Dirichlet series
| L(s) = 1 | − 8·3-s + 4·4-s + 14·9-s − 32·12-s + 46·17-s + 36·23-s + 40·25-s + 60·27-s − 30·29-s + 56·36-s + 36·43-s − 6·49-s − 368·51-s − 50·53-s + 32·61-s − 19·64-s + 184·68-s − 288·69-s − 320·75-s + 4·79-s − 257·81-s + 240·87-s + 144·92-s + 160·100-s + 64·101-s + 30·103-s − 22·107-s + ⋯ |
| L(s) = 1 | − 4.61·3-s + 2·4-s + 14/3·9-s − 9.23·12-s + 11.1·17-s + 7.50·23-s + 8·25-s + 11.5·27-s − 5.57·29-s + 28/3·36-s + 5.48·43-s − 6/7·49-s − 51.5·51-s − 6.86·53-s + 4.09·61-s − 2.37·64-s + 22.3·68-s − 34.6·69-s − 36.9·75-s + 0.450·79-s − 28.5·81-s + 25.7·87-s + 15.0·92-s + 16·100-s + 6.36·101-s + 2.95·103-s − 2.12·107-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{12} \cdot 13^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{12} \cdot 13^{24}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(7^{12} \cdot 13^{24}\) |
| Sign: | $1$ |
| Analytic conductor: | \(5.04826\times 10^{11}\) |
| Root analytic conductor: | \(3.07348\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 7^{12} \cdot 13^{24} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(6.806416785\) |
| \(L(\frac12)\) | \(\approx\) | \(6.806416785\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 7 | \( ( 1 + T^{2} )^{6} \) |
| 13 | \( 1 \) | |
| good | 2 | \( 1 - p^{2} T^{2} + p^{4} T^{4} - 45 T^{6} + 13 p^{3} T^{8} - 7 p^{5} T^{10} + 449 T^{12} - 7 p^{7} T^{14} + 13 p^{7} T^{16} - 45 p^{6} T^{18} + p^{12} T^{20} - p^{12} T^{22} + p^{12} T^{24} \) |
| 3 | \( ( 1 + 4 T + 17 T^{2} + 46 T^{3} + 122 T^{4} + 245 T^{5} + 481 T^{6} + 245 p T^{7} + 122 p^{2} T^{8} + 46 p^{3} T^{9} + 17 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 5 | \( 1 - 8 p T^{2} + 802 T^{4} - 10533 T^{6} + 100289 T^{8} - 726767 T^{10} + 4102769 T^{12} - 726767 p^{2} T^{14} + 100289 p^{4} T^{16} - 10533 p^{6} T^{18} + 802 p^{8} T^{20} - 8 p^{11} T^{22} + p^{12} T^{24} \) | |
| 11 | \( 1 - 36 T^{2} + 778 T^{4} - 9677 T^{6} + 61129 T^{8} + 158753 T^{10} - 6464135 T^{12} + 158753 p^{2} T^{14} + 61129 p^{4} T^{16} - 9677 p^{6} T^{18} + 778 p^{8} T^{20} - 36 p^{10} T^{22} + p^{12} T^{24} \) | |
| 17 | \( ( 1 - 23 T + 292 T^{2} - 2540 T^{3} + 16936 T^{4} - 91115 T^{5} + 409523 T^{6} - 91115 p T^{7} + 16936 p^{2} T^{8} - 2540 p^{3} T^{9} + 292 p^{4} T^{10} - 23 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 19 | \( 1 - 153 T^{2} + 11661 T^{4} - 581113 T^{6} + 20995645 T^{8} - 577631081 T^{10} + 12387419253 T^{12} - 577631081 p^{2} T^{14} + 20995645 p^{4} T^{16} - 581113 p^{6} T^{18} + 11661 p^{8} T^{20} - 153 p^{10} T^{22} + p^{12} T^{24} \) | |
| 23 | \( ( 1 - 18 T + 235 T^{2} - 2091 T^{3} + 15661 T^{4} - 94020 T^{5} + 495523 T^{6} - 94020 p T^{7} + 15661 p^{2} T^{8} - 2091 p^{3} T^{9} + 235 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 29 | \( ( 1 + 15 T + 183 T^{2} + 1577 T^{3} + 12663 T^{4} + 80831 T^{5} + 478995 T^{6} + 80831 p T^{7} + 12663 p^{2} T^{8} + 1577 p^{3} T^{9} + 183 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 31 | \( 1 - 49 T^{2} + 4165 T^{4} - 195909 T^{6} + 8388357 T^{8} - 336381549 T^{10} + 10204263797 T^{12} - 336381549 p^{2} T^{14} + 8388357 p^{4} T^{16} - 195909 p^{6} T^{18} + 4165 p^{8} T^{20} - 49 p^{10} T^{22} + p^{12} T^{24} \) | |
| 37 | \( 1 - 285 T^{2} + 34589 T^{4} - 2236349 T^{6} + 74489037 T^{8} - 645851633 T^{10} - 24858497607 T^{12} - 645851633 p^{2} T^{14} + 74489037 p^{4} T^{16} - 2236349 p^{6} T^{18} + 34589 p^{8} T^{20} - 285 p^{10} T^{22} + p^{12} T^{24} \) | |
| 41 | \( 1 - 6 p T^{2} + 27141 T^{4} - 1906210 T^{6} + 108622938 T^{8} - 5695308526 T^{10} + 258694117213 T^{12} - 5695308526 p^{2} T^{14} + 108622938 p^{4} T^{16} - 1906210 p^{6} T^{18} + 27141 p^{8} T^{20} - 6 p^{11} T^{22} + p^{12} T^{24} \) | |
| 43 | \( ( 1 - 18 T + 311 T^{2} - 3496 T^{3} + 35744 T^{4} - 284555 T^{5} + 2083101 T^{6} - 284555 p T^{7} + 35744 p^{2} T^{8} - 3496 p^{3} T^{9} + 311 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 47 | \( 1 - 264 T^{2} + 37930 T^{4} - 3831553 T^{6} + 298938493 T^{8} - 18770054015 T^{10} + 969558636241 T^{12} - 18770054015 p^{2} T^{14} + 298938493 p^{4} T^{16} - 3831553 p^{6} T^{18} + 37930 p^{8} T^{20} - 264 p^{10} T^{22} + p^{12} T^{24} \) | |
| 53 | \( ( 1 + 25 T + 429 T^{2} + 5407 T^{3} + 58624 T^{4} + 523460 T^{5} + 4125969 T^{6} + 523460 p T^{7} + 58624 p^{2} T^{8} + 5407 p^{3} T^{9} + 429 p^{4} T^{10} + 25 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 59 | \( 1 - 138 T^{2} + 12547 T^{4} - 1095907 T^{6} + 88932373 T^{8} - 5912540938 T^{10} + 346202653593 T^{12} - 5912540938 p^{2} T^{14} + 88932373 p^{4} T^{16} - 1095907 p^{6} T^{18} + 12547 p^{8} T^{20} - 138 p^{10} T^{22} + p^{12} T^{24} \) | |
| 61 | \( ( 1 - 16 T + 311 T^{2} - 3792 T^{3} + 44666 T^{4} - 408293 T^{5} + 3575773 T^{6} - 408293 p T^{7} + 44666 p^{2} T^{8} - 3792 p^{3} T^{9} + 311 p^{4} T^{10} - 16 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 67 | \( 1 - 336 T^{2} + 56422 T^{4} - 6693414 T^{6} + 653596647 T^{8} - 55020595777 T^{10} + 3984181298567 T^{12} - 55020595777 p^{2} T^{14} + 653596647 p^{4} T^{16} - 6693414 p^{6} T^{18} + 56422 p^{8} T^{20} - 336 p^{10} T^{22} + p^{12} T^{24} \) | |
| 71 | \( 1 - 275 T^{2} + 54086 T^{4} - 7348715 T^{6} + 829812476 T^{8} - 75200037647 T^{10} + 5877883131849 T^{12} - 75200037647 p^{2} T^{14} + 829812476 p^{4} T^{16} - 7348715 p^{6} T^{18} + 54086 p^{8} T^{20} - 275 p^{10} T^{22} + p^{12} T^{24} \) | |
| 73 | \( 1 - 499 T^{2} + 118348 T^{4} - 17649040 T^{6} + 1885663102 T^{8} - 160651072403 T^{10} + 12135777527271 T^{12} - 160651072403 p^{2} T^{14} + 1885663102 p^{4} T^{16} - 17649040 p^{6} T^{18} + 118348 p^{8} T^{20} - 499 p^{10} T^{22} + p^{12} T^{24} \) | |
| 79 | \( ( 1 - 2 T + 307 T^{2} - 689 T^{3} + 48091 T^{4} - 95350 T^{5} + 4742205 T^{6} - 95350 p T^{7} + 48091 p^{2} T^{8} - 689 p^{3} T^{9} + 307 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 83 | \( 1 - 403 T^{2} + 87466 T^{4} - 13409571 T^{6} + 1637073136 T^{8} - 169072833947 T^{10} + 15066038203705 T^{12} - 169072833947 p^{2} T^{14} + 1637073136 p^{4} T^{16} - 13409571 p^{6} T^{18} + 87466 p^{8} T^{20} - 403 p^{10} T^{22} + p^{12} T^{24} \) | |
| 89 | \( 1 - 484 T^{2} + 126726 T^{4} - 22653578 T^{6} + 3093919203 T^{8} - 345421853993 T^{10} + 32970021127819 T^{12} - 345421853993 p^{2} T^{14} + 3093919203 p^{4} T^{16} - 22653578 p^{6} T^{18} + 126726 p^{8} T^{20} - 484 p^{10} T^{22} + p^{12} T^{24} \) | |
| 97 | \( 1 - 547 T^{2} + 147296 T^{4} - 27098713 T^{6} + 3960459647 T^{8} - 489928464772 T^{10} + 51633238123929 T^{12} - 489928464772 p^{2} T^{14} + 3960459647 p^{4} T^{16} - 27098713 p^{6} T^{18} + 147296 p^{8} T^{20} - 547 p^{10} T^{22} + p^{12} T^{24} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.07898164630579663054609555595, −3.05603031927304015637683362691, −3.02117415719902330733054910537, −2.83903052340138230685223863982, −2.80669477229346946083937996503, −2.76373130300890374740397314881, −2.63146140140042848238988950274, −2.51612918559940049103113615350, −2.34501847847089987559158407943, −2.32311467552599234029271852221, −2.20269653993229143718713857905, −1.89229264687998400835839577832, −1.83921944994878094277127535548, −1.51396995073495039760616478053, −1.42160356070551054379468305621, −1.26514363107658484739592929439, −1.23834359975065283940544503859, −1.13486452745248995933080848243, −1.09191133082332155602929412384, −0.931012881130360279583159909137, −0.860582900003912876305567097968, −0.72632906423735141949661504142, −0.65317196269171172881482402367, −0.36546118137145701288431999347, −0.33750611151436626308760498061, 0.33750611151436626308760498061, 0.36546118137145701288431999347, 0.65317196269171172881482402367, 0.72632906423735141949661504142, 0.860582900003912876305567097968, 0.931012881130360279583159909137, 1.09191133082332155602929412384, 1.13486452745248995933080848243, 1.23834359975065283940544503859, 1.26514363107658484739592929439, 1.42160356070551054379468305621, 1.51396995073495039760616478053, 1.83921944994878094277127535548, 1.89229264687998400835839577832, 2.20269653993229143718713857905, 2.32311467552599234029271852221, 2.34501847847089987559158407943, 2.51612918559940049103113615350, 2.63146140140042848238988950274, 2.76373130300890374740397314881, 2.80669477229346946083937996503, 2.83903052340138230685223863982, 3.02117415719902330733054910537, 3.05603031927304015637683362691, 3.07898164630579663054609555595
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.