Dirichlet series
| L(s) = 1 | + 12·3-s − 12·5-s + 78·9-s − 144·15-s − 8·17-s + 8·19-s + 78·25-s + 364·27-s − 936·45-s + 32·49-s − 96·51-s + 96·57-s + 8·59-s − 8·67-s + 32·71-s − 24·73-s + 936·75-s + 16·79-s + 1.36e3·81-s + 96·85-s − 96·95-s + 48·101-s + 40·103-s + 24·107-s + 32·121-s − 364·125-s + 127-s + ⋯ |
| L(s) = 1 | + 6.92·3-s − 5.36·5-s + 26·9-s − 37.1·15-s − 1.94·17-s + 1.83·19-s + 78/5·25-s + 70.0·27-s − 139.·45-s + 32/7·49-s − 13.4·51-s + 12.7·57-s + 1.04·59-s − 0.977·67-s + 3.79·71-s − 2.80·73-s + 108.·75-s + 1.80·79-s + 151.·81-s + 10.4·85-s − 9.84·95-s + 4.77·101-s + 3.94·103-s + 2.32·107-s + 2.90·121-s − 32.5·125-s + 0.0887·127-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{12} \cdot 5^{12} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{12} \cdot 5^{12} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{48} \cdot 3^{12} \cdot 5^{12} \cdot 19^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(5.43129\times 10^{18}\) |
| Root analytic conductor: | \(6.03421\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 2^{48} \cdot 3^{12} \cdot 5^{12} \cdot 19^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(215.5940574\) |
| \(L(\frac12)\) | \(\approx\) | \(215.5940574\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 - T )^{12} \) | |
| 5 | \( ( 1 + T )^{12} \) | |
| 19 | \( 1 - 8 T + 58 T^{2} - 184 T^{3} + 343 T^{4} + 2800 T^{5} - 13588 T^{6} + 2800 p T^{7} + 343 p^{2} T^{8} - 184 p^{3} T^{9} + 58 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \) | |
| good | 7 | \( 1 - 32 T^{2} + 470 T^{4} - 3936 T^{6} + 17251 T^{8} + 1040 T^{10} - 419348 T^{12} + 1040 p^{2} T^{14} + 17251 p^{4} T^{16} - 3936 p^{6} T^{18} + 470 p^{8} T^{20} - 32 p^{10} T^{22} + p^{12} T^{24} \) |
| 11 | \( 1 - 32 T^{2} + 838 T^{4} - 15520 T^{6} + 248755 T^{8} - 3268880 T^{10} + 39224780 T^{12} - 3268880 p^{2} T^{14} + 248755 p^{4} T^{16} - 15520 p^{6} T^{18} + 838 p^{8} T^{20} - 32 p^{10} T^{22} + p^{12} T^{24} \) | |
| 13 | \( 1 - 80 T^{2} + 3158 T^{4} - 81840 T^{6} + 1584403 T^{8} - 25018576 T^{10} + 344004652 T^{12} - 25018576 p^{2} T^{14} + 1584403 p^{4} T^{16} - 81840 p^{6} T^{18} + 3158 p^{8} T^{20} - 80 p^{10} T^{22} + p^{12} T^{24} \) | |
| 17 | \( ( 1 + 4 T + 78 T^{2} + 260 T^{3} + 2851 T^{4} + 7720 T^{5} + 61388 T^{6} + 7720 p T^{7} + 2851 p^{2} T^{8} + 260 p^{3} T^{9} + 78 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 23 | \( 1 - 4 p T^{2} + 4042 T^{4} - 106540 T^{6} + 2040511 T^{8} - 36184952 T^{10} + 763711148 T^{12} - 36184952 p^{2} T^{14} + 2040511 p^{4} T^{16} - 106540 p^{6} T^{18} + 4042 p^{8} T^{20} - 4 p^{11} T^{22} + p^{12} T^{24} \) | |
| 29 | \( 1 - 104 T^{2} + 6310 T^{4} - 268408 T^{6} + 9596179 T^{8} - 302831792 T^{10} + 9084783116 T^{12} - 302831792 p^{2} T^{14} + 9596179 p^{4} T^{16} - 268408 p^{6} T^{18} + 6310 p^{8} T^{20} - 104 p^{10} T^{22} + p^{12} T^{24} \) | |
| 31 | \( ( 1 + 138 T^{2} + 88 T^{3} + 8739 T^{4} + 7608 T^{5} + 336180 T^{6} + 7608 p T^{7} + 8739 p^{2} T^{8} + 88 p^{3} T^{9} + 138 p^{4} T^{10} + p^{6} T^{12} )^{2} \) | |
| 37 | \( 1 - 320 T^{2} + 49862 T^{4} - 135120 p T^{6} + 9707095 p T^{8} - 19468604800 T^{10} + 816115891852 T^{12} - 19468604800 p^{2} T^{14} + 9707095 p^{5} T^{16} - 135120 p^{7} T^{18} + 49862 p^{8} T^{20} - 320 p^{10} T^{22} + p^{12} T^{24} \) | |
| 41 | \( 1 - 240 T^{2} + 28086 T^{4} - 2128976 T^{6} + 119914563 T^{8} - 5587536432 T^{10} + 235895790060 T^{12} - 5587536432 p^{2} T^{14} + 119914563 p^{4} T^{16} - 2128976 p^{6} T^{18} + 28086 p^{8} T^{20} - 240 p^{10} T^{22} + p^{12} T^{24} \) | |
| 43 | \( 1 - 392 T^{2} + 73958 T^{4} - 8888424 T^{6} + 758271955 T^{8} - 48379526176 T^{10} + 2366851212364 T^{12} - 48379526176 p^{2} T^{14} + 758271955 p^{4} T^{16} - 8888424 p^{6} T^{18} + 73958 p^{8} T^{20} - 392 p^{10} T^{22} + p^{12} T^{24} \) | |
| 47 | \( 1 - 236 T^{2} + 31018 T^{4} - 2928508 T^{6} + 218985631 T^{8} - 13444821848 T^{10} + 689689882988 T^{12} - 13444821848 p^{2} T^{14} + 218985631 p^{4} T^{16} - 2928508 p^{6} T^{18} + 31018 p^{8} T^{20} - 236 p^{10} T^{22} + p^{12} T^{24} \) | |
| 53 | \( 1 - 140 T^{2} + 9946 T^{4} - 322300 T^{6} + 6861487 T^{8} - 400275800 T^{10} + 36502822220 T^{12} - 400275800 p^{2} T^{14} + 6861487 p^{4} T^{16} - 322300 p^{6} T^{18} + 9946 p^{8} T^{20} - 140 p^{10} T^{22} + p^{12} T^{24} \) | |
| 59 | \( ( 1 - 4 T + 198 T^{2} - 140 T^{3} + 14599 T^{4} + 41768 T^{5} + 753812 T^{6} + 41768 p T^{7} + 14599 p^{2} T^{8} - 140 p^{3} T^{9} + 198 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 61 | \( ( 1 + 270 T^{2} + 304 T^{3} + 32955 T^{4} + 53088 T^{5} + 2466876 T^{6} + 53088 p T^{7} + 32955 p^{2} T^{8} + 304 p^{3} T^{9} + 270 p^{4} T^{10} + p^{6} T^{12} )^{2} \) | |
| 67 | \( ( 1 + 4 T + 130 T^{2} - 172 T^{3} + 5503 T^{4} - 29504 T^{5} + 347276 T^{6} - 29504 p T^{7} + 5503 p^{2} T^{8} - 172 p^{3} T^{9} + 130 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 71 | \( ( 1 - 16 T + 186 T^{2} - 1256 T^{3} + 7351 T^{4} - 37048 T^{5} + 382076 T^{6} - 37048 p T^{7} + 7351 p^{2} T^{8} - 1256 p^{3} T^{9} + 186 p^{4} T^{10} - 16 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 73 | \( ( 1 + 12 T + 426 T^{2} + 4084 T^{3} + 76839 T^{4} + 7872 p T^{5} + 7456188 T^{6} + 7872 p^{2} T^{7} + 76839 p^{2} T^{8} + 4084 p^{3} T^{9} + 426 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 79 | \( ( 1 - 8 T + 254 T^{2} - 1560 T^{3} + 30751 T^{4} - 134416 T^{5} + 2666116 T^{6} - 134416 p T^{7} + 30751 p^{2} T^{8} - 1560 p^{3} T^{9} + 254 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 83 | \( 1 - 324 T^{2} + 71418 T^{4} - 11500820 T^{6} + 1489251279 T^{8} - 159836113416 T^{10} + 14389649529228 T^{12} - 159836113416 p^{2} T^{14} + 1489251279 p^{4} T^{16} - 11500820 p^{6} T^{18} + 71418 p^{8} T^{20} - 324 p^{10} T^{22} + p^{12} T^{24} \) | |
| 89 | \( 1 - 480 T^{2} + 112374 T^{4} - 18161936 T^{6} + 2365451715 T^{8} - 262397851392 T^{10} + 25106077134828 T^{12} - 262397851392 p^{2} T^{14} + 2365451715 p^{4} T^{16} - 18161936 p^{6} T^{18} + 112374 p^{8} T^{20} - 480 p^{10} T^{22} + p^{12} T^{24} \) | |
| 97 | \( 1 - 576 T^{2} + 170598 T^{4} - 34743088 T^{6} + 5434601763 T^{8} - 688539223968 T^{10} + 72748479971340 T^{12} - 688539223968 p^{2} T^{14} + 5434601763 p^{4} T^{16} - 34743088 p^{6} T^{18} + 170598 p^{8} T^{20} - 576 p^{10} T^{22} + p^{12} T^{24} \) | |
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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
−2.53739950022576723538010145523, −2.42322523791501265416010325924, −2.42261232832788478686943091094, −2.32767630914126471896556659350, −2.17688773866185915701699143212, −2.07617110439199533164346900644, −2.05112877562354069648416555209, −2.01684160722093923568521206902, −1.97137209307088541502278062890, −1.82412695720394957815289226249, −1.81874325317211796570306231403, −1.76031649134197587338873866781, −1.49483383109057590320590985703, −1.32173316540877004785290967124, −1.28723816146179000988413952599, −1.16909366558798210579904657097, −1.00958915331023038534414514177, −0.982566287519755807469338182735, −0.942180415543340960216242941502, −0.71326388615507887075724614456, −0.65453354341484917337886276538, −0.58808383293400238695160053580, −0.48715094650823104774841746700, −0.39872999844512265870289524249, −0.14237984690237156538186712112, 0.14237984690237156538186712112, 0.39872999844512265870289524249, 0.48715094650823104774841746700, 0.58808383293400238695160053580, 0.65453354341484917337886276538, 0.71326388615507887075724614456, 0.942180415543340960216242941502, 0.982566287519755807469338182735, 1.00958915331023038534414514177, 1.16909366558798210579904657097, 1.28723816146179000988413952599, 1.32173316540877004785290967124, 1.49483383109057590320590985703, 1.76031649134197587338873866781, 1.81874325317211796570306231403, 1.82412695720394957815289226249, 1.97137209307088541502278062890, 2.01684160722093923568521206902, 2.05112877562354069648416555209, 2.07617110439199533164346900644, 2.17688773866185915701699143212, 2.32767630914126471896556659350, 2.42261232832788478686943091094, 2.42322523791501265416010325924, 2.53739950022576723538010145523
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.