Dirichlet series
| L(s) = 1 | − 7·4-s − 2·13-s + 28·16-s − 8·17-s − 8·23-s + 28·25-s − 16·29-s + 4·43-s + 44·49-s + 14·52-s + 60·53-s − 28·61-s − 84·64-s + 56·68-s − 28·79-s + 56·92-s − 196·100-s − 16·101-s + 28·103-s − 76·107-s + 26·113-s + 112·116-s + 70·121-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 7/2·4-s − 0.554·13-s + 7·16-s − 1.94·17-s − 1.66·23-s + 28/5·25-s − 2.97·29-s + 0.609·43-s + 44/7·49-s + 1.94·52-s + 8.24·53-s − 3.58·61-s − 10.5·64-s + 6.79·68-s − 3.15·79-s + 5.83·92-s − 19.5·100-s − 1.59·101-s + 2.75·103-s − 7.34·107-s + 2.44·113-s + 10.3·116-s + 6.36·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{14} \cdot 3^{56} \cdot 13^{14}\right)^{s/2} \, \Gamma_{\C}(s)^{14} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{14} \cdot 3^{56} \cdot 13^{14}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{14} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(28\) |
| Conductor: | \(2^{14} \cdot 3^{56} \cdot 13^{14}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.44642\times 10^{17}\) |
| Root analytic conductor: | \(4.10079\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((28,\ 2^{14} \cdot 3^{56} \cdot 13^{14} ,\ ( \ : [1/2]^{14} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(10.90022365\) |
| \(L(\frac12)\) | \(\approx\) | \(10.90022365\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + T^{2} )^{7} \) |
| 3 | \( 1 \) | |
| 13 | \( 1 + 2 T - 17 T^{2} - 76 T^{3} + 313 T^{4} + 134 p T^{5} - 1057 T^{6} - 26216 T^{7} - 1057 p T^{8} + 134 p^{3} T^{9} + 313 p^{3} T^{10} - 76 p^{4} T^{11} - 17 p^{5} T^{12} + 2 p^{6} T^{13} + p^{7} T^{14} \) | |
| good | 5 | \( 1 - 28 T^{2} + 412 T^{4} - 4393 T^{6} + 37402 T^{8} - 264616 T^{10} + 322597 p T^{12} - 8614226 T^{14} + 322597 p^{3} T^{16} - 264616 p^{4} T^{18} + 37402 p^{6} T^{20} - 4393 p^{8} T^{22} + 412 p^{10} T^{24} - 28 p^{12} T^{26} + p^{14} T^{28} \) |
| 7 | \( 1 - 44 T^{2} + 136 p T^{4} - 12993 T^{6} + 122454 T^{8} - 834024 T^{10} + 4488961 T^{12} - 25970006 T^{14} + 4488961 p^{2} T^{16} - 834024 p^{4} T^{18} + 122454 p^{6} T^{20} - 12993 p^{8} T^{22} + 136 p^{11} T^{24} - 44 p^{12} T^{26} + p^{14} T^{28} \) | |
| 11 | \( 1 - 70 T^{2} + 2641 T^{4} - 69764 T^{6} + 1430648 T^{8} - 23956004 T^{10} + 336263838 T^{12} - 4007214276 T^{14} + 336263838 p^{2} T^{16} - 23956004 p^{4} T^{18} + 1430648 p^{6} T^{20} - 69764 p^{8} T^{22} + 2641 p^{10} T^{24} - 70 p^{12} T^{26} + p^{14} T^{28} \) | |
| 17 | \( ( 1 + 4 T + 58 T^{2} + 191 T^{3} + 103 p T^{4} + 5470 T^{5} + 37227 T^{6} + 105949 T^{7} + 37227 p T^{8} + 5470 p^{2} T^{9} + 103 p^{4} T^{10} + 191 p^{4} T^{11} + 58 p^{5} T^{12} + 4 p^{6} T^{13} + p^{7} T^{14} )^{2} \) | |
| 19 | \( 1 - 182 T^{2} + 16417 T^{4} - 968836 T^{6} + 41679224 T^{8} - 1380175780 T^{10} + 36234765726 T^{12} - 765633284004 T^{14} + 36234765726 p^{2} T^{16} - 1380175780 p^{4} T^{18} + 41679224 p^{6} T^{20} - 968836 p^{8} T^{22} + 16417 p^{10} T^{24} - 182 p^{12} T^{26} + p^{14} T^{28} \) | |
| 23 | \( ( 1 + 4 T + 85 T^{2} + 134 T^{3} + 3023 T^{4} - 104 T^{5} + 76203 T^{6} - 54836 T^{7} + 76203 p T^{8} - 104 p^{2} T^{9} + 3023 p^{3} T^{10} + 134 p^{4} T^{11} + 85 p^{5} T^{12} + 4 p^{6} T^{13} + p^{7} T^{14} )^{2} \) | |
| 29 | \( ( 1 + 8 T + 109 T^{2} + 586 T^{3} + 4613 T^{4} + 14768 T^{5} + 106149 T^{6} + 262412 T^{7} + 106149 p T^{8} + 14768 p^{2} T^{9} + 4613 p^{3} T^{10} + 586 p^{4} T^{11} + 109 p^{5} T^{12} + 8 p^{6} T^{13} + p^{7} T^{14} )^{2} \) | |
| 31 | \( 1 - 263 T^{2} + 35674 T^{4} - 3244107 T^{6} + 218856612 T^{8} - 11519598837 T^{10} + 486151273849 T^{12} - 16679108009342 T^{14} + 486151273849 p^{2} T^{16} - 11519598837 p^{4} T^{18} + 218856612 p^{6} T^{20} - 3244107 p^{8} T^{22} + 35674 p^{10} T^{24} - 263 p^{12} T^{26} + p^{14} T^{28} \) | |
| 37 | \( 1 - 284 T^{2} + 37456 T^{4} - 3050493 T^{6} + 172969374 T^{8} - 7429592952 T^{10} + 268983793465 T^{12} - 9578405834366 T^{14} + 268983793465 p^{2} T^{16} - 7429592952 p^{4} T^{18} + 172969374 p^{6} T^{20} - 3050493 p^{8} T^{22} + 37456 p^{10} T^{24} - 284 p^{12} T^{26} + p^{14} T^{28} \) | |
| 41 | \( 1 - 214 T^{2} + 24433 T^{4} - 2043904 T^{6} + 138093520 T^{8} - 7879560508 T^{10} + 391124566286 T^{12} - 17082940877852 T^{14} + 391124566286 p^{2} T^{16} - 7879560508 p^{4} T^{18} + 138093520 p^{6} T^{20} - 2043904 p^{8} T^{22} + 24433 p^{10} T^{24} - 214 p^{12} T^{26} + p^{14} T^{28} \) | |
| 43 | \( ( 1 - 2 T + 94 T^{2} - 15 T^{3} + 7383 T^{4} - 8118 T^{5} + 400579 T^{6} - 177365 T^{7} + 400579 p T^{8} - 8118 p^{2} T^{9} + 7383 p^{3} T^{10} - 15 p^{4} T^{11} + 94 p^{5} T^{12} - 2 p^{6} T^{13} + p^{7} T^{14} )^{2} \) | |
| 47 | \( 1 - 460 T^{2} + 105448 T^{4} - 15801305 T^{6} + 1719474566 T^{8} - 143236278668 T^{10} + 9399341836689 T^{12} - 493014801863130 T^{14} + 9399341836689 p^{2} T^{16} - 143236278668 p^{4} T^{18} + 1719474566 p^{6} T^{20} - 15801305 p^{8} T^{22} + 105448 p^{10} T^{24} - 460 p^{12} T^{26} + p^{14} T^{28} \) | |
| 53 | \( ( 1 - 30 T + 557 T^{2} - 7506 T^{3} + 83961 T^{4} - 823554 T^{5} + 7185121 T^{6} - 55798572 T^{7} + 7185121 p T^{8} - 823554 p^{2} T^{9} + 83961 p^{3} T^{10} - 7506 p^{4} T^{11} + 557 p^{5} T^{12} - 30 p^{6} T^{13} + p^{7} T^{14} )^{2} \) | |
| 59 | \( 1 - 454 T^{2} + 107641 T^{4} - 17374636 T^{6} + 2114118088 T^{8} - 203852238220 T^{10} + 16004602382678 T^{12} - 1037150341753028 T^{14} + 16004602382678 p^{2} T^{16} - 203852238220 p^{4} T^{18} + 2114118088 p^{6} T^{20} - 17374636 p^{8} T^{22} + 107641 p^{10} T^{24} - 454 p^{12} T^{26} + p^{14} T^{28} \) | |
| 61 | \( ( 1 + 14 T + 325 T^{2} + 3402 T^{3} + 48333 T^{4} + 411918 T^{5} + 4473181 T^{6} + 31330004 T^{7} + 4473181 p T^{8} + 411918 p^{2} T^{9} + 48333 p^{3} T^{10} + 3402 p^{4} T^{11} + 325 p^{5} T^{12} + 14 p^{6} T^{13} + p^{7} T^{14} )^{2} \) | |
| 67 | \( 1 - 410 T^{2} + 89185 T^{4} - 13271892 T^{6} + 1513227864 T^{8} - 141429058884 T^{10} + 11345803420894 T^{12} - 12008375417492 p T^{14} + 11345803420894 p^{2} T^{16} - 141429058884 p^{4} T^{18} + 1513227864 p^{6} T^{20} - 13271892 p^{8} T^{22} + 89185 p^{10} T^{24} - 410 p^{12} T^{26} + p^{14} T^{28} \) | |
| 71 | \( 1 - 352 T^{2} + 58744 T^{4} - 6619505 T^{6} + 605142734 T^{8} - 47940177920 T^{10} + 3360968223801 T^{12} - 230643268785402 T^{14} + 3360968223801 p^{2} T^{16} - 47940177920 p^{4} T^{18} + 605142734 p^{6} T^{20} - 6619505 p^{8} T^{22} + 58744 p^{10} T^{24} - 352 p^{12} T^{26} + p^{14} T^{28} \) | |
| 73 | \( 1 - 686 T^{2} + 226609 T^{4} - 48066708 T^{6} + 7378595280 T^{8} - 877142110920 T^{10} + 1155068610382 p T^{12} - 6726936288556364 T^{14} + 1155068610382 p^{3} T^{16} - 877142110920 p^{4} T^{18} + 7378595280 p^{6} T^{20} - 48066708 p^{8} T^{22} + 226609 p^{10} T^{24} - 686 p^{12} T^{26} + p^{14} T^{28} \) | |
| 79 | \( ( 1 + 14 T + 409 T^{2} + 3662 T^{3} + 62815 T^{4} + 387590 T^{5} + 5721599 T^{6} + 29512780 T^{7} + 5721599 p T^{8} + 387590 p^{2} T^{9} + 62815 p^{3} T^{10} + 3662 p^{4} T^{11} + 409 p^{5} T^{12} + 14 p^{6} T^{13} + p^{7} T^{14} )^{2} \) | |
| 83 | \( 1 - 754 T^{2} + 268795 T^{4} - 60872116 T^{6} + 9964991185 T^{8} - 1274968363054 T^{10} + 134331539047331 T^{12} - 12029284235840408 T^{14} + 134331539047331 p^{2} T^{16} - 1274968363054 p^{4} T^{18} + 9964991185 p^{6} T^{20} - 60872116 p^{8} T^{22} + 268795 p^{10} T^{24} - 754 p^{12} T^{26} + p^{14} T^{28} \) | |
| 89 | \( 1 - 886 T^{2} + 387667 T^{4} - 110427116 T^{6} + 22794467681 T^{8} - 3598208391434 T^{10} + 447224808902379 T^{12} - 44422413444004776 T^{14} + 447224808902379 p^{2} T^{16} - 3598208391434 p^{4} T^{18} + 22794467681 p^{6} T^{20} - 110427116 p^{8} T^{22} + 387667 p^{10} T^{24} - 886 p^{12} T^{26} + p^{14} T^{28} \) | |
| 97 | \( 1 - 446 T^{2} + 112945 T^{4} - 21578872 T^{6} + 3350182448 T^{8} - 446829447508 T^{10} + 52197266729598 T^{12} - 5370626561858796 T^{14} + 52197266729598 p^{2} T^{16} - 446829447508 p^{4} T^{18} + 3350182448 p^{6} T^{20} - 21578872 p^{8} T^{22} + 112945 p^{10} T^{24} - 446 p^{12} T^{26} + p^{14} T^{28} \) | |
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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
−2.51381975290325287023175468866, −2.50755521670693431054037399866, −2.39683169415982810527888883454, −2.32742162902172972331686175217, −2.16933650951062821783418491182, −2.09131179743859861765268409900, −2.02455679453877819148868626329, −1.98982720990560698794380704990, −1.83021610647557888280265901496, −1.70679614174622875374804064353, −1.68604661611783210103668539087, −1.56249263616319379880253369980, −1.33231402080072245719313731567, −1.27607909397622443872378925540, −1.27323657062525533544181152644, −1.24577918011282488785839721481, −0.967611201171790896109092920500, −0.873635241660791296933792791131, −0.77911457653075616313504486792, −0.72273309653754651009159717455, −0.57454855781378304451277422801, −0.50800699784169712351824859870, −0.39177580629702724686844223348, −0.30230167203703979110949961289, −0.27891564831407636127431493596, 0.27891564831407636127431493596, 0.30230167203703979110949961289, 0.39177580629702724686844223348, 0.50800699784169712351824859870, 0.57454855781378304451277422801, 0.72273309653754651009159717455, 0.77911457653075616313504486792, 0.873635241660791296933792791131, 0.967611201171790896109092920500, 1.24577918011282488785839721481, 1.27323657062525533544181152644, 1.27607909397622443872378925540, 1.33231402080072245719313731567, 1.56249263616319379880253369980, 1.68604661611783210103668539087, 1.70679614174622875374804064353, 1.83021610647557888280265901496, 1.98982720990560698794380704990, 2.02455679453877819148868626329, 2.09131179743859861765268409900, 2.16933650951062821783418491182, 2.32742162902172972331686175217, 2.39683169415982810527888883454, 2.50755521670693431054037399866, 2.51381975290325287023175468866
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.