Dirichlet series
| L(s) = 1 | + 3·4-s − 2·5-s + 8·11-s + 3·16-s − 14·19-s − 6·20-s − 12·29-s + 8·31-s − 4·41-s + 24·44-s + 32·49-s − 16·55-s − 36·59-s + 24·61-s + 9·64-s + 36·71-s − 42·76-s + 8·79-s − 6·80-s − 16·89-s + 28·95-s + 80·101-s − 12·109-s − 36·116-s − 20·121-s + 24·124-s + 16·125-s + ⋯ |
| L(s) = 1 | + 3/2·4-s − 0.894·5-s + 2.41·11-s + 3/4·16-s − 3.21·19-s − 1.34·20-s − 2.22·29-s + 1.43·31-s − 0.624·41-s + 3.61·44-s + 32/7·49-s − 2.15·55-s − 4.68·59-s + 3.07·61-s + 9/8·64-s + 4.27·71-s − 4.81·76-s + 0.900·79-s − 0.670·80-s − 1.69·89-s + 2.87·95-s + 7.96·101-s − 1.14·109-s − 3.34·116-s − 1.81·121-s + 2.15·124-s + 1.43·125-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{28} \cdot 5^{14} \cdot 19^{14}\right)^{s/2} \, \Gamma_{\C}(s)^{14} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{28} \cdot 5^{14} \cdot 19^{14}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{14} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(28\) |
| Conductor: | \(3^{28} \cdot 5^{14} \cdot 19^{14}\) |
| Sign: | $1$ |
| Analytic conductor: | \(4.77974\times 10^{11}\) |
| Root analytic conductor: | \(2.61289\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((28,\ 3^{28} \cdot 5^{14} \cdot 19^{14} ,\ ( \ : [1/2]^{14} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(11.38602745\) |
| \(L(\frac12)\) | \(\approx\) | \(11.38602745\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + 2 T + 4 T^{2} - 8 T^{3} - 22 T^{4} - 82 T^{5} + 9 p T^{6} + 16 T^{7} + 9 p^{2} T^{8} - 82 p^{2} T^{9} - 22 p^{3} T^{10} - 8 p^{4} T^{11} + 4 p^{5} T^{12} + 2 p^{6} T^{13} + p^{7} T^{14} \) | |
| 19 | \( ( 1 + T )^{14} \) | |
| good | 2 | \( 1 - 3 T^{2} + 3 p T^{4} - 9 p T^{6} + 37 T^{8} - 91 T^{10} + 43 p^{2} T^{12} - 27 p^{3} T^{14} + 43 p^{4} T^{16} - 91 p^{4} T^{18} + 37 p^{6} T^{20} - 9 p^{9} T^{22} + 3 p^{11} T^{24} - 3 p^{12} T^{26} + p^{14} T^{28} \) |
| 7 | \( 1 - 32 T^{2} + 600 T^{4} - 8262 T^{6} + 92558 T^{8} - 18076 p^{2} T^{10} + 7422489 T^{12} - 55166684 T^{14} + 7422489 p^{2} T^{16} - 18076 p^{6} T^{18} + 92558 p^{6} T^{20} - 8262 p^{8} T^{22} + 600 p^{10} T^{24} - 32 p^{12} T^{26} + p^{14} T^{28} \) | |
| 11 | \( ( 1 - 4 T + 34 T^{2} - 152 T^{3} + 856 T^{4} - 2878 T^{5} + 12979 T^{6} - 40412 T^{7} + 12979 p T^{8} - 2878 p^{2} T^{9} + 856 p^{3} T^{10} - 152 p^{4} T^{11} + 34 p^{5} T^{12} - 4 p^{6} T^{13} + p^{7} T^{14} )^{2} \) | |
| 13 | \( 1 - 74 T^{2} + 227 p T^{4} - 80168 T^{6} + 1648733 T^{8} - 27556262 T^{10} + 30729303 p T^{12} - 5339524752 T^{14} + 30729303 p^{3} T^{16} - 27556262 p^{4} T^{18} + 1648733 p^{6} T^{20} - 80168 p^{8} T^{22} + 227 p^{11} T^{24} - 74 p^{12} T^{26} + p^{14} T^{28} \) | |
| 17 | \( 1 - 56 T^{2} + 2324 T^{4} - 65330 T^{6} + 1671582 T^{8} - 36003056 T^{10} + 43706141 p T^{12} - 13081258956 T^{14} + 43706141 p^{3} T^{16} - 36003056 p^{4} T^{18} + 1671582 p^{6} T^{20} - 65330 p^{8} T^{22} + 2324 p^{10} T^{24} - 56 p^{12} T^{26} + p^{14} T^{28} \) | |
| 23 | \( 1 - 126 T^{2} + 8651 T^{4} - 427068 T^{6} + 16797001 T^{8} - 551780834 T^{10} + 15575022795 T^{12} - 383045399624 T^{14} + 15575022795 p^{2} T^{16} - 551780834 p^{4} T^{18} + 16797001 p^{6} T^{20} - 427068 p^{8} T^{22} + 8651 p^{10} T^{24} - 126 p^{12} T^{26} + p^{14} T^{28} \) | |
| 29 | \( ( 1 + 6 T + 115 T^{2} + 702 T^{3} + 7541 T^{4} + 39778 T^{5} + 316975 T^{6} + 1439028 T^{7} + 316975 p T^{8} + 39778 p^{2} T^{9} + 7541 p^{3} T^{10} + 702 p^{4} T^{11} + 115 p^{5} T^{12} + 6 p^{6} T^{13} + p^{7} T^{14} )^{2} \) | |
| 31 | \( ( 1 - 4 T + 133 T^{2} - 668 T^{3} + 9417 T^{4} - 44348 T^{5} + 439237 T^{6} - 1717576 T^{7} + 439237 p T^{8} - 44348 p^{2} T^{9} + 9417 p^{3} T^{10} - 668 p^{4} T^{11} + 133 p^{5} T^{12} - 4 p^{6} T^{13} + p^{7} T^{14} )^{2} \) | |
| 37 | \( 1 - 238 T^{2} + 30175 T^{4} - 2665120 T^{6} + 182330181 T^{8} - 10180420098 T^{10} + 477219189835 T^{12} - 19074672062368 T^{14} + 477219189835 p^{2} T^{16} - 10180420098 p^{4} T^{18} + 182330181 p^{6} T^{20} - 2665120 p^{8} T^{22} + 30175 p^{10} T^{24} - 238 p^{12} T^{26} + p^{14} T^{28} \) | |
| 41 | \( ( 1 + 2 T + 207 T^{2} + 610 T^{3} + 19693 T^{4} + 68974 T^{5} + 1162547 T^{6} + 3880604 T^{7} + 1162547 p T^{8} + 68974 p^{2} T^{9} + 19693 p^{3} T^{10} + 610 p^{4} T^{11} + 207 p^{5} T^{12} + 2 p^{6} T^{13} + p^{7} T^{14} )^{2} \) | |
| 43 | \( 1 - 348 T^{2} + 60832 T^{4} - 7128154 T^{6} + 626871502 T^{8} - 43718855872 T^{10} + 2488671504137 T^{12} - 117324725836532 T^{14} + 2488671504137 p^{2} T^{16} - 43718855872 p^{4} T^{18} + 626871502 p^{6} T^{20} - 7128154 p^{8} T^{22} + 60832 p^{10} T^{24} - 348 p^{12} T^{26} + p^{14} T^{28} \) | |
| 47 | \( 1 - 8 p T^{2} + 74556 T^{4} - 10013738 T^{6} + 1005388406 T^{8} - 79230933744 T^{10} + 5032682210925 T^{12} - 261155791825244 T^{14} + 5032682210925 p^{2} T^{16} - 79230933744 p^{4} T^{18} + 1005388406 p^{6} T^{20} - 10013738 p^{8} T^{22} + 74556 p^{10} T^{24} - 8 p^{13} T^{26} + p^{14} T^{28} \) | |
| 53 | \( 1 - 614 T^{2} + 179811 T^{4} - 33328300 T^{6} + 4372089721 T^{8} - 429690483994 T^{10} + 32640522600115 T^{12} - 1947272706448104 T^{14} + 32640522600115 p^{2} T^{16} - 429690483994 p^{4} T^{18} + 4372089721 p^{6} T^{20} - 33328300 p^{8} T^{22} + 179811 p^{10} T^{24} - 614 p^{12} T^{26} + p^{14} T^{28} \) | |
| 59 | \( ( 1 + 18 T + 313 T^{2} + 3004 T^{3} + 28433 T^{4} + 157182 T^{5} + 1104449 T^{6} + 5068232 T^{7} + 1104449 p T^{8} + 157182 p^{2} T^{9} + 28433 p^{3} T^{10} + 3004 p^{4} T^{11} + 313 p^{5} T^{12} + 18 p^{6} T^{13} + p^{7} T^{14} )^{2} \) | |
| 61 | \( ( 1 - 12 T + 136 T^{2} - 1326 T^{3} + 16514 T^{4} - 156640 T^{5} + 1332393 T^{6} - 9048172 T^{7} + 1332393 p T^{8} - 156640 p^{2} T^{9} + 16514 p^{3} T^{10} - 1326 p^{4} T^{11} + 136 p^{5} T^{12} - 12 p^{6} T^{13} + p^{7} T^{14} )^{2} \) | |
| 67 | \( 1 - 478 T^{2} + 117043 T^{4} - 19527020 T^{6} + 2479606329 T^{8} - 253914050978 T^{10} + 21679405397251 T^{12} - 1571653455427048 T^{14} + 21679405397251 p^{2} T^{16} - 253914050978 p^{4} T^{18} + 2479606329 p^{6} T^{20} - 19527020 p^{8} T^{22} + 117043 p^{10} T^{24} - 478 p^{12} T^{26} + p^{14} T^{28} \) | |
| 71 | \( ( 1 - 18 T + 429 T^{2} - 4556 T^{3} + 62585 T^{4} - 456014 T^{5} + 5090621 T^{6} - 31694952 T^{7} + 5090621 p T^{8} - 456014 p^{2} T^{9} + 62585 p^{3} T^{10} - 4556 p^{4} T^{11} + 429 p^{5} T^{12} - 18 p^{6} T^{13} + p^{7} T^{14} )^{2} \) | |
| 73 | \( 1 - 668 T^{2} + 221508 T^{4} - 660902 p T^{6} + 7710129326 T^{8} - 956480133164 T^{10} + 94989347045565 T^{12} - 7671125278066244 T^{14} + 94989347045565 p^{2} T^{16} - 956480133164 p^{4} T^{18} + 7710129326 p^{6} T^{20} - 660902 p^{9} T^{22} + 221508 p^{10} T^{24} - 668 p^{12} T^{26} + p^{14} T^{28} \) | |
| 79 | \( ( 1 - 4 T + 225 T^{2} - 600 T^{3} + 17885 T^{4} + 516 T^{5} + 443613 T^{6} + 3708336 T^{7} + 443613 p T^{8} + 516 p^{2} T^{9} + 17885 p^{3} T^{10} - 600 p^{4} T^{11} + 225 p^{5} T^{12} - 4 p^{6} T^{13} + p^{7} T^{14} )^{2} \) | |
| 83 | \( 1 - 670 T^{2} + 217891 T^{4} - 46074108 T^{6} + 7186165641 T^{8} - 890976408098 T^{10} + 92251473530755 T^{12} - 8215297964746568 T^{14} + 92251473530755 p^{2} T^{16} - 890976408098 p^{4} T^{18} + 7186165641 p^{6} T^{20} - 46074108 p^{8} T^{22} + 217891 p^{10} T^{24} - 670 p^{12} T^{26} + p^{14} T^{28} \) | |
| 89 | \( ( 1 + 8 T + 295 T^{2} + 4354 T^{3} + 54701 T^{4} + 726512 T^{5} + 8547675 T^{6} + 70970572 T^{7} + 8547675 p T^{8} + 726512 p^{2} T^{9} + 54701 p^{3} T^{10} + 4354 p^{4} T^{11} + 295 p^{5} T^{12} + 8 p^{6} T^{13} + p^{7} T^{14} )^{2} \) | |
| 97 | \( 1 - 922 T^{2} + 416447 T^{4} - 122496072 T^{6} + 26270420381 T^{8} - 4353416249334 T^{10} + 575826811363875 T^{12} - 61829382860447824 T^{14} + 575826811363875 p^{2} T^{16} - 4353416249334 p^{4} T^{18} + 26270420381 p^{6} T^{20} - 122496072 p^{8} T^{22} + 416447 p^{10} T^{24} - 922 p^{12} T^{26} + p^{14} T^{28} \) | |
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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
−2.82925767864963353732391708099, −2.82698299667760794009024950474, −2.66834768524640797283160961611, −2.65172072503438398001605964730, −2.61444743923774402578148143393, −2.55244067529809864458742099683, −2.31429359148562095952995007176, −2.19663119321528483743556905012, −2.17307084637925482729786469552, −2.14425250106591269976714674762, −2.10613620588282786341810743423, −1.81487289617263923547742101067, −1.79812051496429152689743486740, −1.79358863411132419659552274856, −1.78164422475892104607033217078, −1.62199580226324057033006035322, −1.31663649179184423456979966240, −1.31641431888425883124885170147, −1.08376812764315747731126865350, −0.965165061762396227855908382266, −0.867078431670072747726241439742, −0.70624675174005692316613364532, −0.55504117899014873961843373517, −0.41881261991320426040993513060, −0.20537953966229600357669917833, 0.20537953966229600357669917833, 0.41881261991320426040993513060, 0.55504117899014873961843373517, 0.70624675174005692316613364532, 0.867078431670072747726241439742, 0.965165061762396227855908382266, 1.08376812764315747731126865350, 1.31641431888425883124885170147, 1.31663649179184423456979966240, 1.62199580226324057033006035322, 1.78164422475892104607033217078, 1.79358863411132419659552274856, 1.79812051496429152689743486740, 1.81487289617263923547742101067, 2.10613620588282786341810743423, 2.14425250106591269976714674762, 2.17307084637925482729786469552, 2.19663119321528483743556905012, 2.31429359148562095952995007176, 2.55244067529809864458742099683, 2.61444743923774402578148143393, 2.65172072503438398001605964730, 2.66834768524640797283160961611, 2.82698299667760794009024950474, 2.82925767864963353732391708099
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.