Dirichlet series
| L(s) = 1 | + 2·2-s + 3-s + 6·4-s − 2·5-s + 2·6-s + 10·8-s + 11·9-s − 4·10-s + 4·11-s + 6·12-s − 2·13-s − 2·15-s + 22·16-s + 5·17-s + 22·18-s − 19-s − 12·20-s + 8·22-s − 23-s + 10·24-s − 35·25-s − 4·26-s + 10·27-s + 3·29-s − 4·30-s − 32·31-s + 34·32-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 0.577·3-s + 3·4-s − 0.894·5-s + 0.816·6-s + 3.53·8-s + 11/3·9-s − 1.26·10-s + 1.20·11-s + 1.73·12-s − 0.554·13-s − 0.516·15-s + 11/2·16-s + 1.21·17-s + 5.18·18-s − 0.229·19-s − 2.68·20-s + 1.70·22-s − 0.208·23-s + 2.04·24-s − 7·25-s − 0.784·26-s + 1.92·27-s + 0.557·29-s − 0.730·30-s − 5.74·31-s + 6.01·32-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
| \(L(1)\) | \(\approx\) | \(19.67318521\) |
| \(L(\frac12)\) | \(\approx\) | \(19.67318521\) |
| \(L(\frac{3}{2})\) | 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 | \( 1 + 2 T - 16 T^{2} + 3 T^{3} + 607 T^{4} + 433 T^{5} - 5615 T^{6} + 433 p T^{7} + 607 p^{2} T^{8} + 3 p^{3} T^{9} - 16 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \) | |
| good | 2 | \( 1 - p T - p T^{2} + 3 p T^{3} - p T^{4} - p T^{5} + 7 T^{6} - p T^{7} - 7 p T^{8} + 3 p T^{9} - 5 p^{2} T^{10} - 5 p^{2} T^{11} + 153 T^{12} - 5 p^{3} T^{13} - 5 p^{4} T^{14} + 3 p^{4} T^{15} - 7 p^{5} T^{16} - p^{6} T^{17} + 7 p^{6} T^{18} - p^{8} T^{19} - p^{9} T^{20} + 3 p^{10} T^{21} - p^{11} T^{22} - p^{12} T^{23} + p^{12} T^{24} \) |
| 3 | \( 1 - T - 10 T^{2} + 11 T^{3} + 53 T^{4} - 62 T^{5} - 167 T^{6} + 221 T^{7} + 98 p T^{8} - 535 T^{9} + 79 T^{10} + 604 T^{11} - 1559 T^{12} + 604 p T^{13} + 79 p^{2} T^{14} - 535 p^{3} T^{15} + 98 p^{5} T^{16} + 221 p^{5} T^{17} - 167 p^{6} T^{18} - 62 p^{7} T^{19} + 53 p^{8} T^{20} + 11 p^{9} T^{21} - 10 p^{10} T^{22} - p^{11} T^{23} + p^{12} T^{24} \) | |
| 5 | \( ( 1 + T + 19 T^{2} + 7 T^{3} + 161 T^{4} - 3 T^{5} + 913 T^{6} - 3 p T^{7} + 161 p^{2} T^{8} + 7 p^{3} T^{9} + 19 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 11 | \( 1 - 4 T - 29 T^{2} + 108 T^{3} + 477 T^{4} - 113 p T^{5} - 6686 T^{6} + 7665 T^{7} + 89323 T^{8} + 423 T^{9} - 1282040 T^{10} - 249219 T^{11} + 16505087 T^{12} - 249219 p T^{13} - 1282040 p^{2} T^{14} + 423 p^{3} T^{15} + 89323 p^{4} T^{16} + 7665 p^{5} T^{17} - 6686 p^{6} T^{18} - 113 p^{8} T^{19} + 477 p^{8} T^{20} + 108 p^{9} T^{21} - 29 p^{10} T^{22} - 4 p^{11} T^{23} + p^{12} T^{24} \) | |
| 17 | \( 1 - 5 T - 65 T^{2} + 372 T^{3} + 2506 T^{4} - 15344 T^{5} - 62063 T^{6} + 395128 T^{7} + 1158376 T^{8} - 6526599 T^{9} - 17123414 T^{10} + 46016896 T^{11} + 268434807 T^{12} + 46016896 p T^{13} - 17123414 p^{2} T^{14} - 6526599 p^{3} T^{15} + 1158376 p^{4} T^{16} + 395128 p^{5} T^{17} - 62063 p^{6} T^{18} - 15344 p^{7} T^{19} + 2506 p^{8} T^{20} + 372 p^{9} T^{21} - 65 p^{10} T^{22} - 5 p^{11} T^{23} + p^{12} T^{24} \) | |
| 19 | \( 1 + T - 49 T^{2} + 82 T^{3} + 1336 T^{4} - 4335 T^{5} - 10907 T^{6} + 99626 T^{7} - 263580 T^{8} - 1110690 T^{9} + 9684539 T^{10} + 2414194 T^{11} - 215227743 T^{12} + 2414194 p T^{13} + 9684539 p^{2} T^{14} - 1110690 p^{3} T^{15} - 263580 p^{4} T^{16} + 99626 p^{5} T^{17} - 10907 p^{6} T^{18} - 4335 p^{7} T^{19} + 1336 p^{8} T^{20} + 82 p^{9} T^{21} - 49 p^{10} T^{22} + p^{11} T^{23} + p^{12} T^{24} \) | |
| 23 | \( 1 + T - 31 T^{2} - 72 T^{3} - 242 T^{4} + 619 T^{5} + 6343 T^{6} + 28360 T^{7} + 121372 T^{8} - 9024 p T^{9} + 5143119 T^{10} - 4729028 T^{11} - 273608319 T^{12} - 4729028 p T^{13} + 5143119 p^{2} T^{14} - 9024 p^{4} T^{15} + 121372 p^{4} T^{16} + 28360 p^{5} T^{17} + 6343 p^{6} T^{18} + 619 p^{7} T^{19} - 242 p^{8} T^{20} - 72 p^{9} T^{21} - 31 p^{10} T^{22} + p^{11} T^{23} + p^{12} T^{24} \) | |
| 29 | \( 1 - 3 T - 3 p T^{2} + 94 T^{3} + 158 p T^{4} + 904 T^{5} - 123467 T^{6} - 308042 T^{7} + 1215550 T^{8} + 10832851 T^{9} + 73693658 T^{10} - 174959016 T^{11} - 3324017493 T^{12} - 174959016 p T^{13} + 73693658 p^{2} T^{14} + 10832851 p^{3} T^{15} + 1215550 p^{4} T^{16} - 308042 p^{5} T^{17} - 123467 p^{6} T^{18} + 904 p^{7} T^{19} + 158 p^{9} T^{20} + 94 p^{9} T^{21} - 3 p^{11} T^{22} - 3 p^{11} T^{23} + p^{12} T^{24} \) | |
| 31 | \( ( 1 + 16 T + 236 T^{2} + 2185 T^{3} + 18573 T^{4} + 122325 T^{5} + 755039 T^{6} + 122325 p T^{7} + 18573 p^{2} T^{8} + 2185 p^{3} T^{9} + 236 p^{4} T^{10} + 16 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 37 | \( 1 + 13 T - 15 T^{2} + 284 T^{3} + 12996 T^{4} + 18401 T^{5} - 116147 T^{6} + 5523346 T^{7} + 19538810 T^{8} - 71463812 T^{9} + 1452640399 T^{10} + 7689412934 T^{11} - 18842100883 T^{12} + 7689412934 p T^{13} + 1452640399 p^{2} T^{14} - 71463812 p^{3} T^{15} + 19538810 p^{4} T^{16} + 5523346 p^{5} T^{17} - 116147 p^{6} T^{18} + 18401 p^{7} T^{19} + 12996 p^{8} T^{20} + 284 p^{9} T^{21} - 15 p^{10} T^{22} + 13 p^{11} T^{23} + p^{12} T^{24} \) | |
| 41 | \( 1 + 8 T - 161 T^{2} - 924 T^{3} + 18241 T^{4} + 64367 T^{5} - 1502654 T^{6} - 3175261 T^{7} + 96068491 T^{8} + 105301221 T^{9} - 5078164754 T^{10} - 1647875431 T^{11} + 226350132753 T^{12} - 1647875431 p T^{13} - 5078164754 p^{2} T^{14} + 105301221 p^{3} T^{15} + 96068491 p^{4} T^{16} - 3175261 p^{5} T^{17} - 1502654 p^{6} T^{18} + 64367 p^{7} T^{19} + 18241 p^{8} T^{20} - 924 p^{9} T^{21} - 161 p^{10} T^{22} + 8 p^{11} T^{23} + p^{12} T^{24} \) | |
| 43 | \( 1 + 11 T - 138 T^{2} - 1349 T^{3} + 16370 T^{4} + 106653 T^{5} - 1472431 T^{6} - 5757651 T^{7} + 106708219 T^{8} + 224797058 T^{9} - 6088028976 T^{10} - 3777766292 T^{11} + 288640495545 T^{12} - 3777766292 p T^{13} - 6088028976 p^{2} T^{14} + 224797058 p^{3} T^{15} + 106708219 p^{4} T^{16} - 5757651 p^{5} T^{17} - 1472431 p^{6} T^{18} + 106653 p^{7} T^{19} + 16370 p^{8} T^{20} - 1349 p^{9} T^{21} - 138 p^{10} T^{22} + 11 p^{11} T^{23} + p^{12} T^{24} \) | |
| 47 | \( ( 1 - T + 105 T^{2} - 147 T^{3} + 5543 T^{4} - 3359 T^{5} + 246951 T^{6} - 3359 p T^{7} + 5543 p^{2} T^{8} - 147 p^{3} T^{9} + 105 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 53 | \( ( 1 - 2 T + 218 T^{2} - 344 T^{3} + 22040 T^{4} - 26940 T^{5} + 1409201 T^{6} - 26940 p T^{7} + 22040 p^{2} T^{8} - 344 p^{3} T^{9} + 218 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 59 | \( 1 - 13 T - 126 T^{2} + 1843 T^{3} + 11161 T^{4} - 119322 T^{5} - 1337447 T^{6} + 7367025 T^{7} + 123366322 T^{8} - 382593671 T^{9} - 9090177085 T^{10} + 8156701016 T^{11} + 592237594305 T^{12} + 8156701016 p T^{13} - 9090177085 p^{2} T^{14} - 382593671 p^{3} T^{15} + 123366322 p^{4} T^{16} + 7367025 p^{5} T^{17} - 1337447 p^{6} T^{18} - 119322 p^{7} T^{19} + 11161 p^{8} T^{20} + 1843 p^{9} T^{21} - 126 p^{10} T^{22} - 13 p^{11} T^{23} + p^{12} T^{24} \) | |
| 61 | \( 1 + 5 T - 140 T^{2} - 373 T^{3} + 8487 T^{4} - 5202 T^{5} - 147441 T^{6} + 963135 T^{7} - 4711566 T^{8} - 13690661 T^{9} - 1296684385 T^{10} - 689962304 T^{11} + 162150963097 T^{12} - 689962304 p T^{13} - 1296684385 p^{2} T^{14} - 13690661 p^{3} T^{15} - 4711566 p^{4} T^{16} + 963135 p^{5} T^{17} - 147441 p^{6} T^{18} - 5202 p^{7} T^{19} + 8487 p^{8} T^{20} - 373 p^{9} T^{21} - 140 p^{10} T^{22} + 5 p^{11} T^{23} + p^{12} T^{24} \) | |
| 67 | \( 1 + 11 T - 175 T^{2} - 2336 T^{3} + 15663 T^{4} + 247450 T^{5} - 15954 p T^{6} - 18125445 T^{7} + 60512732 T^{8} + 977936543 T^{9} - 2490157221 T^{10} - 26393757979 T^{11} + 95373451231 T^{12} - 26393757979 p T^{13} - 2490157221 p^{2} T^{14} + 977936543 p^{3} T^{15} + 60512732 p^{4} T^{16} - 18125445 p^{5} T^{17} - 15954 p^{7} T^{18} + 247450 p^{7} T^{19} + 15663 p^{8} T^{20} - 2336 p^{9} T^{21} - 175 p^{10} T^{22} + 11 p^{11} T^{23} + p^{12} T^{24} \) | |
| 71 | \( 1 - 6 T - 249 T^{2} + 278 T^{3} + 39793 T^{4} + 68141 T^{5} - 3761552 T^{6} - 15648583 T^{7} + 241594531 T^{8} + 1275513473 T^{9} - 10122739162 T^{10} - 44683203723 T^{11} + 523547364015 T^{12} - 44683203723 p T^{13} - 10122739162 p^{2} T^{14} + 1275513473 p^{3} T^{15} + 241594531 p^{4} T^{16} - 15648583 p^{5} T^{17} - 3761552 p^{6} T^{18} + 68141 p^{7} T^{19} + 39793 p^{8} T^{20} + 278 p^{9} T^{21} - 249 p^{10} T^{22} - 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 73 | \( ( 1 - 30 T + 676 T^{2} - 10699 T^{3} + 141027 T^{4} - 1519265 T^{5} + 14149139 T^{6} - 1519265 p T^{7} + 141027 p^{2} T^{8} - 10699 p^{3} T^{9} + 676 p^{4} T^{10} - 30 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 79 | \( ( 1 + 7 T + 326 T^{2} + 2455 T^{3} + 54096 T^{4} + 344443 T^{5} + 5474643 T^{6} + 344443 p T^{7} + 54096 p^{2} T^{8} + 2455 p^{3} T^{9} + 326 p^{4} T^{10} + 7 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 83 | \( ( 1 + 27 T + 656 T^{2} + 10802 T^{3} + 153994 T^{4} + 1760871 T^{5} + 17670883 T^{6} + 1760871 p T^{7} + 153994 p^{2} T^{8} + 10802 p^{3} T^{9} + 656 p^{4} T^{10} + 27 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 89 | \( 1 - 4 T - 151 T^{2} + 2628 T^{3} + 262 T^{4} - 309046 T^{5} + 2802769 T^{6} + 6034970 T^{7} - 281402495 T^{8} + 1666391304 T^{9} + 5367237150 T^{10} - 95837476354 T^{11} + 701675320941 T^{12} - 95837476354 p T^{13} + 5367237150 p^{2} T^{14} + 1666391304 p^{3} T^{15} - 281402495 p^{4} T^{16} + 6034970 p^{5} T^{17} + 2802769 p^{6} T^{18} - 309046 p^{7} T^{19} + 262 p^{8} T^{20} + 2628 p^{9} T^{21} - 151 p^{10} T^{22} - 4 p^{11} T^{23} + p^{12} T^{24} \) | |
| 97 | \( 1 + 35 T + 278 T^{2} - 3177 T^{3} - 20496 T^{4} + 1111333 T^{5} + 13328183 T^{6} - 54713297 T^{7} - 1182920923 T^{8} + 11775176076 T^{9} + 251445486222 T^{10} + 186060844192 T^{11} - 17274836413101 T^{12} + 186060844192 p T^{13} + 251445486222 p^{2} T^{14} + 11775176076 p^{3} T^{15} - 1182920923 p^{4} T^{16} - 54713297 p^{5} T^{17} + 13328183 p^{6} T^{18} + 1111333 p^{7} T^{19} - 20496 p^{8} T^{20} - 3177 p^{9} T^{21} + 278 p^{10} T^{22} + 35 p^{11} T^{23} + p^{12} T^{24} \) | |
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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
−3.65184674771719494053593861782, −3.48434163207391919639476741067, −3.40862682353548383573994079784, −3.26612599083269979103031227625, −3.04892806531106091612971625469, −2.93681885303542192752732179998, −2.86929364820579453002392010866, −2.77443250128593687561481278172, −2.68869827197297701579190980266, −2.59819125750010852674556210931, −2.34330626996258981422204234691, −2.09719734668459183692693502641, −1.94699979096427311493647825960, −1.92974246710099264969435630888, −1.91465519376068749623184461451, −1.85744123356645939367478951197, −1.83244579021385370850900693684, −1.57132682732571764607067727754, −1.54423023462123840947950044016, −1.44152213732022047672273748379, −1.42096509711317658192033369289, −1.11186390317145017501669058865, −0.67176075289715635544913054514, −0.40731117100996054017637362686, −0.25451494206916139887286050188, 0.25451494206916139887286050188, 0.40731117100996054017637362686, 0.67176075289715635544913054514, 1.11186390317145017501669058865, 1.42096509711317658192033369289, 1.44152213732022047672273748379, 1.54423023462123840947950044016, 1.57132682732571764607067727754, 1.83244579021385370850900693684, 1.85744123356645939367478951197, 1.91465519376068749623184461451, 1.92974246710099264969435630888, 1.94699979096427311493647825960, 2.09719734668459183692693502641, 2.34330626996258981422204234691, 2.59819125750010852674556210931, 2.68869827197297701579190980266, 2.77443250128593687561481278172, 2.86929364820579453002392010866, 2.93681885303542192752732179998, 3.04892806531106091612971625469, 3.26612599083269979103031227625, 3.40862682353548383573994079784, 3.48434163207391919639476741067, 3.65184674771719494053593861782
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.