42\!\cdots\!76\)\( p^{6} T^{18} + \)\(39\!\cdots\!64\)\( p^{12} T^{20} - \)\(71\!\cdots\!68\)\( p^{19} T^{22} + \)\(25\!\cdots\!46\)\( p^{24} T^{24} - 16795335458693168 p^{30} T^{26} + 90535322508 p^{36} T^{28} - 358312 p^{42} T^{30} + p^{48} T^{32} \) | |
| 53 | \( 1 + 506674 T^{2} + 85866141819 T^{4} + 10208516436240318 T^{6} + \)\(32\!\cdots\!49\)\( T^{8} + \)\(63\!\cdots\!00\)\( T^{10} + \)\(61\!\cdots\!74\)\( T^{12} + \)\(12\!\cdots\!24\)\( T^{14} + \)\(28\!\cdots\!38\)\( T^{16} + \)\(12\!\cdots\!24\)\( p^{6} T^{18} + \)\(61\!\cdots\!74\)\( p^{12} T^{20} + \)\(63\!\cdots\!00\)\( p^{18} T^{22} + \)\(32\!\cdots\!49\)\( p^{24} T^{24} + 10208516436240318 p^{30} T^{26} + 85866141819 p^{36} T^{28} + 506674 p^{42} T^{30} + p^{48} T^{32} \) | |
| 59 | \( 1 - 399442 T^{2} + 89994325083 T^{4} - 32591407040941742 T^{6} + \)\(78\!\cdots\!45\)\( T^{8} - \)\(13\!\cdots\!56\)\( T^{10} + \)\(43\!\cdots\!26\)\( T^{12} - \)\(98\!\cdots\!64\)\( T^{14} + \)\(16\!\cdots\!22\)\( T^{16} - \)\(98\!\cdots\!64\)\( p^{6} T^{18} + \)\(43\!\cdots\!26\)\( p^{12} T^{20} - \)\(13\!\cdots\!56\)\( p^{18} T^{22} + \)\(78\!\cdots\!45\)\( p^{24} T^{24} - 32591407040941742 p^{30} T^{26} + 89994325083 p^{36} T^{28} - 399442 p^{42} T^{30} + p^{48} T^{32} \) | |
| 61 | \( ( 1 + 816 T + 837478 T^{2} + 502269216 T^{3} + 333755818642 T^{4} + 188353178559720 T^{5} + 93910648157586304 T^{6} + 50370689927417233344 T^{7} + \)\(21\!\cdots\!87\)\( T^{8} + 50370689927417233344 p^{3} T^{9} + 93910648157586304 p^{6} T^{10} + 188353178559720 p^{9} T^{11} + 333755818642 p^{12} T^{12} + 502269216 p^{15} T^{13} + 837478 p^{18} T^{14} + 816 p^{21} T^{15} + p^{24} T^{16} )^{2} \) | |
| 67 | \( ( 1 + 154 T - 1009305 T^{2} - 101871994 T^{3} + 602011276205 T^{4} + 36961847361936 T^{5} - 261850611517253414 T^{6} - 66134569555683020 p T^{7} + \)\(89\!\cdots\!66\)\( T^{8} - 66134569555683020 p^{4} T^{9} - 261850611517253414 p^{6} T^{10} + 36961847361936 p^{9} T^{11} + 602011276205 p^{12} T^{12} - 101871994 p^{15} T^{13} - 1009305 p^{18} T^{14} + 154 p^{21} T^{15} + p^{24} T^{16} )^{2} \) | |
| 71 | \( ( 1 - 2546056 T^{2} + 2933736697204 T^{4} - 1997708473027518488 T^{6} + \)\(87\!\cdots\!74\)\( T^{8} - 1997708473027518488 p^{6} T^{10} + 2933736697204 p^{12} T^{12} - 2546056 p^{18} T^{14} + p^{24} T^{16} )^{2} \) | |
| 73 | \( ( 1 - 2034 T + 2450047 T^{2} - 2178403830 T^{3} + 1347031106821 T^{4} - 430919574332148 T^{5} - 113107388808572534 T^{6} + \)\(30\!\cdots\!80\)\( T^{7} - \)\(25\!\cdots\!10\)\( T^{8} + \)\(30\!\cdots\!80\)\( p^{3} T^{9} - 113107388808572534 p^{6} T^{10} - 430919574332148 p^{9} T^{11} + 1347031106821 p^{12} T^{12} - 2178403830 p^{15} T^{13} + 2450047 p^{18} T^{14} - 2034 p^{21} T^{15} + p^{24} T^{16} )^{2} \) | |
| 79 | \( ( 1 - 1088 T - 269028 T^{2} + 1109682680 T^{3} - 391267378201 T^{4} - 358084200932532 T^{5} + 285753619209423700 T^{6} + 15705523238748286852 T^{7} - \)\(10\!\cdots\!68\)\( T^{8} + 15705523238748286852 p^{3} T^{9} + 285753619209423700 p^{6} T^{10} - 358084200932532 p^{9} T^{11} - 391267378201 p^{12} T^{12} + 1109682680 p^{15} T^{13} - 269028 p^{18} T^{14} - 1088 p^{21} T^{15} + p^{24} T^{16} )^{2} \) | |
| 83 | \( ( 1 + 2632486 T^{2} + 3636170382349 T^{4} + 3366834686358200962 T^{6} + \)\(22\!\cdots\!84\)\( T^{8} + 3366834686358200962 p^{6} T^{10} + 3636170382349 p^{12} T^{12} + 2632486 p^{18} T^{14} + p^{24} T^{16} )^{2} \) | |
| 89 | \( 1 - 2548864 T^{2} + 4462766350596 T^{4} - 5306850100326352640 T^{6} + \)\(51\!\cdots\!26\)\( T^{8} - \)\(40\!\cdots\!68\)\( T^{10} + \)\(27\!\cdots\!80\)\( T^{12} - \)\(17\!\cdots\!04\)\( T^{14} + \)\(11\!\cdots\!27\)\( T^{16} - \)\(17\!\cdots\!04\)\( p^{6} T^{18} + \)\(27\!\cdots\!80\)\( p^{12} T^{20} - \)\(40\!\cdots\!68\)\( p^{18} T^{22} + \)\(51\!\cdots\!26\)\( p^{24} T^{24} - 5306850100326352640 p^{30} T^{26} + 4462766350596 p^{36} T^{28} - 2548864 p^{42} T^{30} + p^{48} T^{32} \) | |
| 97 | \( ( 1 - 2978570 T^{2} + 5898756336049 T^{4} - 7901916867899068322 T^{6} + \)\(83\!\cdots\!24\)\( T^{8} - 7901916867899068322 p^{6} T^{10} + 5898756336049 p^{12} T^{12} - 2978570 p^{18} T^{14} + p^{24} T^{16} )^{2} \) | |
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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
−2.05922466842020152335237969274, −2.00177056284670617495097603608, −1.97747986044552037952024774533, −1.95466454795089383538458624482, −1.87245469361410310846176952064, −1.77989376435194340466176270890, −1.66911639679948832155136775931, −1.62089161398446707522444157410, −1.50674713455854152383680908063, −1.23790527154334893311560746227, −1.22052193089095717599548625978, −1.13108315112134765924231403713, −1.11731389101779815269325248988, −1.08155147237021322342472759024, −1.05543795034336738480659032940, −0.944109884448999755622101263169, −0.869984835922590963280884899809, −0.74192405137990296685819672562, −0.62061449709242920647395653309, −0.43953373218952204793102742408, −0.33905237056715601717056760680, −0.27920985540581251978144196456, −0.23049720313961390378609315516, −0.20329511529711058526391770846, −0.17801668914702054666568827992, 0.17801668914702054666568827992, 0.20329511529711058526391770846, 0.23049720313961390378609315516, 0.27920985540581251978144196456, 0.33905237056715601717056760680, 0.43953373218952204793102742408, 0.62061449709242920647395653309, 0.74192405137990296685819672562, 0.869984835922590963280884899809, 0.944109884448999755622101263169, 1.05543795034336738480659032940, 1.08155147237021322342472759024, 1.11731389101779815269325248988, 1.13108315112134765924231403713, 1.22052193089095717599548625978, 1.23790527154334893311560746227, 1.50674713455854152383680908063, 1.62089161398446707522444157410, 1.66911639679948832155136775931, 1.77989376435194340466176270890, 1.87245469361410310846176952064, 1.95466454795089383538458624482, 1.97747986044552037952024774533, 2.00177056284670617495097603608, 2.05922466842020152335237969274
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.