Dirichlet series
| L(s) = 1 | + 4·2-s + 2·3-s + 9·4-s + 8·6-s + 12·8-s + 2·9-s − 2·11-s + 18·12-s − 4·13-s + 9·16-s − 8·17-s + 8·18-s − 14·19-s − 8·22-s + 24·24-s − 16·26-s − 4·31-s − 4·32-s − 4·33-s − 32·34-s + 18·36-s + 8·37-s − 56·38-s − 8·39-s − 18·44-s + 8·47-s + 18·48-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 1.15·3-s + 9/2·4-s + 3.26·6-s + 4.24·8-s + 2/3·9-s − 0.603·11-s + 5.19·12-s − 1.10·13-s + 9/4·16-s − 1.94·17-s + 1.88·18-s − 3.21·19-s − 1.70·22-s + 4.89·24-s − 3.13·26-s − 0.718·31-s − 0.707·32-s − 0.696·33-s − 5.48·34-s + 3·36-s + 1.31·37-s − 9.08·38-s − 1.28·39-s − 2.71·44-s + 1.16·47-s + 2.59·48-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 5^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 5^{24}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{48} \cdot 5^{24}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.12731\times 10^{6}\) |
| Root analytic conductor: | \(1.78718\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 2^{48} \cdot 5^{24} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(8.746575483\) |
| \(L(\frac12)\) | \(\approx\) | \(8.746575483\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 - p^{2} T + 7 T^{2} - p^{2} T^{3} - p^{3} T^{4} + 3 p^{3} T^{5} - 19 p T^{6} + 3 p^{4} T^{7} - p^{5} T^{8} - p^{5} T^{9} + 7 p^{4} T^{10} - p^{7} T^{11} + p^{6} T^{12} \) |
| 5 | \( 1 \) | |
| good | 3 | \( 1 - 2 T + 2 T^{2} - p T^{4} + 4 p T^{5} - 2 p^{2} T^{6} + 74 T^{7} - 10 p^{2} T^{8} + 2 p^{2} T^{9} + 2 p^{4} T^{10} - 28 p^{2} T^{11} + 937 T^{12} - 28 p^{3} T^{13} + 2 p^{6} T^{14} + 2 p^{5} T^{15} - 10 p^{6} T^{16} + 74 p^{5} T^{17} - 2 p^{8} T^{18} + 4 p^{8} T^{19} - p^{9} T^{20} + 2 p^{10} T^{22} - 2 p^{11} T^{23} + p^{12} T^{24} \) |
| 7 | \( 1 - 44 T^{2} + 918 T^{4} - 12220 T^{6} + 119871 T^{8} - 967000 T^{10} + 7009300 T^{12} - 967000 p^{2} T^{14} + 119871 p^{4} T^{16} - 12220 p^{6} T^{18} + 918 p^{8} T^{20} - 44 p^{10} T^{22} + p^{12} T^{24} \) | |
| 11 | \( 1 + 2 T + 2 T^{2} - 72 T^{3} - 283 T^{4} + 52 p T^{5} + 4302 T^{6} + 19126 T^{7} - 29610 T^{8} - 266114 T^{9} - 469022 T^{10} + 1099492 T^{11} + 13555393 T^{12} + 1099492 p T^{13} - 469022 p^{2} T^{14} - 266114 p^{3} T^{15} - 29610 p^{4} T^{16} + 19126 p^{5} T^{17} + 4302 p^{6} T^{18} + 52 p^{8} T^{19} - 283 p^{8} T^{20} - 72 p^{9} T^{21} + 2 p^{10} T^{22} + 2 p^{11} T^{23} + p^{12} T^{24} \) | |
| 13 | \( 1 + 4 T + 8 T^{2} - 60 T^{3} + 46 T^{4} + 1540 T^{5} + 584 p T^{6} - 1052 T^{7} - 20945 T^{8} + 10312 p T^{9} + 9488 p^{2} T^{10} + 119240 p T^{11} + 1108324 T^{12} + 119240 p^{2} T^{13} + 9488 p^{4} T^{14} + 10312 p^{4} T^{15} - 20945 p^{4} T^{16} - 1052 p^{5} T^{17} + 584 p^{7} T^{18} + 1540 p^{7} T^{19} + 46 p^{8} T^{20} - 60 p^{9} T^{21} + 8 p^{10} T^{22} + 4 p^{11} T^{23} + p^{12} T^{24} \) | |
| 17 | \( ( 1 + 4 T + 53 T^{2} + 188 T^{3} + 1634 T^{4} + 5220 T^{5} + 33925 T^{6} + 5220 p T^{7} + 1634 p^{2} T^{8} + 188 p^{3} T^{9} + 53 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 19 | \( 1 + 14 T + 98 T^{2} + 520 T^{3} + 2181 T^{4} + 4980 T^{5} - 8818 T^{6} - 161206 T^{7} - 941002 T^{8} - 3498718 T^{9} - 8958718 T^{10} - 3269620 T^{11} + 76280161 T^{12} - 3269620 p T^{13} - 8958718 p^{2} T^{14} - 3498718 p^{3} T^{15} - 941002 p^{4} T^{16} - 161206 p^{5} T^{17} - 8818 p^{6} T^{18} + 4980 p^{7} T^{19} + 2181 p^{8} T^{20} + 520 p^{9} T^{21} + 98 p^{10} T^{22} + 14 p^{11} T^{23} + p^{12} T^{24} \) | |
| 23 | \( 1 - 4 p T^{2} + 194 p T^{4} - 176140 T^{6} + 6115855 T^{8} - 173125400 T^{10} + 4178547204 T^{12} - 173125400 p^{2} T^{14} + 6115855 p^{4} T^{16} - 176140 p^{6} T^{18} + 194 p^{9} T^{20} - 4 p^{11} T^{22} + p^{12} T^{24} \) | |
| 29 | \( 1 - 32 T^{3} - 914 T^{4} + 576 T^{5} + 512 T^{6} + 1120 p T^{7} + 1670127 T^{8} - 363616 T^{9} - 405504 T^{10} - 17708576 T^{11} - 898620444 T^{12} - 17708576 p T^{13} - 405504 p^{2} T^{14} - 363616 p^{3} T^{15} + 1670127 p^{4} T^{16} + 1120 p^{6} T^{17} + 512 p^{6} T^{18} + 576 p^{7} T^{19} - 914 p^{8} T^{20} - 32 p^{9} T^{21} + p^{12} T^{24} \) | |
| 31 | \( ( 1 + 2 T + 104 T^{2} + 2 p T^{3} + 5347 T^{4} - 156 T^{5} + 193360 T^{6} - 156 p T^{7} + 5347 p^{2} T^{8} + 2 p^{4} T^{9} + 104 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 37 | \( 1 - 8 T + 32 T^{2} + 216 T^{3} - 1454 T^{4} - 9736 T^{5} + 147744 T^{6} - 667944 T^{7} - 104049 T^{8} + 9992272 T^{9} + 41330112 T^{10} - 644241520 T^{11} + 6131713436 T^{12} - 644241520 p T^{13} + 41330112 p^{2} T^{14} + 9992272 p^{3} T^{15} - 104049 p^{4} T^{16} - 667944 p^{5} T^{17} + 147744 p^{6} T^{18} - 9736 p^{7} T^{19} - 1454 p^{8} T^{20} + 216 p^{9} T^{21} + 32 p^{10} T^{22} - 8 p^{11} T^{23} + p^{12} T^{24} \) | |
| 41 | \( 1 - 230 T^{2} + 28213 T^{4} - 2424754 T^{6} + 161424378 T^{8} - 8712385310 T^{10} + 390373428269 T^{12} - 8712385310 p^{2} T^{14} + 161424378 p^{4} T^{16} - 2424754 p^{6} T^{18} + 28213 p^{8} T^{20} - 230 p^{10} T^{22} + p^{12} T^{24} \) | |
| 43 | \( 1 + 128 T^{3} - 718 T^{4} - 12672 T^{5} + 8192 T^{6} - 789248 T^{7} - 4747697 T^{8} + 34261248 T^{9} - 14852096 T^{10} + 523527168 T^{11} + 11610577628 T^{12} + 523527168 p T^{13} - 14852096 p^{2} T^{14} + 34261248 p^{3} T^{15} - 4747697 p^{4} T^{16} - 789248 p^{5} T^{17} + 8192 p^{6} T^{18} - 12672 p^{7} T^{19} - 718 p^{8} T^{20} + 128 p^{9} T^{21} + p^{12} T^{24} \) | |
| 47 | \( ( 1 - 4 T + 146 T^{2} - 468 T^{3} + 11635 T^{4} - 31600 T^{5} + 634292 T^{6} - 31600 p T^{7} + 11635 p^{2} T^{8} - 468 p^{3} T^{9} + 146 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 53 | \( 1 + 16 T + 128 T^{2} + 1200 T^{3} + 5210 T^{4} - 30160 T^{5} - 429440 T^{6} - 5512176 T^{7} - 42537697 T^{8} - 34310496 T^{9} + 973441280 T^{10} + 17458522336 T^{11} + 202410048492 T^{12} + 17458522336 p T^{13} + 973441280 p^{2} T^{14} - 34310496 p^{3} T^{15} - 42537697 p^{4} T^{16} - 5512176 p^{5} T^{17} - 429440 p^{6} T^{18} - 30160 p^{7} T^{19} + 5210 p^{8} T^{20} + 1200 p^{9} T^{21} + 128 p^{10} T^{22} + 16 p^{11} T^{23} + p^{12} T^{24} \) | |
| 59 | \( 1 - 20 T + 200 T^{2} - 1892 T^{3} + 6866 T^{4} + 82788 T^{5} - 1239128 T^{6} + 13899380 T^{7} - 98937457 T^{8} + 169449240 T^{9} + 2035658960 T^{10} - 40572655368 T^{11} + 458024033436 T^{12} - 40572655368 p T^{13} + 2035658960 p^{2} T^{14} + 169449240 p^{3} T^{15} - 98937457 p^{4} T^{16} + 13899380 p^{5} T^{17} - 1239128 p^{6} T^{18} + 82788 p^{7} T^{19} + 6866 p^{8} T^{20} - 1892 p^{9} T^{21} + 200 p^{10} T^{22} - 20 p^{11} T^{23} + p^{12} T^{24} \) | |
| 61 | \( 1 - 4 T + 8 T^{2} - 172 T^{3} + 5938 T^{4} - 17388 T^{5} + 36840 T^{6} - 733732 T^{7} - 89637 p T^{8} + 50846808 T^{9} - 101027120 T^{10} + 3426409992 T^{11} - 107884090788 T^{12} + 3426409992 p T^{13} - 101027120 p^{2} T^{14} + 50846808 p^{3} T^{15} - 89637 p^{5} T^{16} - 733732 p^{5} T^{17} + 36840 p^{6} T^{18} - 17388 p^{7} T^{19} + 5938 p^{8} T^{20} - 172 p^{9} T^{21} + 8 p^{10} T^{22} - 4 p^{11} T^{23} + p^{12} T^{24} \) | |
| 67 | \( 1 - 50 T + 1250 T^{2} - 22736 T^{3} + 354061 T^{4} - 4890436 T^{5} + 60408398 T^{6} - 686964646 T^{7} + 7328407750 T^{8} - 73141121374 T^{9} + 685298903426 T^{10} - 6094145044748 T^{11} + 51370207376281 T^{12} - 6094145044748 p T^{13} + 685298903426 p^{2} T^{14} - 73141121374 p^{3} T^{15} + 7328407750 p^{4} T^{16} - 686964646 p^{5} T^{17} + 60408398 p^{6} T^{18} - 4890436 p^{7} T^{19} + 354061 p^{8} T^{20} - 22736 p^{9} T^{21} + 1250 p^{10} T^{22} - 50 p^{11} T^{23} + p^{12} T^{24} \) | |
| 71 | \( 1 - 596 T^{2} + 174954 T^{4} - 33295492 T^{6} + 4561416063 T^{8} - 472931841704 T^{10} + 38004556050028 T^{12} - 472931841704 p^{2} T^{14} + 4561416063 p^{4} T^{16} - 33295492 p^{6} T^{18} + 174954 p^{8} T^{20} - 596 p^{10} T^{22} + p^{12} T^{24} \) | |
| 73 | \( 1 - 538 T^{2} + 147485 T^{4} - 26955150 T^{6} + 3625911418 T^{8} - 376343620610 T^{10} + 30824747951477 T^{12} - 376343620610 p^{2} T^{14} + 3625911418 p^{4} T^{16} - 26955150 p^{6} T^{18} + 147485 p^{8} T^{20} - 538 p^{10} T^{22} + p^{12} T^{24} \) | |
| 79 | \( ( 1 - 6 T + 24 T^{2} + 1606 T^{3} - 2953 T^{4} - 3268 T^{5} + 1470256 T^{6} - 3268 p T^{7} - 2953 p^{2} T^{8} + 1606 p^{3} T^{9} + 24 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 83 | \( 1 + 2 T + 2 T^{2} - 328 T^{3} + 11541 T^{4} + 37420 T^{5} + 105550 T^{6} - 4780250 T^{7} - 17553450 T^{8} - 38041522 T^{9} + 1008921506 T^{10} - 33532735084 T^{11} - 653736173231 T^{12} - 33532735084 p T^{13} + 1008921506 p^{2} T^{14} - 38041522 p^{3} T^{15} - 17553450 p^{4} T^{16} - 4780250 p^{5} T^{17} + 105550 p^{6} T^{18} + 37420 p^{7} T^{19} + 11541 p^{8} T^{20} - 328 p^{9} T^{21} + 2 p^{10} T^{22} + 2 p^{11} T^{23} + p^{12} T^{24} \) | |
| 89 | \( 1 - 538 T^{2} + 147933 T^{4} - 27536782 T^{6} + 3885648250 T^{8} - 443968555266 T^{10} + 42748473802229 T^{12} - 443968555266 p^{2} T^{14} + 3885648250 p^{4} T^{16} - 27536782 p^{6} T^{18} + 147933 p^{8} T^{20} - 538 p^{10} T^{22} + p^{12} T^{24} \) | |
| 97 | \( ( 1 + 258 T^{2} + 1088 T^{3} + 29759 T^{4} + 253120 T^{5} + 2781052 T^{6} + 253120 p T^{7} + 29759 p^{2} T^{8} + 1088 p^{3} T^{9} + 258 p^{4} T^{10} + p^{6} T^{12} )^{2} \) | |
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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
−3.84428435021896002203205559320, −3.80020830423017680902026643815, −3.71579863359804070657718299467, −3.63105834799326307034136316902, −3.43095856593442772349631385634, −3.33387143943533306356645488013, −3.11469625764528404665319218582, −3.00364219663611210965892883271, −2.78255673275285316057354356820, −2.75729115148102497889878853465, −2.63127935758387584787797836777, −2.57090940656335871593130085736, −2.42707570903185804227806504943, −2.29791616688543587884940284555, −2.27030186678085329566053132776, −2.22740689906662645881754816442, −2.22126381900086882861725382774, −2.09720915865478400325619856065, −1.84707809157056461338657992796, −1.47824700453599774792846152482, −1.39459954823878975391345142607, −1.06811647502963565388532643658, −0.961020376227969477885819242782, −0.57669193101252604785343834824, −0.18905210829248421287678108721, 0.18905210829248421287678108721, 0.57669193101252604785343834824, 0.961020376227969477885819242782, 1.06811647502963565388532643658, 1.39459954823878975391345142607, 1.47824700453599774792846152482, 1.84707809157056461338657992796, 2.09720915865478400325619856065, 2.22126381900086882861725382774, 2.22740689906662645881754816442, 2.27030186678085329566053132776, 2.29791616688543587884940284555, 2.42707570903185804227806504943, 2.57090940656335871593130085736, 2.63127935758387584787797836777, 2.75729115148102497889878853465, 2.78255673275285316057354356820, 3.00364219663611210965892883271, 3.11469625764528404665319218582, 3.33387143943533306356645488013, 3.43095856593442772349631385634, 3.63105834799326307034136316902, 3.71579863359804070657718299467, 3.80020830423017680902026643815, 3.84428435021896002203205559320
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.