Dirichlet series
| L(s) = 1 | − 6·4-s − 12·13-s + 21·16-s + 12·23-s − 12·29-s + 12·31-s − 12·37-s + 24·47-s + 72·52-s − 12·61-s − 56·64-s − 12·67-s + 36·71-s − 36·73-s + 48·79-s − 12·89-s − 72·92-s + 36·97-s + 12·103-s + 36·107-s + 12·109-s − 36·113-s + 72·116-s − 72·124-s + 16·125-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 3·4-s − 3.32·13-s + 21/4·16-s + 2.50·23-s − 2.22·29-s + 2.15·31-s − 1.97·37-s + 3.50·47-s + 9.98·52-s − 1.53·61-s − 7·64-s − 1.46·67-s + 4.27·71-s − 4.21·73-s + 5.40·79-s − 1.27·89-s − 7.50·92-s + 3.65·97-s + 1.18·103-s + 3.48·107-s + 1.14·109-s − 3.38·113-s + 6.68·116-s − 6.46·124-s + 1.43·125-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{24} \cdot 7^{12} \cdot 17^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{24} \cdot 7^{12} \cdot 17^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{12} \cdot 3^{24} \cdot 7^{12} \cdot 17^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(6.26843\times 10^{14}\) |
| Root analytic conductor: | \(4.13569\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 2^{12} \cdot 3^{24} \cdot 7^{12} \cdot 17^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.2415245574\) |
| \(L(\frac12)\) | \(\approx\) | \(0.2415245574\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + T^{2} )^{6} \) |
| 3 | \( 1 \) | |
| 7 | \( ( 1 + T^{4} )^{3} \) | |
| 17 | \( ( 1 + 33 T^{2} - 16 T^{3} + 33 p T^{4} + p^{3} T^{6} )^{2} \) | |
| good | 5 | \( 1 - 16 T^{3} + 3 T^{4} + 128 T^{6} + 48 T^{7} + 627 T^{8} - 1392 T^{9} - 1152 T^{10} - 2928 T^{11} + 18914 T^{12} - 2928 p T^{13} - 1152 p^{2} T^{14} - 1392 p^{3} T^{15} + 627 p^{4} T^{16} + 48 p^{5} T^{17} + 128 p^{6} T^{18} + 3 p^{8} T^{20} - 16 p^{9} T^{21} + p^{12} T^{24} \) |
| 11 | \( 1 + 4 T^{3} - 321 T^{4} - 48 T^{5} + 8 T^{6} - 972 T^{7} + 67695 T^{8} + 2308 T^{9} - 168 T^{10} + 53388 T^{11} - 9306974 T^{12} + 53388 p T^{13} - 168 p^{2} T^{14} + 2308 p^{3} T^{15} + 67695 p^{4} T^{16} - 972 p^{5} T^{17} + 8 p^{6} T^{18} - 48 p^{7} T^{19} - 321 p^{8} T^{20} + 4 p^{9} T^{21} + p^{12} T^{24} \) | |
| 13 | \( ( 1 + 6 T + 63 T^{2} + 334 T^{3} + 1929 T^{4} + 7812 T^{5} + 33286 T^{6} + 7812 p T^{7} + 1929 p^{2} T^{8} + 334 p^{3} T^{9} + 63 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 19 | \( 1 - 108 T^{2} + 270 p T^{4} - 138204 T^{6} + 2271375 T^{8} - 23535576 T^{10} + 254027756 T^{12} - 23535576 p^{2} T^{14} + 2271375 p^{4} T^{16} - 138204 p^{6} T^{18} + 270 p^{9} T^{20} - 108 p^{10} T^{22} + p^{12} T^{24} \) | |
| 23 | \( ( 1 - 4 T + 8 T^{2} - 84 T^{3} + 878 T^{4} - 84 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{3} \) | |
| 29 | \( 1 + 12 T + 72 T^{2} + 292 T^{3} - 750 T^{4} - 22716 T^{5} - 175960 T^{6} - 924852 T^{7} - 3069393 T^{8} + 1782168 T^{9} + 98363856 T^{10} + 821014344 T^{11} + 5049993884 T^{12} + 821014344 p T^{13} + 98363856 p^{2} T^{14} + 1782168 p^{3} T^{15} - 3069393 p^{4} T^{16} - 924852 p^{5} T^{17} - 175960 p^{6} T^{18} - 22716 p^{7} T^{19} - 750 p^{8} T^{20} + 292 p^{9} T^{21} + 72 p^{10} T^{22} + 12 p^{11} T^{23} + p^{12} T^{24} \) | |
| 31 | \( 1 - 12 T + 72 T^{2} - 636 T^{3} + 4482 T^{4} - 13524 T^{5} + 41832 T^{6} - 158532 T^{7} - 71775 p T^{8} + 22612296 T^{9} - 106361136 T^{10} + 773315880 T^{11} - 5565888772 T^{12} + 773315880 p T^{13} - 106361136 p^{2} T^{14} + 22612296 p^{3} T^{15} - 71775 p^{5} T^{16} - 158532 p^{5} T^{17} + 41832 p^{6} T^{18} - 13524 p^{7} T^{19} + 4482 p^{8} T^{20} - 636 p^{9} T^{21} + 72 p^{10} T^{22} - 12 p^{11} T^{23} + p^{12} T^{24} \) | |
| 37 | \( 1 + 12 T + 72 T^{2} + 596 T^{3} + 7791 T^{4} + 59664 T^{5} + 332624 T^{6} + 2696016 T^{7} + 25884591 T^{8} + 155618148 T^{9} + 798975624 T^{10} + 6024660060 T^{11} + 45225637058 T^{12} + 6024660060 p T^{13} + 798975624 p^{2} T^{14} + 155618148 p^{3} T^{15} + 25884591 p^{4} T^{16} + 2696016 p^{5} T^{17} + 332624 p^{6} T^{18} + 59664 p^{7} T^{19} + 7791 p^{8} T^{20} + 596 p^{9} T^{21} + 72 p^{10} T^{22} + 12 p^{11} T^{23} + p^{12} T^{24} \) | |
| 41 | \( 1 - 64 T^{3} - 2586 T^{4} - 576 T^{5} + 2048 T^{6} - 76032 T^{7} + 4660719 T^{8} + 5865216 T^{9} + 6144 p^{2} T^{10} - 261504 p T^{11} - 10043949868 T^{12} - 261504 p^{2} T^{13} + 6144 p^{4} T^{14} + 5865216 p^{3} T^{15} + 4660719 p^{4} T^{16} - 76032 p^{5} T^{17} + 2048 p^{6} T^{18} - 576 p^{7} T^{19} - 2586 p^{8} T^{20} - 64 p^{9} T^{21} + p^{12} T^{24} \) | |
| 43 | \( 1 - 138 T^{2} + 9975 T^{4} - 373954 T^{6} + 2415915 T^{8} + 677955996 T^{10} - 43540581254 T^{12} + 677955996 p^{2} T^{14} + 2415915 p^{4} T^{16} - 373954 p^{6} T^{18} + 9975 p^{8} T^{20} - 138 p^{10} T^{22} + p^{12} T^{24} \) | |
| 47 | \( ( 1 - 12 T + 204 T^{2} - 1876 T^{3} + 18279 T^{4} - 139272 T^{5} + 1022424 T^{6} - 139272 p T^{7} + 18279 p^{2} T^{8} - 1876 p^{3} T^{9} + 204 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 53 | \( 1 - 270 T^{2} + 38007 T^{4} - 3808550 T^{6} + 302535051 T^{8} - 20098140252 T^{10} + 1146522319546 T^{12} - 20098140252 p^{2} T^{14} + 302535051 p^{4} T^{16} - 3808550 p^{6} T^{18} + 38007 p^{8} T^{20} - 270 p^{10} T^{22} + p^{12} T^{24} \) | |
| 59 | \( 1 - 240 T^{2} + 31230 T^{4} - 3058064 T^{6} + 250948815 T^{8} - 17931119616 T^{10} + 1128945718084 T^{12} - 17931119616 p^{2} T^{14} + 250948815 p^{4} T^{16} - 3058064 p^{6} T^{18} + 31230 p^{8} T^{20} - 240 p^{10} T^{22} + p^{12} T^{24} \) | |
| 61 | \( 1 + 12 T + 72 T^{2} + 1156 T^{3} + 13266 T^{4} + 47076 T^{5} + 277928 T^{6} + 4050828 T^{7} - 13805649 T^{8} - 407477032 T^{9} - 1791878448 T^{10} - 26318936760 T^{11} - 366560027876 T^{12} - 26318936760 p T^{13} - 1791878448 p^{2} T^{14} - 407477032 p^{3} T^{15} - 13805649 p^{4} T^{16} + 4050828 p^{5} T^{17} + 277928 p^{6} T^{18} + 47076 p^{7} T^{19} + 13266 p^{8} T^{20} + 1156 p^{9} T^{21} + 72 p^{10} T^{22} + 12 p^{11} T^{23} + p^{12} T^{24} \) | |
| 67 | \( ( 1 + 6 T + 75 T^{2} + 138 T^{3} + 4683 T^{4} + 5964 T^{5} + 340914 T^{6} + 5964 p T^{7} + 4683 p^{2} T^{8} + 138 p^{3} T^{9} + 75 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 71 | \( 1 - 36 T + 648 T^{2} - 7524 T^{3} + 62106 T^{4} - 424620 T^{5} + 3346920 T^{6} - 32444460 T^{7} + 306534063 T^{8} - 2828980584 T^{9} + 29608603920 T^{10} - 327677395176 T^{11} + 3111847928620 T^{12} - 327677395176 p T^{13} + 29608603920 p^{2} T^{14} - 2828980584 p^{3} T^{15} + 306534063 p^{4} T^{16} - 32444460 p^{5} T^{17} + 3346920 p^{6} T^{18} - 424620 p^{7} T^{19} + 62106 p^{8} T^{20} - 7524 p^{9} T^{21} + 648 p^{10} T^{22} - 36 p^{11} T^{23} + p^{12} T^{24} \) | |
| 73 | \( 1 + 36 T + 648 T^{2} + 6840 T^{3} + 42699 T^{4} + 151608 T^{5} + 1181736 T^{6} + 21532884 T^{7} + 231283587 T^{8} + 1626697224 T^{9} + 17007809616 T^{10} + 252023745264 T^{11} + 2738343299954 T^{12} + 252023745264 p T^{13} + 17007809616 p^{2} T^{14} + 1626697224 p^{3} T^{15} + 231283587 p^{4} T^{16} + 21532884 p^{5} T^{17} + 1181736 p^{6} T^{18} + 151608 p^{7} T^{19} + 42699 p^{8} T^{20} + 6840 p^{9} T^{21} + 648 p^{10} T^{22} + 36 p^{11} T^{23} + p^{12} T^{24} \) | |
| 79 | \( 1 - 48 T + 1152 T^{2} - 17496 T^{3} + 195867 T^{4} - 2012928 T^{5} + 24036768 T^{6} - 306692952 T^{7} + 3462532803 T^{8} - 33672308040 T^{9} + 311262813792 T^{10} - 2932181360904 T^{11} + 26991049865042 T^{12} - 2932181360904 p T^{13} + 311262813792 p^{2} T^{14} - 33672308040 p^{3} T^{15} + 3462532803 p^{4} T^{16} - 306692952 p^{5} T^{17} + 24036768 p^{6} T^{18} - 2012928 p^{7} T^{19} + 195867 p^{8} T^{20} - 17496 p^{9} T^{21} + 1152 p^{10} T^{22} - 48 p^{11} T^{23} + p^{12} T^{24} \) | |
| 83 | \( 1 - 198 T^{2} + 15243 T^{4} - 1517858 T^{6} + 230556231 T^{8} - 19851680760 T^{10} + 1333329894394 T^{12} - 19851680760 p^{2} T^{14} + 230556231 p^{4} T^{16} - 1517858 p^{6} T^{18} + 15243 p^{8} T^{20} - 198 p^{10} T^{22} + p^{12} T^{24} \) | |
| 89 | \( ( 1 + 6 T + 195 T^{2} + 1822 T^{3} + 21411 T^{4} + 274836 T^{5} + 1679910 T^{6} + 274836 p T^{7} + 21411 p^{2} T^{8} + 1822 p^{3} T^{9} + 195 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 97 | \( 1 - 36 T + 648 T^{2} - 10184 T^{3} + 148923 T^{4} - 1668216 T^{5} + 15410600 T^{6} - 128762724 T^{7} + 644100003 T^{8} + 1791055400 T^{9} - 74055687600 T^{10} + 1224522981984 T^{11} - 14884423469102 T^{12} + 1224522981984 p T^{13} - 74055687600 p^{2} T^{14} + 1791055400 p^{3} T^{15} + 644100003 p^{4} T^{16} - 128762724 p^{5} T^{17} + 15410600 p^{6} T^{18} - 1668216 p^{7} T^{19} + 148923 p^{8} T^{20} - 10184 p^{9} T^{21} + 648 p^{10} T^{22} - 36 p^{11} T^{23} + p^{12} T^{24} \) | |
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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
−2.75001927803555355961167078870, −2.73481968776920297138294618312, −2.60457144379581529129981315295, −2.49348695571244359133062202200, −2.49107355972018666161144700750, −2.43555765849090124111092360903, −2.15618346645995979172371320862, −2.14215216438315719862290939122, −2.12130208780422026936624569967, −2.11203303854760651479298168653, −1.98682520608626050200933523249, −1.78604336725707422947717033590, −1.48182347243210572243099119016, −1.47662432396858037329418164206, −1.38051960161547937973467940630, −1.34180604302148849627957225202, −1.31244891290642754626370053967, −1.06528455540292011923908204887, −0.802371115273029744421241048772, −0.77257656442340687703073348019, −0.65003243647718191036989796876, −0.60549007821615097700521064029, −0.36298001154382649360345686294, −0.34679152947444789392246393985, −0.05072996970698370408233591046, 0.05072996970698370408233591046, 0.34679152947444789392246393985, 0.36298001154382649360345686294, 0.60549007821615097700521064029, 0.65003243647718191036989796876, 0.77257656442340687703073348019, 0.802371115273029744421241048772, 1.06528455540292011923908204887, 1.31244891290642754626370053967, 1.34180604302148849627957225202, 1.38051960161547937973467940630, 1.47662432396858037329418164206, 1.48182347243210572243099119016, 1.78604336725707422947717033590, 1.98682520608626050200933523249, 2.11203303854760651479298168653, 2.12130208780422026936624569967, 2.14215216438315719862290939122, 2.15618346645995979172371320862, 2.43555765849090124111092360903, 2.49107355972018666161144700750, 2.49348695571244359133062202200, 2.60457144379581529129981315295, 2.73481968776920297138294618312, 2.75001927803555355961167078870
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.