Dirichlet series
| L(s) = 1 | + 14·4-s + 20·7-s + 45·9-s − 4·11-s + 79·16-s − 22·17-s + 39·19-s − 12·23-s + 280·28-s + 630·36-s − 90·43-s − 56·44-s − 148·47-s − 26·49-s + 222·61-s + 900·63-s + 179·64-s − 308·68-s + 228·73-s + 546·76-s − 80·77-s + 1.03e3·81-s + 280·83-s − 168·92-s − 180·99-s − 348·101-s + 1.58e3·112-s + ⋯ |
| L(s) = 1 | + 7/2·4-s + 20/7·7-s + 5·9-s − 0.363·11-s + 4.93·16-s − 1.29·17-s + 2.05·19-s − 0.521·23-s + 10·28-s + 35/2·36-s − 2.09·43-s − 1.27·44-s − 3.14·47-s − 0.530·49-s + 3.63·61-s + 14.2·63-s + 2.79·64-s − 4.52·68-s + 3.12·73-s + 7.18·76-s − 1.03·77-s + 12.7·81-s + 3.37·83-s − 1.82·92-s − 1.81·99-s − 3.44·101-s + 14.1·112-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{28} \cdot 19^{14}\right)^{s/2} \, \Gamma_{\C}(s)^{14} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{28} \cdot 19^{14}\right)^{s/2} \, \Gamma_{\C}(s+1)^{14} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(28\) |
| Conductor: | \(5^{28} \cdot 19^{14}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.70170\times 10^{15}\) |
| Root analytic conductor: | \(3.59761\) |
| Motivic weight: | \(2\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((28,\ 5^{28} \cdot 19^{14} ,\ ( \ : [1]^{14} ),\ 1 )\) |
Particular Values
| \(L(\frac{3}{2})\) | \(\approx\) | \(30.84570870\) |
| \(L(\frac12)\) | \(\approx\) | \(30.84570870\) |
| \(L(2)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 5 | \( 1 \) |
| 19 | \( 1 - 39 T + 1269 T^{2} - 22218 T^{3} + 375147 T^{4} - 354723 p T^{5} + 436423 p^{2} T^{6} - 511268 p^{3} T^{7} + 436423 p^{4} T^{8} - 354723 p^{5} T^{9} + 375147 p^{6} T^{10} - 22218 p^{8} T^{11} + 1269 p^{10} T^{12} - 39 p^{12} T^{13} + p^{14} T^{14} \) | |
| good | 2 | \( 1 - 7 p T^{2} + 117 T^{4} - 711 T^{6} + 3445 T^{8} - 6903 p T^{10} + 50037 T^{12} - 189453 T^{14} + 50037 p^{4} T^{16} - 6903 p^{9} T^{18} + 3445 p^{12} T^{20} - 711 p^{16} T^{22} + 117 p^{20} T^{24} - 7 p^{25} T^{26} + p^{28} T^{28} \) |
| 3 | \( 1 - 5 p^{2} T^{2} + 331 p T^{4} - 14122 T^{6} + 148096 T^{8} - 1306930 T^{10} + 11137675 T^{12} - 10930229 p^{2} T^{14} + 11137675 p^{4} T^{16} - 1306930 p^{8} T^{18} + 148096 p^{12} T^{20} - 14122 p^{16} T^{22} + 331 p^{21} T^{24} - 5 p^{26} T^{26} + p^{28} T^{28} \) | |
| 7 | \( ( 1 - 10 T + 163 T^{2} - 1664 T^{3} + 19326 T^{4} - 143254 T^{5} + 1324275 T^{6} - 183956 p^{2} T^{7} + 1324275 p^{2} T^{8} - 143254 p^{4} T^{9} + 19326 p^{6} T^{10} - 1664 p^{8} T^{11} + 163 p^{10} T^{12} - 10 p^{12} T^{13} + p^{14} T^{14} )^{2} \) | |
| 11 | \( ( 1 + 2 T + 456 T^{2} + 1097 T^{3} + 102859 T^{4} + 227398 T^{5} + 15626384 T^{6} + 31988126 T^{7} + 15626384 p^{2} T^{8} + 227398 p^{4} T^{9} + 102859 p^{6} T^{10} + 1097 p^{8} T^{11} + 456 p^{10} T^{12} + 2 p^{12} T^{13} + p^{14} T^{14} )^{2} \) | |
| 13 | \( 1 - 1115 T^{2} + 628418 T^{4} - 239269917 T^{6} + 69444783231 T^{8} - 16467546282910 T^{10} + 3344471650732775 T^{12} - 598940529705960586 T^{14} + 3344471650732775 p^{4} T^{16} - 16467546282910 p^{8} T^{18} + 69444783231 p^{12} T^{20} - 239269917 p^{16} T^{22} + 628418 p^{20} T^{24} - 1115 p^{24} T^{26} + p^{28} T^{28} \) | |
| 17 | \( ( 1 + 11 T + 955 T^{2} + 15526 T^{3} + 32348 p T^{4} + 8508652 T^{5} + 235017005 T^{6} + 2872256407 T^{7} + 235017005 p^{2} T^{8} + 8508652 p^{4} T^{9} + 32348 p^{7} T^{10} + 15526 p^{8} T^{11} + 955 p^{10} T^{12} + 11 p^{12} T^{13} + p^{14} T^{14} )^{2} \) | |
| 23 | \( ( 1 + 6 T + 1745 T^{2} + 4820 T^{3} + 1496236 T^{4} + 2793256 T^{5} + 949275905 T^{6} + 1988169526 T^{7} + 949275905 p^{2} T^{8} + 2793256 p^{4} T^{9} + 1496236 p^{6} T^{10} + 4820 p^{8} T^{11} + 1745 p^{10} T^{12} + 6 p^{12} T^{13} + p^{14} T^{14} )^{2} \) | |
| 29 | \( 1 - 4641 T^{2} + 13061348 T^{4} - 26156988711 T^{6} + 41307448414601 T^{8} - 53173274890523642 T^{10} + 57389145367195002147 T^{12} - \)\(52\!\cdots\!46\)\( T^{14} + 57389145367195002147 p^{4} T^{16} - 53173274890523642 p^{8} T^{18} + 41307448414601 p^{12} T^{20} - 26156988711 p^{16} T^{22} + 13061348 p^{20} T^{24} - 4641 p^{24} T^{26} + p^{28} T^{28} \) | |
| 31 | \( 1 - 5661 T^{2} + 14872243 T^{4} - 24995360921 T^{6} + 32496929351181 T^{8} - 38374459458546367 T^{10} + 43705600440104911487 T^{12} - \)\(45\!\cdots\!86\)\( T^{14} + 43705600440104911487 p^{4} T^{16} - 38374459458546367 p^{8} T^{18} + 32496929351181 p^{12} T^{20} - 24995360921 p^{16} T^{22} + 14872243 p^{20} T^{24} - 5661 p^{24} T^{26} + p^{28} T^{28} \) | |
| 37 | \( 1 - 9084 T^{2} + 46132233 T^{4} - 162073113579 T^{6} + 436108896713531 T^{8} - 941102729673065551 T^{10} + \)\(16\!\cdots\!00\)\( T^{12} - \)\(25\!\cdots\!22\)\( T^{14} + \)\(16\!\cdots\!00\)\( p^{4} T^{16} - 941102729673065551 p^{8} T^{18} + 436108896713531 p^{12} T^{20} - 162073113579 p^{16} T^{22} + 46132233 p^{20} T^{24} - 9084 p^{24} T^{26} + p^{28} T^{28} \) | |
| 41 | \( 1 - 12508 T^{2} + 77404416 T^{4} - 313868900873 T^{6} + 937770422551181 T^{8} - 2222986188107558762 T^{10} + \)\(10\!\cdots\!86\)\( p T^{12} - \)\(78\!\cdots\!82\)\( T^{14} + \)\(10\!\cdots\!86\)\( p^{5} T^{16} - 2222986188107558762 p^{8} T^{18} + 937770422551181 p^{12} T^{20} - 313868900873 p^{16} T^{22} + 77404416 p^{20} T^{24} - 12508 p^{24} T^{26} + p^{28} T^{28} \) | |
| 43 | \( ( 1 + 45 T + 5298 T^{2} + 36709 T^{3} + 9892696 T^{4} - 2834341 T^{5} + 34324700565 T^{6} + 496161099574 T^{7} + 34324700565 p^{2} T^{8} - 2834341 p^{4} T^{9} + 9892696 p^{6} T^{10} + 36709 p^{8} T^{11} + 5298 p^{10} T^{12} + 45 p^{12} T^{13} + p^{14} T^{14} )^{2} \) | |
| 47 | \( ( 1 + 74 T + 201 p T^{2} + 11451 p T^{3} + 45459121 T^{4} + 2217367221 T^{5} + 141969399010 T^{6} + 5798967082006 T^{7} + 141969399010 p^{2} T^{8} + 2217367221 p^{4} T^{9} + 45459121 p^{6} T^{10} + 11451 p^{9} T^{11} + 201 p^{11} T^{12} + 74 p^{12} T^{13} + p^{14} T^{14} )^{2} \) | |
| 53 | \( 1 - 25045 T^{2} + 314849573 T^{4} - 2623153398987 T^{6} + 16117486022333931 T^{8} - 77038491111561956055 T^{10} + \)\(29\!\cdots\!80\)\( T^{12} - \)\(91\!\cdots\!16\)\( T^{14} + \)\(29\!\cdots\!80\)\( p^{4} T^{16} - 77038491111561956055 p^{8} T^{18} + 16117486022333931 p^{12} T^{20} - 2623153398987 p^{16} T^{22} + 314849573 p^{20} T^{24} - 25045 p^{24} T^{26} + p^{28} T^{28} \) | |
| 59 | \( 1 - 39080 T^{2} + 735930580 T^{4} - 8835219377045 T^{6} + 75452181799294346 T^{8} - \)\(48\!\cdots\!52\)\( T^{10} + \)\(24\!\cdots\!25\)\( T^{12} - \)\(94\!\cdots\!10\)\( T^{14} + \)\(24\!\cdots\!25\)\( p^{4} T^{16} - \)\(48\!\cdots\!52\)\( p^{8} T^{18} + 75452181799294346 p^{12} T^{20} - 8835219377045 p^{16} T^{22} + 735930580 p^{20} T^{24} - 39080 p^{24} T^{26} + p^{28} T^{28} \) | |
| 61 | \( ( 1 - 111 T + 20753 T^{2} - 1652101 T^{3} + 179478171 T^{4} - 11573155689 T^{5} + 951563310235 T^{6} - 51936749212358 T^{7} + 951563310235 p^{2} T^{8} - 11573155689 p^{4} T^{9} + 179478171 p^{6} T^{10} - 1652101 p^{8} T^{11} + 20753 p^{10} T^{12} - 111 p^{12} T^{13} + p^{14} T^{14} )^{2} \) | |
| 67 | \( 1 - 32722 T^{2} + 518919431 T^{4} - 5469674112052 T^{6} + 657687540154637 p T^{8} - \)\(29\!\cdots\!18\)\( T^{10} + \)\(16\!\cdots\!79\)\( T^{12} - \)\(79\!\cdots\!81\)\( T^{14} + \)\(16\!\cdots\!79\)\( p^{4} T^{16} - \)\(29\!\cdots\!18\)\( p^{8} T^{18} + 657687540154637 p^{13} T^{20} - 5469674112052 p^{16} T^{22} + 518919431 p^{20} T^{24} - 32722 p^{24} T^{26} + p^{28} T^{28} \) | |
| 71 | \( 1 - 18519 T^{2} + 209621958 T^{4} - 1992490776979 T^{6} + 15940635661328336 T^{8} - \)\(10\!\cdots\!77\)\( T^{10} + \)\(65\!\cdots\!53\)\( T^{12} - \)\(35\!\cdots\!66\)\( T^{14} + \)\(65\!\cdots\!53\)\( p^{4} T^{16} - \)\(10\!\cdots\!77\)\( p^{8} T^{18} + 15940635661328336 p^{12} T^{20} - 1992490776979 p^{16} T^{22} + 209621958 p^{20} T^{24} - 18519 p^{24} T^{26} + p^{28} T^{28} \) | |
| 73 | \( ( 1 - 114 T + 15355 T^{2} - 1094820 T^{3} + 74132171 T^{4} - 2824218054 T^{5} + 166440461775 T^{6} - 1728417774559 T^{7} + 166440461775 p^{2} T^{8} - 2824218054 p^{4} T^{9} + 74132171 p^{6} T^{10} - 1094820 p^{8} T^{11} + 15355 p^{10} T^{12} - 114 p^{12} T^{13} + p^{14} T^{14} )^{2} \) | |
| 79 | \( 1 - 25994 T^{2} + 462190554 T^{4} - 5857864185249 T^{6} + 61938632030589436 T^{8} - \)\(54\!\cdots\!02\)\( T^{10} + \)\(42\!\cdots\!73\)\( T^{12} - \)\(27\!\cdots\!58\)\( T^{14} + \)\(42\!\cdots\!73\)\( p^{4} T^{16} - \)\(54\!\cdots\!02\)\( p^{8} T^{18} + 61938632030589436 p^{12} T^{20} - 5857864185249 p^{16} T^{22} + 462190554 p^{20} T^{24} - 25994 p^{24} T^{26} + p^{28} T^{28} \) | |
| 83 | \( ( 1 - 140 T + 28764 T^{2} - 2377647 T^{3} + 303590661 T^{4} - 15487490172 T^{5} + 1820521880750 T^{6} - 73438861276762 T^{7} + 1820521880750 p^{2} T^{8} - 15487490172 p^{4} T^{9} + 303590661 p^{6} T^{10} - 2377647 p^{8} T^{11} + 28764 p^{10} T^{12} - 140 p^{12} T^{13} + p^{14} T^{14} )^{2} \) | |
| 89 | \( 1 - 47527 T^{2} + 979525420 T^{4} - 11880881724252 T^{6} + 97816510727876161 T^{8} - \)\(54\!\cdots\!07\)\( T^{10} + \)\(13\!\cdots\!74\)\( T^{12} + \)\(17\!\cdots\!60\)\( T^{14} + \)\(13\!\cdots\!74\)\( p^{4} T^{16} - \)\(54\!\cdots\!07\)\( p^{8} T^{18} + 97816510727876161 p^{12} T^{20} - 11880881724252 p^{16} T^{22} + 979525420 p^{20} T^{24} - 47527 p^{24} T^{26} + p^{28} T^{28} \) | |
| 97 | \( 1 - 90269 T^{2} + 3842554255 T^{4} - 102884451130745 T^{6} + 1958666581531251273 T^{8} - \)\(28\!\cdots\!51\)\( T^{10} + \)\(34\!\cdots\!31\)\( T^{12} - \)\(34\!\cdots\!10\)\( T^{14} + \)\(34\!\cdots\!31\)\( p^{4} T^{16} - \)\(28\!\cdots\!51\)\( p^{8} T^{18} + 1958666581531251273 p^{12} T^{20} - 102884451130745 p^{16} T^{22} + 3842554255 p^{20} T^{24} - 90269 p^{24} T^{26} + p^{28} T^{28} \) | |
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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
−3.07712384817446194930492830451, −2.81148432091632335266116971643, −2.75127860551323293618825632556, −2.60053642185780321708299708485, −2.49710897983064565252455433320, −2.37233387635282767297902704370, −2.34803982924418696537807792395, −2.29440796573169627379128848843, −2.24575065261220222288493668385, −2.06859536402877141309207289369, −1.90433120272541963114814880198, −1.88959513983101926547257399768, −1.83723809005254032651728135305, −1.56248802651072059072692550163, −1.53412073728380711841352642248, −1.52732556949284498795146647539, −1.51702476068134777396681420290, −1.23073414611091768667503507799, −1.17525717350210098615795233537, −1.14228508347233088482074425667, −1.09688405466874915297776229672, −0.69133985938167676438258758575, −0.63900820182564887457083661737, −0.25914113419313076584262371059, −0.10599902770801514052725432942, 0.10599902770801514052725432942, 0.25914113419313076584262371059, 0.63900820182564887457083661737, 0.69133985938167676438258758575, 1.09688405466874915297776229672, 1.14228508347233088482074425667, 1.17525717350210098615795233537, 1.23073414611091768667503507799, 1.51702476068134777396681420290, 1.52732556949284498795146647539, 1.53412073728380711841352642248, 1.56248802651072059072692550163, 1.83723809005254032651728135305, 1.88959513983101926547257399768, 1.90433120272541963114814880198, 2.06859536402877141309207289369, 2.24575065261220222288493668385, 2.29440796573169627379128848843, 2.34803982924418696537807792395, 2.37233387635282767297902704370, 2.49710897983064565252455433320, 2.60053642185780321708299708485, 2.75127860551323293618825632556, 2.81148432091632335266116971643, 3.07712384817446194930492830451
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.