Dirichlet series
| L(s) = 1 | + 2·3-s − 12·7-s + 2·9-s + 2·11-s + 4·13-s − 14·19-s − 24·21-s − 12·23-s + 4·31-s + 4·33-s − 8·37-s + 8·39-s + 28·49-s − 16·53-s − 28·57-s + 20·59-s + 4·61-s − 24·63-s − 50·67-s − 24·69-s + 40·73-s − 24·77-s + 12·79-s − 81-s − 2·83-s − 48·91-s + 8·93-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 4.53·7-s + 2/3·9-s + 0.603·11-s + 1.10·13-s − 3.21·19-s − 5.23·21-s − 2.50·23-s + 0.718·31-s + 0.696·33-s − 1.31·37-s + 1.28·39-s + 4·49-s − 2.19·53-s − 3.70·57-s + 2.60·59-s + 0.512·61-s − 3.02·63-s − 6.10·67-s − 2.88·69-s + 4.68·73-s − 2.73·77-s + 1.35·79-s − 1/9·81-s − 0.219·83-s − 5.03·91-s + 0.829·93-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{72} \cdot 5^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{72} \cdot 5^{24}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{72} \cdot 5^{24}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.89132\times 10^{13}\) |
| Root analytic conductor: | \(3.57436\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 2^{72} \cdot 5^{24} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.08746575483\) |
| \(L(\frac12)\) | \(\approx\) | \(0.08746575483\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 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 | \( ( 1 + 6 T + 40 T^{2} + 170 T^{3} + 97 p T^{4} + 2148 T^{5} + 6288 T^{6} + 2148 p T^{7} + 97 p^{3} T^{8} + 170 p^{3} T^{9} + 40 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 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 | \( 1 - 90 T^{2} + 269 p T^{4} - 163950 T^{6} + 4537818 T^{8} - 101772290 T^{10} + 1892465141 T^{12} - 101772290 p^{2} T^{14} + 4537818 p^{4} T^{16} - 163950 p^{6} T^{18} + 269 p^{9} T^{20} - 90 p^{10} T^{22} + p^{12} T^{24} \) | |
| 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 | \( ( 1 + 6 T + 64 T^{2} + 362 T^{3} + 2355 T^{4} + 9868 T^{5} + 62080 T^{6} + 9868 p T^{7} + 2355 p^{2} T^{8} + 362 p^{3} T^{9} + 64 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 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 | \( 1 - 276 T^{2} + 40842 T^{4} - 4194180 T^{6} + 330530719 T^{8} - 20912321960 T^{10} + 1083311102636 T^{12} - 20912321960 p^{2} T^{14} + 330530719 p^{4} T^{16} - 4194180 p^{6} T^{18} + 40842 p^{8} T^{20} - 276 p^{10} T^{22} + p^{12} T^{24} \) | |
| 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 | \( ( 1 - 20 T + 469 T^{2} - 5676 T^{3} + 77282 T^{4} - 685396 T^{5} + 7048725 T^{6} - 685396 p T^{7} + 77282 p^{2} T^{8} - 5676 p^{3} T^{9} + 469 p^{4} T^{10} - 20 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 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 | \( 1 - 516 T^{2} + 126082 T^{4} - 19734004 T^{6} + 2329836655 T^{8} - 238188737288 T^{10} + 23260834196764 T^{12} - 238188737288 p^{2} T^{14} + 2329836655 p^{4} T^{16} - 19734004 p^{6} T^{18} + 126082 p^{8} T^{20} - 516 p^{10} T^{22} + p^{12} T^{24} \) | |
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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
−3.07468820213272397212264122418, −2.88666675471376490006339571695, −2.68888488479086327140892801430, −2.66529234325091548974960394004, −2.65265568736742455409684922352, −2.62242668607749000224286073951, −2.33445762673132776451860395137, −2.26048693967147118290024005660, −2.18980816910076260825881116346, −2.01675140941942830762297463908, −1.97035711402294519668305790717, −1.96785576757625421090635895161, −1.95283784222547811018843486380, −1.71710988995705557045789812862, −1.62964553079020196268941106914, −1.44310364564656867723493734719, −1.36540931063682102989564774489, −1.23752913406789116194745354591, −1.18776013242185260190915097868, −0.874795083498088180037686368775, −0.57041123053864873518910872855, −0.51936673615164126059253469438, −0.36132436477682076613212565828, −0.33708375924044294898250289078, −0.03513718367214094994801254941, 0.03513718367214094994801254941, 0.33708375924044294898250289078, 0.36132436477682076613212565828, 0.51936673615164126059253469438, 0.57041123053864873518910872855, 0.874795083498088180037686368775, 1.18776013242185260190915097868, 1.23752913406789116194745354591, 1.36540931063682102989564774489, 1.44310364564656867723493734719, 1.62964553079020196268941106914, 1.71710988995705557045789812862, 1.95283784222547811018843486380, 1.96785576757625421090635895161, 1.97035711402294519668305790717, 2.01675140941942830762297463908, 2.18980816910076260825881116346, 2.26048693967147118290024005660, 2.33445762673132776451860395137, 2.62242668607749000224286073951, 2.65265568736742455409684922352, 2.66529234325091548974960394004, 2.68888488479086327140892801430, 2.88666675471376490006339571695, 3.07468820213272397212264122418
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.