Dirichlet series
| L(s) = 1 | − 4·3-s + 16·4-s − 4·7-s + 4·9-s − 64·12-s + 122·16-s + 8·19-s + 16·21-s + 194·25-s + 26·27-s − 64·28-s − 144·31-s + 64·36-s + 48·37-s − 48·43-s − 488·48-s − 350·49-s − 32·57-s − 140·61-s − 16·63-s + 577·64-s + 204·67-s − 224·73-s − 776·75-s + 128·76-s − 344·79-s − 154·81-s + ⋯ |
| L(s) = 1 | − 4/3·3-s + 4·4-s − 4/7·7-s + 4/9·9-s − 5.33·12-s + 61/8·16-s + 8/19·19-s + 0.761·21-s + 7.75·25-s + 0.962·27-s − 2.28·28-s − 4.64·31-s + 16/9·36-s + 1.29·37-s − 1.11·43-s − 10.1·48-s − 7.14·49-s − 0.561·57-s − 2.29·61-s − 0.253·63-s + 9.01·64-s + 3.04·67-s − 3.06·73-s − 10.3·75-s + 1.68·76-s − 4.35·79-s − 1.90·81-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{14} \cdot 23^{14}\right)^{s/2} \, \Gamma_{\C}(s)^{14} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{14} \cdot 23^{14}\right)^{s/2} \, \Gamma_{\C}(s+1)^{14} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(28\) |
| Conductor: | \(3^{14} \cdot 23^{14}\) |
| Sign: | $1$ |
| Analytic conductor: | \(6895.69\) |
| Root analytic conductor: | \(1.37117\) |
| Motivic weight: | \(2\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((28,\ 3^{14} \cdot 23^{14} ,\ ( \ : [1]^{14} ),\ 1 )\) |
Particular Values
| \(L(\frac{3}{2})\) | \(\approx\) | \(4.615303892\) |
| \(L(\frac12)\) | \(\approx\) | \(4.615303892\) |
| \(L(2)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 + 4 T + 4 p T^{2} + 2 p T^{3} + 26 T^{4} - 196 T^{5} - 7 p^{2} T^{6} - 140 p^{2} T^{7} - 7 p^{4} T^{8} - 196 p^{4} T^{9} + 26 p^{6} T^{10} + 2 p^{9} T^{11} + 4 p^{11} T^{12} + 4 p^{12} T^{13} + p^{14} T^{14} \) |
| 23 | \( ( 1 + p T^{2} )^{7} \) | |
| good | 2 | \( 1 - p^{4} T^{2} + 67 p T^{4} - 769 T^{6} + 3457 T^{8} - 6263 p T^{10} + 4841 p^{3} T^{12} - 130001 T^{14} + 4841 p^{7} T^{16} - 6263 p^{9} T^{18} + 3457 p^{12} T^{20} - 769 p^{16} T^{22} + 67 p^{21} T^{24} - p^{28} T^{26} + p^{28} T^{28} \) |
| 5 | \( 1 - 194 T^{2} + 18151 T^{4} - 1083772 T^{6} + 46424549 T^{8} - 308845958 p T^{10} + 43226962579 T^{12} - 1107119968712 T^{14} + 43226962579 p^{4} T^{16} - 308845958 p^{9} T^{18} + 46424549 p^{12} T^{20} - 1083772 p^{16} T^{22} + 18151 p^{20} T^{24} - 194 p^{24} T^{26} + p^{28} T^{28} \) | |
| 7 | \( ( 1 + 2 T + 181 T^{2} + 16 T^{3} + 15455 T^{4} - 24338 T^{5} + 128845 p T^{6} - 2028832 T^{7} + 128845 p^{3} T^{8} - 24338 p^{4} T^{9} + 15455 p^{6} T^{10} + 16 p^{8} T^{11} + 181 p^{10} T^{12} + 2 p^{12} T^{13} + p^{14} T^{14} )^{2} \) | |
| 11 | \( 1 - 730 T^{2} + 27449 p T^{4} - 88858900 T^{6} + 20398664257 T^{8} - 3812880778342 T^{10} + 594242496842603 T^{12} - 78133659691801304 T^{14} + 594242496842603 p^{4} T^{16} - 3812880778342 p^{8} T^{18} + 20398664257 p^{12} T^{20} - 88858900 p^{16} T^{22} + 27449 p^{21} T^{24} - 730 p^{24} T^{26} + p^{28} T^{28} \) | |
| 13 | \( ( 1 + 920 T^{2} - 330 T^{3} + 397130 T^{4} - 181608 T^{5} + 103425381 T^{6} - 43373052 T^{7} + 103425381 p^{2} T^{8} - 181608 p^{4} T^{9} + 397130 p^{6} T^{10} - 330 p^{8} T^{11} + 920 p^{10} T^{12} + p^{14} T^{14} )^{2} \) | |
| 17 | \( 1 - 1538 T^{2} + 1221463 T^{4} - 678463164 T^{6} + 302970276293 T^{8} - 116589964335390 T^{10} + 39799466785618531 T^{12} - 12150896167996880968 T^{14} + 39799466785618531 p^{4} T^{16} - 116589964335390 p^{8} T^{18} + 302970276293 p^{12} T^{20} - 678463164 p^{16} T^{22} + 1221463 p^{20} T^{24} - 1538 p^{24} T^{26} + p^{28} T^{28} \) | |
| 19 | \( ( 1 - 4 T + 701 T^{2} - 1848 T^{3} + 434431 T^{4} + 33860 p T^{5} + 175953059 T^{6} + 200136160 T^{7} + 175953059 p^{2} T^{8} + 33860 p^{5} T^{9} + 434431 p^{6} T^{10} - 1848 p^{8} T^{11} + 701 p^{10} T^{12} - 4 p^{12} T^{13} + p^{14} T^{14} )^{2} \) | |
| 29 | \( 1 - 6592 T^{2} + 20819652 T^{4} - 41592620218 T^{6} + 59120779259070 T^{8} - 64850476486591160 T^{10} + 59847078622390648877 T^{12} - \)\(51\!\cdots\!68\)\( T^{14} + 59847078622390648877 p^{4} T^{16} - 64850476486591160 p^{8} T^{18} + 59120779259070 p^{12} T^{20} - 41592620218 p^{16} T^{22} + 20819652 p^{20} T^{24} - 6592 p^{24} T^{26} + p^{28} T^{28} \) | |
| 31 | \( ( 1 + 72 T + 7184 T^{2} + 359842 T^{3} + 20994182 T^{4} + 800853016 T^{5} + 33681974265 T^{6} + 1000486816476 T^{7} + 33681974265 p^{2} T^{8} + 800853016 p^{4} T^{9} + 20994182 p^{6} T^{10} + 359842 p^{8} T^{11} + 7184 p^{10} T^{12} + 72 p^{12} T^{13} + p^{14} T^{14} )^{2} \) | |
| 37 | \( ( 1 - 24 T + 4911 T^{2} - 102720 T^{3} + 13669849 T^{4} - 264405272 T^{5} + 26182389687 T^{6} - 430953976896 T^{7} + 26182389687 p^{2} T^{8} - 264405272 p^{4} T^{9} + 13669849 p^{6} T^{10} - 102720 p^{8} T^{11} + 4911 p^{10} T^{12} - 24 p^{12} T^{13} + p^{14} T^{14} )^{2} \) | |
| 41 | \( 1 - 7848 T^{2} + 34514020 T^{4} - 115131253090 T^{6} + 312429195575406 T^{8} - 17573526246954320 p T^{10} + \)\(14\!\cdots\!85\)\( T^{12} - \)\(25\!\cdots\!36\)\( T^{14} + \)\(14\!\cdots\!85\)\( p^{4} T^{16} - 17573526246954320 p^{9} T^{18} + 312429195575406 p^{12} T^{20} - 115131253090 p^{16} T^{22} + 34514020 p^{20} T^{24} - 7848 p^{24} T^{26} + p^{28} T^{28} \) | |
| 43 | \( ( 1 + 24 T + 7633 T^{2} + 246272 T^{3} + 30265947 T^{4} + 1000329200 T^{5} + 80876557187 T^{6} + 2327556833808 T^{7} + 80876557187 p^{2} T^{8} + 1000329200 p^{4} T^{9} + 30265947 p^{6} T^{10} + 246272 p^{8} T^{11} + 7633 p^{10} T^{12} + 24 p^{12} T^{13} + p^{14} T^{14} )^{2} \) | |
| 47 | \( 1 - 15024 T^{2} + 116349468 T^{4} - 624405979762 T^{6} + 2589836385989446 T^{8} - 185913905884648248 p T^{10} + \)\(24\!\cdots\!17\)\( T^{12} - \)\(59\!\cdots\!64\)\( T^{14} + \)\(24\!\cdots\!17\)\( p^{4} T^{16} - 185913905884648248 p^{9} T^{18} + 2589836385989446 p^{12} T^{20} - 624405979762 p^{16} T^{22} + 116349468 p^{20} T^{24} - 15024 p^{24} T^{26} + p^{28} T^{28} \) | |
| 53 | \( 1 - 19422 T^{2} + 170207959 T^{4} - 864481199364 T^{6} + 2677199069145685 T^{8} - 4421554998703903426 T^{10} - \)\(51\!\cdots\!01\)\( T^{12} + \)\(18\!\cdots\!40\)\( T^{14} - \)\(51\!\cdots\!01\)\( p^{4} T^{16} - 4421554998703903426 p^{8} T^{18} + 2677199069145685 p^{12} T^{20} - 864481199364 p^{16} T^{22} + 170207959 p^{20} T^{24} - 19422 p^{24} T^{26} + p^{28} T^{28} \) | |
| 59 | \( 1 - 21542 T^{2} + 247178683 T^{4} - 1980054800572 T^{6} + 12380069815275545 T^{8} - 63833748248890309498 T^{10} + \)\(27\!\cdots\!75\)\( T^{12} - \)\(10\!\cdots\!32\)\( T^{14} + \)\(27\!\cdots\!75\)\( p^{4} T^{16} - 63833748248890309498 p^{8} T^{18} + 12380069815275545 p^{12} T^{20} - 1980054800572 p^{16} T^{22} + 247178683 p^{20} T^{24} - 21542 p^{24} T^{26} + p^{28} T^{28} \) | |
| 61 | \( ( 1 + 70 T + 11803 T^{2} + 751044 T^{3} + 84847797 T^{4} + 3971650370 T^{5} + 384968019327 T^{6} + 280671100488 p T^{7} + 384968019327 p^{2} T^{8} + 3971650370 p^{4} T^{9} + 84847797 p^{6} T^{10} + 751044 p^{8} T^{11} + 11803 p^{10} T^{12} + 70 p^{12} T^{13} + p^{14} T^{14} )^{2} \) | |
| 67 | \( ( 1 - 102 T + 24525 T^{2} - 2396736 T^{3} + 287545471 T^{4} - 24714457802 T^{5} + 2018020701027 T^{6} - 143920041601536 T^{7} + 2018020701027 p^{2} T^{8} - 24714457802 p^{4} T^{9} + 287545471 p^{6} T^{10} - 2396736 p^{8} T^{11} + 24525 p^{10} T^{12} - 102 p^{12} T^{13} + p^{14} T^{14} )^{2} \) | |
| 71 | \( 1 - 43232 T^{2} + 833718804 T^{4} - 9212823268858 T^{6} + 60714264448795598 T^{8} - \)\(20\!\cdots\!48\)\( T^{10} - \)\(15\!\cdots\!99\)\( T^{12} + \)\(39\!\cdots\!76\)\( T^{14} - \)\(15\!\cdots\!99\)\( p^{4} T^{16} - \)\(20\!\cdots\!48\)\( p^{8} T^{18} + 60714264448795598 p^{12} T^{20} - 9212823268858 p^{16} T^{22} + 833718804 p^{20} T^{24} - 43232 p^{24} T^{26} + p^{28} T^{28} \) | |
| 73 | \( ( 1 + 112 T + 24760 T^{2} + 1235278 T^{3} + 155151610 T^{4} - 3024988408 T^{5} + 78112816781 T^{6} - 67579683772588 T^{7} + 78112816781 p^{2} T^{8} - 3024988408 p^{4} T^{9} + 155151610 p^{6} T^{10} + 1235278 p^{8} T^{11} + 24760 p^{10} T^{12} + 112 p^{12} T^{13} + p^{14} T^{14} )^{2} \) | |
| 79 | \( ( 1 + 172 T + 41221 T^{2} + 4643840 T^{3} + 655328367 T^{4} + 56066202076 T^{5} + 6016461228491 T^{6} + 422292224401008 T^{7} + 6016461228491 p^{2} T^{8} + 56066202076 p^{4} T^{9} + 655328367 p^{6} T^{10} + 4643840 p^{8} T^{11} + 41221 p^{10} T^{12} + 172 p^{12} T^{13} + p^{14} T^{14} )^{2} \) | |
| 83 | \( 1 - 69226 T^{2} + 2314079635 T^{4} - 49673157621812 T^{6} + 768438177989251713 T^{8} - \)\(91\!\cdots\!46\)\( T^{10} + \)\(85\!\cdots\!47\)\( T^{12} - \)\(65\!\cdots\!28\)\( T^{14} + \)\(85\!\cdots\!47\)\( p^{4} T^{16} - \)\(91\!\cdots\!46\)\( p^{8} T^{18} + 768438177989251713 p^{12} T^{20} - 49673157621812 p^{16} T^{22} + 2314079635 p^{20} T^{24} - 69226 p^{24} T^{26} + p^{28} T^{28} \) | |
| 89 | \( 1 - 50966 T^{2} + 1511448343 T^{4} - 31249433353332 T^{6} + 497309084376747557 T^{8} - \)\(63\!\cdots\!26\)\( T^{10} + \)\(66\!\cdots\!55\)\( T^{12} - \)\(57\!\cdots\!44\)\( T^{14} + \)\(66\!\cdots\!55\)\( p^{4} T^{16} - \)\(63\!\cdots\!26\)\( p^{8} T^{18} + 497309084376747557 p^{12} T^{20} - 31249433353332 p^{16} T^{22} + 1511448343 p^{20} T^{24} - 50966 p^{24} T^{26} + p^{28} T^{28} \) | |
| 97 | \( ( 1 + 12 T + 40623 T^{2} + 863184 T^{3} + 811228941 T^{4} + 22011788564 T^{5} + 10550220835011 T^{6} + 281519321178144 T^{7} + 10550220835011 p^{2} T^{8} + 22011788564 p^{4} T^{9} + 811228941 p^{6} T^{10} + 863184 p^{8} T^{11} + 40623 p^{10} T^{12} + 12 p^{12} T^{13} + p^{14} T^{14} )^{2} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{28} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.83177356880290715636907029697, −4.74946863839051563646004565652, −4.52957917663968344499956127758, −4.51759954463942933635290356333, −4.39028654791890244680864344198, −4.35468980641529923293689222758, −4.07887140836595090161424937969, −3.59239744971554407590353622452, −3.43882822669262336197125282354, −3.43334603817897659111892600415, −3.40695198071030186814313196283, −3.15931393937891969078153888837, −3.11783679466742362282451583378, −3.06799659296372391278382621368, −2.82978296591482075591427947792, −2.82320886301845776404984718018, −2.64128097419712183634720494822, −2.21303242629458173859764704959, −2.16489426123888751896543122692, −1.84070447263795107187432695738, −1.66748491692940475399602463307, −1.63464243621635988174400551265, −1.26518399156166327293133845770, −1.06200692506277104737476807478, −0.40903665836117021261856123089, 0.40903665836117021261856123089, 1.06200692506277104737476807478, 1.26518399156166327293133845770, 1.63464243621635988174400551265, 1.66748491692940475399602463307, 1.84070447263795107187432695738, 2.16489426123888751896543122692, 2.21303242629458173859764704959, 2.64128097419712183634720494822, 2.82320886301845776404984718018, 2.82978296591482075591427947792, 3.06799659296372391278382621368, 3.11783679466742362282451583378, 3.15931393937891969078153888837, 3.40695198071030186814313196283, 3.43334603817897659111892600415, 3.43882822669262336197125282354, 3.59239744971554407590353622452, 4.07887140836595090161424937969, 4.35468980641529923293689222758, 4.39028654791890244680864344198, 4.51759954463942933635290356333, 4.52957917663968344499956127758, 4.74946863839051563646004565652, 4.83177356880290715636907029697
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.