Dirichlet series
| L(s) = 1 | + 8·4-s − 14·9-s + 32·16-s + 8·17-s + 24·23-s + 20·25-s + 4·27-s − 16·29-s − 112·36-s − 4·43-s − 6·49-s − 44·53-s − 28·61-s + 82·64-s + 64·68-s − 56·79-s + 91·81-s + 192·92-s + 160·100-s − 20·101-s − 36·103-s − 44·107-s + 32·108-s − 20·113-s − 128·116-s + 82·121-s + 127-s + ⋯ |
| L(s) = 1 | + 4·4-s − 4.66·9-s + 8·16-s + 1.94·17-s + 5.00·23-s + 4·25-s + 0.769·27-s − 2.97·29-s − 18.6·36-s − 0.609·43-s − 6/7·49-s − 6.04·53-s − 3.58·61-s + 41/4·64-s + 7.76·68-s − 6.30·79-s + 91/9·81-s + 20.0·92-s + 16·100-s − 1.99·101-s − 3.54·103-s − 4.25·107-s + 3.07·108-s − 1.88·113-s − 11.8·116-s + 7.45·121-s + 0.0887·127-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
| \(L(1)\) | \(\approx\) | \(3.199313168\) |
| \(L(\frac12)\) | \(\approx\) | \(3.199313168\) |
| \(L(\frac{3}{2})\) | 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 | \( 1 - p^{3} T^{2} + p^{5} T^{4} - 41 p T^{6} + 19 p^{3} T^{8} - 15 p^{4} T^{10} + 417 T^{12} - 15 p^{6} T^{14} + 19 p^{7} T^{16} - 41 p^{7} T^{18} + p^{13} T^{20} - p^{13} T^{22} + p^{12} T^{24} \) |
| 3 | \( ( 1 + 7 T^{2} - 2 T^{3} + 28 T^{4} + 2 T^{5} + 100 T^{6} + 2 p T^{7} + 28 p^{2} T^{8} - 2 p^{3} T^{9} + 7 p^{4} T^{10} + p^{6} T^{12} )^{2} \) | |
| 5 | \( 1 - 4 p T^{2} + 2 p^{3} T^{4} - 2354 T^{6} + 3607 p T^{8} - 114902 T^{10} + 620501 T^{12} - 114902 p^{2} T^{14} + 3607 p^{5} T^{16} - 2354 p^{6} T^{18} + 2 p^{11} T^{20} - 4 p^{11} T^{22} + p^{12} T^{24} \) | |
| 11 | \( 1 - 82 T^{2} + 3073 T^{4} - 69688 T^{6} + 1086920 T^{8} - 1196198 p T^{10} + 144840644 T^{12} - 1196198 p^{3} T^{14} + 1086920 p^{4} T^{16} - 69688 p^{6} T^{18} + 3073 p^{8} T^{20} - 82 p^{10} T^{22} + p^{12} T^{24} \) | |
| 17 | \( ( 1 - 4 T + 81 T^{2} - 280 T^{3} + 3074 T^{4} - 8724 T^{5} + 67033 T^{6} - 8724 p T^{7} + 3074 p^{2} T^{8} - 280 p^{3} T^{9} + 81 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 19 | \( 1 - 170 T^{2} + 13833 T^{4} - 714484 T^{6} + 26229656 T^{8} - 725388054 T^{10} + 15570602740 T^{12} - 725388054 p^{2} T^{14} + 26229656 p^{4} T^{16} - 714484 p^{6} T^{18} + 13833 p^{8} T^{20} - 170 p^{10} T^{22} + p^{12} T^{24} \) | |
| 23 | \( ( 1 - 12 T + 118 T^{2} - 772 T^{3} + 4479 T^{4} - 1000 p T^{5} + 111732 T^{6} - 1000 p^{2} T^{7} + 4479 p^{2} T^{8} - 772 p^{3} T^{9} + 118 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 29 | \( ( 1 + 8 T + 130 T^{2} + 594 T^{3} + 6099 T^{4} + 18374 T^{5} + 187029 T^{6} + 18374 p T^{7} + 6099 p^{2} T^{8} + 594 p^{3} T^{9} + 130 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 31 | \( 1 - 236 T^{2} + 27120 T^{4} - 2027596 T^{6} + 111018776 T^{8} - 4737151116 T^{10} + 162780593398 T^{12} - 4737151116 p^{2} T^{14} + 111018776 p^{4} T^{16} - 2027596 p^{6} T^{18} + 27120 p^{8} T^{20} - 236 p^{10} T^{22} + p^{12} T^{24} \) | |
| 37 | \( 1 - 126 T^{2} + 11151 T^{4} - 716278 T^{6} + 38725515 T^{8} - 1787180508 T^{10} + 71085787530 T^{12} - 1787180508 p^{2} T^{14} + 38725515 p^{4} T^{16} - 716278 p^{6} T^{18} + 11151 p^{8} T^{20} - 126 p^{10} T^{22} + p^{12} T^{24} \) | |
| 41 | \( 1 - 222 T^{2} + 24223 T^{4} - 1612294 T^{6} + 71421755 T^{8} - 2303212796 T^{10} + 76913171402 T^{12} - 2303212796 p^{2} T^{14} + 71421755 p^{4} T^{16} - 1612294 p^{6} T^{18} + 24223 p^{8} T^{20} - 222 p^{10} T^{22} + p^{12} T^{24} \) | |
| 43 | \( ( 1 + 2 T + 149 T^{2} + 340 T^{3} + 11408 T^{4} + 21594 T^{5} + 590652 T^{6} + 21594 p T^{7} + 11408 p^{2} T^{8} + 340 p^{3} T^{9} + 149 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 47 | \( 1 - 292 T^{2} + 39736 T^{4} - 3270700 T^{6} + 180363824 T^{8} - 7478971060 T^{10} + 309914184182 T^{12} - 7478971060 p^{2} T^{14} + 180363824 p^{4} T^{16} - 3270700 p^{6} T^{18} + 39736 p^{8} T^{20} - 292 p^{10} T^{22} + p^{12} T^{24} \) | |
| 53 | \( ( 1 + 22 T + 409 T^{2} + 5130 T^{3} + 58074 T^{4} + 513982 T^{5} + 4153497 T^{6} + 513982 p T^{7} + 58074 p^{2} T^{8} + 5130 p^{3} T^{9} + 409 p^{4} T^{10} + 22 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 59 | \( 1 - 380 T^{2} + 77000 T^{4} - 10623268 T^{6} + 1101557216 T^{8} - 89823025164 T^{10} + 5891147678118 T^{12} - 89823025164 p^{2} T^{14} + 1101557216 p^{4} T^{16} - 10623268 p^{6} T^{18} + 77000 p^{8} T^{20} - 380 p^{10} T^{22} + p^{12} T^{24} \) | |
| 61 | \( ( 1 + 14 T + 279 T^{2} + 2854 T^{3} + 32699 T^{4} + 263412 T^{5} + 2369290 T^{6} + 263412 p T^{7} + 32699 p^{2} T^{8} + 2854 p^{3} T^{9} + 279 p^{4} T^{10} + 14 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 67 | \( 1 - 416 T^{2} + 89024 T^{4} - 12805108 T^{6} + 1386054656 T^{8} - 120600192288 T^{10} + 8774166644694 T^{12} - 120600192288 p^{2} T^{14} + 1386054656 p^{4} T^{16} - 12805108 p^{6} T^{18} + 89024 p^{8} T^{20} - 416 p^{10} T^{22} + p^{12} T^{24} \) | |
| 71 | \( 1 - 700 T^{2} + 233602 T^{4} - 49022476 T^{6} + 7193838095 T^{8} - 776915839096 T^{10} + 63282468701852 T^{12} - 776915839096 p^{2} T^{14} + 7193838095 p^{4} T^{16} - 49022476 p^{6} T^{18} + 233602 p^{8} T^{20} - 700 p^{10} T^{22} + p^{12} T^{24} \) | |
| 73 | \( 1 - 542 T^{2} + 148367 T^{4} - 26886646 T^{6} + 3579906587 T^{8} - 368619908940 T^{10} + 30101790766122 T^{12} - 368619908940 p^{2} T^{14} + 3579906587 p^{4} T^{16} - 26886646 p^{6} T^{18} + 148367 p^{8} T^{20} - 542 p^{10} T^{22} + p^{12} T^{24} \) | |
| 79 | \( ( 1 + 28 T + 686 T^{2} + 11252 T^{3} + 159023 T^{4} + 1794648 T^{5} + 17548548 T^{6} + 1794648 p T^{7} + 159023 p^{2} T^{8} + 11252 p^{3} T^{9} + 686 p^{4} T^{10} + 28 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 83 | \( 1 - 692 T^{2} + 237440 T^{4} - 52943716 T^{6} + 8500384712 T^{8} - 1032128260020 T^{10} + 97020933033606 T^{12} - 1032128260020 p^{2} T^{14} + 8500384712 p^{4} T^{16} - 52943716 p^{6} T^{18} + 237440 p^{8} T^{20} - 692 p^{10} T^{22} + p^{12} T^{24} \) | |
| 89 | \( 1 - 410 T^{2} + 83665 T^{4} - 12720140 T^{6} + 1658742032 T^{8} - 183597109502 T^{10} + 17425244531684 T^{12} - 183597109502 p^{2} T^{14} + 1658742032 p^{4} T^{16} - 12720140 p^{6} T^{18} + 83665 p^{8} T^{20} - 410 p^{10} T^{22} + p^{12} T^{24} \) | |
| 97 | \( 1 - 782 T^{2} + 297065 T^{4} - 72936892 T^{6} + 12980062424 T^{8} - 1776005237946 T^{10} + 192588073428228 T^{12} - 1776005237946 p^{2} T^{14} + 12980062424 p^{4} T^{16} - 72936892 p^{6} T^{18} + 297065 p^{8} T^{20} - 782 p^{10} T^{22} + p^{12} T^{24} \) | |
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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
−3.01724193965807580858588447689, −3.01632878461577448276048762676, −2.92161641936650031078282337252, −2.84331039404909633437384373153, −2.84162298418150619930541691132, −2.80735080674473259056041298352, −2.65063975360568580529352310621, −2.40601194456677691036876494188, −2.37205340676392858162588763742, −2.35933546707841189401181961071, −2.33701501286827731354639171623, −1.94486429756595630177129316879, −1.73877110092837697578794050686, −1.68954950825539643068392092515, −1.67007901761001347956875416871, −1.60563026683544439854038155034, −1.55553770622918272157799088115, −1.28590327515186817096668358902, −1.18965994910072296294388742785, −1.16172641029127390943573080499, −1.00975201449042475412682878366, −0.940578765282255785283123403771, −0.45722400072079209930621035243, −0.19604297392388980089224172554, −0.16605927014247876806878233182, 0.16605927014247876806878233182, 0.19604297392388980089224172554, 0.45722400072079209930621035243, 0.940578765282255785283123403771, 1.00975201449042475412682878366, 1.16172641029127390943573080499, 1.18965994910072296294388742785, 1.28590327515186817096668358902, 1.55553770622918272157799088115, 1.60563026683544439854038155034, 1.67007901761001347956875416871, 1.68954950825539643068392092515, 1.73877110092837697578794050686, 1.94486429756595630177129316879, 2.33701501286827731354639171623, 2.35933546707841189401181961071, 2.37205340676392858162588763742, 2.40601194456677691036876494188, 2.65063975360568580529352310621, 2.80735080674473259056041298352, 2.84162298418150619930541691132, 2.84331039404909633437384373153, 2.92161641936650031078282337252, 3.01632878461577448276048762676, 3.01724193965807580858588447689
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.