Dirichlet series
| L(s) = 1 | − 6·3-s − 4·4-s − 3·5-s + 9-s + 24·12-s + 2·13-s + 18·15-s + 8·16-s − 17·17-s + 12·20-s + 3·23-s − 13·25-s + 68·27-s − 29-s + 18·31-s − 4·36-s + 15·37-s − 12·39-s + 6·41-s + 11·43-s − 3·45-s − 15·47-s − 48·48-s + 102·51-s − 8·52-s − 8·53-s − 27·59-s + ⋯ |
| L(s) = 1 | − 3.46·3-s − 2·4-s − 1.34·5-s + 1/3·9-s + 6.92·12-s + 0.554·13-s + 4.64·15-s + 2·16-s − 4.12·17-s + 2.68·20-s + 0.625·23-s − 2.59·25-s + 13.0·27-s − 0.185·29-s + 3.23·31-s − 2/3·36-s + 2.46·37-s − 1.92·39-s + 0.937·41-s + 1.67·43-s − 0.447·45-s − 2.18·47-s − 6.92·48-s + 14.2·51-s − 1.10·52-s − 1.09·53-s − 3.51·59-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{24} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{24} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(7^{24} \cdot 13^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.99915\times 10^{8}\) |
| Root analytic conductor: | \(2.25532\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 7^{24} \cdot 13^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.05029273889\) |
| \(L(\frac12)\) | \(\approx\) | \(0.05029273889\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 7 | \( 1 \) |
| 13 | \( 1 - 2 T - 18 T^{2} + 17 T^{3} + 341 T^{4} + 63 T^{5} - 6395 T^{6} + 63 p T^{7} + 341 p^{2} T^{8} + 17 p^{3} T^{9} - 18 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \) | |
| good | 2 | \( 1 + p^{2} T^{2} + p^{3} T^{4} + 5 T^{6} - p^{4} T^{8} - 3 p^{4} T^{10} - 3 p^{2} T^{11} - 111 T^{12} - 3 p^{3} T^{13} - 3 p^{6} T^{14} - p^{8} T^{16} + 5 p^{6} T^{18} + p^{11} T^{20} + p^{12} T^{22} + p^{12} T^{24} \) |
| 3 | \( ( 1 + p T + 13 T^{2} + 29 T^{3} + 79 T^{4} + 145 T^{5} + 301 T^{6} + 145 p T^{7} + 79 p^{2} T^{8} + 29 p^{3} T^{9} + 13 p^{4} T^{10} + p^{6} T^{11} + p^{6} T^{12} )^{2} \) | |
| 5 | \( 1 + 3 T + 22 T^{2} + 57 T^{3} + 241 T^{4} + 18 p^{2} T^{5} + 1411 T^{6} + 1311 T^{7} + 3032 T^{8} - 7749 T^{9} - 3571 p T^{10} - 104772 T^{11} - 169399 T^{12} - 104772 p T^{13} - 3571 p^{3} T^{14} - 7749 p^{3} T^{15} + 3032 p^{4} T^{16} + 1311 p^{5} T^{17} + 1411 p^{6} T^{18} + 18 p^{9} T^{19} + 241 p^{8} T^{20} + 57 p^{9} T^{21} + 22 p^{10} T^{22} + 3 p^{11} T^{23} + p^{12} T^{24} \) | |
| 11 | \( 1 - 70 T^{2} + 2521 T^{4} - 61186 T^{6} + 1119878 T^{8} - 16383787 T^{10} + 197579915 T^{12} - 16383787 p^{2} T^{14} + 1119878 p^{4} T^{16} - 61186 p^{6} T^{18} + 2521 p^{8} T^{20} - 70 p^{10} T^{22} + p^{12} T^{24} \) | |
| 17 | \( 1 + p T + 91 T^{2} + 80 T^{3} + 354 T^{4} + 13150 T^{5} + 66387 T^{6} + 80838 T^{7} + 350534 T^{8} + 5285257 T^{9} + 24796664 T^{10} + 64859774 T^{11} + 191483227 T^{12} + 64859774 p T^{13} + 24796664 p^{2} T^{14} + 5285257 p^{3} T^{15} + 350534 p^{4} T^{16} + 80838 p^{5} T^{17} + 66387 p^{6} T^{18} + 13150 p^{7} T^{19} + 354 p^{8} T^{20} + 80 p^{9} T^{21} + 91 p^{10} T^{22} + p^{12} T^{23} + p^{12} T^{24} \) | |
| 19 | \( 1 - 149 T^{2} + 10800 T^{4} - 511360 T^{6} + 17748002 T^{8} - 476368077 T^{10} + 10131618103 T^{12} - 476368077 p^{2} T^{14} + 17748002 p^{4} T^{16} - 511360 p^{6} T^{18} + 10800 p^{8} T^{20} - 149 p^{10} T^{22} + p^{12} T^{24} \) | |
| 23 | \( 1 - 3 T - 79 T^{2} - 4 p T^{3} + 4336 T^{4} + 12687 T^{5} - 111893 T^{6} - 708984 T^{7} + 1416466 T^{8} + 17335200 T^{9} + 27721271 T^{10} - 199047772 T^{11} - 1123052895 T^{12} - 199047772 p T^{13} + 27721271 p^{2} T^{14} + 17335200 p^{3} T^{15} + 1416466 p^{4} T^{16} - 708984 p^{5} T^{17} - 111893 p^{6} T^{18} + 12687 p^{7} T^{19} + 4336 p^{8} T^{20} - 4 p^{10} T^{21} - 79 p^{10} T^{22} - 3 p^{11} T^{23} + p^{12} T^{24} \) | |
| 29 | \( 1 + T - 3 p T^{2} + 178 T^{3} + 128 p T^{4} - 15194 T^{5} - 71197 T^{6} + 410532 T^{7} + 447076 T^{8} - 2082223 T^{9} - 28038240 T^{10} - 61539492 T^{11} + 1484469159 T^{12} - 61539492 p T^{13} - 28038240 p^{2} T^{14} - 2082223 p^{3} T^{15} + 447076 p^{4} T^{16} + 410532 p^{5} T^{17} - 71197 p^{6} T^{18} - 15194 p^{7} T^{19} + 128 p^{9} T^{20} + 178 p^{9} T^{21} - 3 p^{11} T^{22} + p^{11} T^{23} + p^{12} T^{24} \) | |
| 31 | \( 1 - 18 T + 232 T^{2} - 72 p T^{3} + 18099 T^{4} - 118611 T^{5} + 671327 T^{6} - 3244986 T^{7} + 13790561 T^{8} - 53577873 T^{9} + 222950535 T^{10} - 1049414532 T^{11} + 5476860865 T^{12} - 1049414532 p T^{13} + 222950535 p^{2} T^{14} - 53577873 p^{3} T^{15} + 13790561 p^{4} T^{16} - 3244986 p^{5} T^{17} + 671327 p^{6} T^{18} - 118611 p^{7} T^{19} + 18099 p^{8} T^{20} - 72 p^{10} T^{21} + 232 p^{10} T^{22} - 18 p^{11} T^{23} + p^{12} T^{24} \) | |
| 37 | \( 1 - 15 T + 261 T^{2} - 2790 T^{3} + 30330 T^{4} - 259743 T^{5} + 2192045 T^{6} - 15890796 T^{7} + 113579334 T^{8} - 728456778 T^{9} + 4731329259 T^{10} - 28450921716 T^{11} + 178599894873 T^{12} - 28450921716 p T^{13} + 4731329259 p^{2} T^{14} - 728456778 p^{3} T^{15} + 113579334 p^{4} T^{16} - 15890796 p^{5} T^{17} + 2192045 p^{6} T^{18} - 259743 p^{7} T^{19} + 30330 p^{8} T^{20} - 2790 p^{9} T^{21} + 261 p^{10} T^{22} - 15 p^{11} T^{23} + p^{12} T^{24} \) | |
| 41 | \( 1 - 6 T + 87 T^{2} - 450 T^{3} + 3253 T^{4} - 22749 T^{5} + 119000 T^{6} - 655563 T^{7} + 2153315 T^{8} + 1902885 T^{9} - 21527678 T^{10} + 258610773 T^{11} - 1564835407 T^{12} + 258610773 p T^{13} - 21527678 p^{2} T^{14} + 1902885 p^{3} T^{15} + 2153315 p^{4} T^{16} - 655563 p^{5} T^{17} + 119000 p^{6} T^{18} - 22749 p^{7} T^{19} + 3253 p^{8} T^{20} - 450 p^{9} T^{21} + 87 p^{10} T^{22} - 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 43 | \( 1 - 11 T - 88 T^{2} + 799 T^{3} + 8712 T^{4} - 35151 T^{5} - 615381 T^{6} + 1493311 T^{7} + 26239709 T^{8} - 29370632 T^{9} - 1062722582 T^{10} - 167088622 T^{11} + 50469301069 T^{12} - 167088622 p T^{13} - 1062722582 p^{2} T^{14} - 29370632 p^{3} T^{15} + 26239709 p^{4} T^{16} + 1493311 p^{5} T^{17} - 615381 p^{6} T^{18} - 35151 p^{7} T^{19} + 8712 p^{8} T^{20} + 799 p^{9} T^{21} - 88 p^{10} T^{22} - 11 p^{11} T^{23} + p^{12} T^{24} \) | |
| 47 | \( 1 + 15 T + 374 T^{2} + 4485 T^{3} + 70585 T^{4} + 717852 T^{5} + 8690861 T^{6} + 76990419 T^{7} + 777130526 T^{8} + 6072726129 T^{9} + 52865829659 T^{10} + 366568670022 T^{11} + 2805436179305 T^{12} + 366568670022 p T^{13} + 52865829659 p^{2} T^{14} + 6072726129 p^{3} T^{15} + 777130526 p^{4} T^{16} + 76990419 p^{5} T^{17} + 8690861 p^{6} T^{18} + 717852 p^{7} T^{19} + 70585 p^{8} T^{20} + 4485 p^{9} T^{21} + 374 p^{10} T^{22} + 15 p^{11} T^{23} + p^{12} T^{24} \) | |
| 53 | \( 1 + 8 T - 216 T^{2} - 1192 T^{3} + 31204 T^{4} + 105740 T^{5} - 3167758 T^{6} - 5889708 T^{7} + 250320748 T^{8} + 208010656 T^{9} - 16435908756 T^{10} - 4149850428 T^{11} + 923207251827 T^{12} - 4149850428 p T^{13} - 16435908756 p^{2} T^{14} + 208010656 p^{3} T^{15} + 250320748 p^{4} T^{16} - 5889708 p^{5} T^{17} - 3167758 p^{6} T^{18} + 105740 p^{7} T^{19} + 31204 p^{8} T^{20} - 1192 p^{9} T^{21} - 216 p^{10} T^{22} + 8 p^{11} T^{23} + p^{12} T^{24} \) | |
| 59 | \( 1 + 27 T + 442 T^{2} + 5373 T^{3} + 47243 T^{4} + 314460 T^{5} + 1356131 T^{6} + 82953 T^{7} - 42334024 T^{8} - 284989455 T^{9} + 1152047535 T^{10} + 39755713086 T^{11} + 391751656497 T^{12} + 39755713086 p T^{13} + 1152047535 p^{2} T^{14} - 284989455 p^{3} T^{15} - 42334024 p^{4} T^{16} + 82953 p^{5} T^{17} + 1356131 p^{6} T^{18} + 314460 p^{7} T^{19} + 47243 p^{8} T^{20} + 5373 p^{9} T^{21} + 442 p^{10} T^{22} + 27 p^{11} T^{23} + p^{12} T^{24} \) | |
| 61 | \( ( 1 - 5 T + 291 T^{2} - 1171 T^{3} + 38615 T^{4} - 126093 T^{5} + 3001147 T^{6} - 126093 p T^{7} + 38615 p^{2} T^{8} - 1171 p^{3} T^{9} + 291 p^{4} T^{10} - 5 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 67 | \( 1 - 365 T^{2} + 70430 T^{4} - 9408718 T^{6} + 14623229 p T^{8} - 84195136134 T^{10} + 6117380213691 T^{12} - 84195136134 p^{2} T^{14} + 14623229 p^{5} T^{16} - 9408718 p^{6} T^{18} + 70430 p^{8} T^{20} - 365 p^{10} T^{22} + p^{12} T^{24} \) | |
| 71 | \( 1 - 30 T + 653 T^{2} - 10590 T^{3} + 145585 T^{4} - 1771113 T^{5} + 19362440 T^{6} - 197718039 T^{7} + 1886667311 T^{8} - 17315970285 T^{9} + 153782228042 T^{10} - 1330750537803 T^{11} + 11347613768747 T^{12} - 1330750537803 p T^{13} + 153782228042 p^{2} T^{14} - 17315970285 p^{3} T^{15} + 1886667311 p^{4} T^{16} - 197718039 p^{5} T^{17} + 19362440 p^{6} T^{18} - 1771113 p^{7} T^{19} + 145585 p^{8} T^{20} - 10590 p^{9} T^{21} + 653 p^{10} T^{22} - 30 p^{11} T^{23} + p^{12} T^{24} \) | |
| 73 | \( 1 - 42 T + 1120 T^{2} - 22344 T^{3} + 367763 T^{4} - 5157765 T^{5} + 63488279 T^{6} - 699015660 T^{7} + 7009629269 T^{8} - 65233277895 T^{9} + 575574698283 T^{10} - 4944431307318 T^{11} + 42178220425467 T^{12} - 4944431307318 p T^{13} + 575574698283 p^{2} T^{14} - 65233277895 p^{3} T^{15} + 7009629269 p^{4} T^{16} - 699015660 p^{5} T^{17} + 63488279 p^{6} T^{18} - 5157765 p^{7} T^{19} + 367763 p^{8} T^{20} - 22344 p^{9} T^{21} + 1120 p^{10} T^{22} - 42 p^{11} T^{23} + p^{12} T^{24} \) | |
| 79 | \( 1 + 35 T + 407 T^{2} + 1988 T^{3} + 21243 T^{4} + 209523 T^{5} - 2661180 T^{6} - 47561854 T^{7} - 79767523 T^{8} + 341374970 T^{9} - 12037425323 T^{10} + 160472632948 T^{11} + 4058664204571 T^{12} + 160472632948 p T^{13} - 12037425323 p^{2} T^{14} + 341374970 p^{3} T^{15} - 79767523 p^{4} T^{16} - 47561854 p^{5} T^{17} - 2661180 p^{6} T^{18} + 209523 p^{7} T^{19} + 21243 p^{8} T^{20} + 1988 p^{9} T^{21} + 407 p^{10} T^{22} + 35 p^{11} T^{23} + p^{12} T^{24} \) | |
| 83 | \( 1 - 533 T^{2} + 130220 T^{4} - 19632136 T^{6} + 2123453318 T^{8} - 188332543809 T^{10} + 15642556571895 T^{12} - 188332543809 p^{2} T^{14} + 2123453318 p^{4} T^{16} - 19632136 p^{6} T^{18} + 130220 p^{8} T^{20} - 533 p^{10} T^{22} + p^{12} T^{24} \) | |
| 89 | \( 1 + 48 T + 1471 T^{2} + 33744 T^{3} + 649840 T^{4} + 10865652 T^{5} + 162607993 T^{6} + 2211532644 T^{7} + 27724965467 T^{8} + 323293945128 T^{9} + 3530567707708 T^{10} + 36297047904324 T^{11} + 352202680354553 T^{12} + 36297047904324 p T^{13} + 3530567707708 p^{2} T^{14} + 323293945128 p^{3} T^{15} + 27724965467 p^{4} T^{16} + 2211532644 p^{5} T^{17} + 162607993 p^{6} T^{18} + 10865652 p^{7} T^{19} + 649840 p^{8} T^{20} + 33744 p^{9} T^{21} + 1471 p^{10} T^{22} + 48 p^{11} T^{23} + p^{12} T^{24} \) | |
| 97 | \( 1 - 3 T + 406 T^{2} - 1209 T^{3} + 93212 T^{4} - 404427 T^{5} + 13886771 T^{6} - 91101243 T^{7} + 1491993611 T^{8} - 15578221158 T^{9} + 128844085584 T^{10} - 1984216560294 T^{11} + 11260776987123 T^{12} - 1984216560294 p T^{13} + 128844085584 p^{2} T^{14} - 15578221158 p^{3} T^{15} + 1491993611 p^{4} T^{16} - 91101243 p^{5} T^{17} + 13886771 p^{6} T^{18} - 404427 p^{7} T^{19} + 93212 p^{8} T^{20} - 1209 p^{9} T^{21} + 406 p^{10} T^{22} - 3 p^{11} T^{23} + p^{12} T^{24} \) | |
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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
−3.49109951150249281240096978747, −3.37527833821535965690534311782, −3.33796888647646719149177098271, −3.24062612781280193779829139892, −3.15668939565791682070067643918, −3.00300509953736237249361608329, −2.81248973037168621416710412182, −2.76581210648882862981858299235, −2.53464438834220998502540587092, −2.48759101245439122364118352947, −2.41761650688533262407628021225, −2.39305808741371959166309843252, −2.13700246779837290931008583285, −2.10826383204110396035777892323, −1.96979330899060420768622882537, −1.85584206817488073364070633813, −1.48468076077077028196617772411, −1.45972576202296593720686490771, −1.17185208351360676201559859086, −0.70366555032453540690329072153, −0.69385510279080677710331791503, −0.63969520050714243277928791517, −0.46236738645008438634653233665, −0.36813195881527378916700734866, −0.16941092966093580448331921964, 0.16941092966093580448331921964, 0.36813195881527378916700734866, 0.46236738645008438634653233665, 0.63969520050714243277928791517, 0.69385510279080677710331791503, 0.70366555032453540690329072153, 1.17185208351360676201559859086, 1.45972576202296593720686490771, 1.48468076077077028196617772411, 1.85584206817488073364070633813, 1.96979330899060420768622882537, 2.10826383204110396035777892323, 2.13700246779837290931008583285, 2.39305808741371959166309843252, 2.41761650688533262407628021225, 2.48759101245439122364118352947, 2.53464438834220998502540587092, 2.76581210648882862981858299235, 2.81248973037168621416710412182, 3.00300509953736237249361608329, 3.15668939565791682070067643918, 3.24062612781280193779829139892, 3.33796888647646719149177098271, 3.37527833821535965690534311782, 3.49109951150249281240096978747
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.