Dirichlet series
| L(s) = 1 | + 2·2-s − 2·3-s + 6·4-s + 5-s − 4·6-s + 9·7-s + 10·8-s − 19·9-s + 2·10-s − 8·11-s − 12·12-s − 2·13-s + 18·14-s − 2·15-s + 22·16-s + 5·17-s − 38·18-s + 2·19-s + 6·20-s − 18·21-s − 16·22-s − 23-s − 20·24-s + 19·25-s − 4·26-s + 40·27-s + 54·28-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1.15·3-s + 3·4-s + 0.447·5-s − 1.63·6-s + 3.40·7-s + 3.53·8-s − 6.33·9-s + 0.632·10-s − 2.41·11-s − 3.46·12-s − 0.554·13-s + 4.81·14-s − 0.516·15-s + 11/2·16-s + 1.21·17-s − 8.95·18-s + 0.458·19-s + 1.34·20-s − 3.92·21-s − 3.41·22-s − 0.208·23-s − 4.08·24-s + 19/5·25-s − 0.784·26-s + 7.69·27-s + 10.2·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{12} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{12} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(7^{12} \cdot 13^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(0.0216681\) |
| Root analytic conductor: | \(0.852431\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 7^{12} \cdot 13^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(1.326283944\) |
| \(L(\frac12)\) | \(\approx\) | \(1.326283944\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 7 | \( 1 - 9 T + 6 p T^{2} - 99 T^{3} - 15 T^{4} + 1026 T^{5} - 3971 T^{6} + 1026 p T^{7} - 15 p^{2} T^{8} - 99 p^{3} T^{9} + 6 p^{5} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \) |
| 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 | \( ( 1 + T + 11 T^{2} + 11 T^{3} + 19 p T^{4} + 55 T^{5} + 197 T^{6} + 55 p T^{7} + 19 p^{3} T^{8} + 11 p^{3} T^{9} + 11 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 5 | \( 1 - T - 18 T^{2} - p T^{3} + 193 T^{4} + 192 T^{5} - 1181 T^{6} - 2139 T^{7} + 908 p T^{8} + 451 p^{2} T^{9} - 6679 T^{10} - 26266 T^{11} + 249 T^{12} - 26266 p T^{13} - 6679 p^{2} T^{14} + 451 p^{5} T^{15} + 908 p^{5} T^{16} - 2139 p^{5} T^{17} - 1181 p^{6} T^{18} + 192 p^{7} T^{19} + 193 p^{8} T^{20} - p^{10} T^{21} - 18 p^{10} T^{22} - p^{11} T^{23} + p^{12} T^{24} \) | |
| 11 | \( ( 1 + 4 T + 45 T^{2} + 144 T^{3} + 972 T^{4} + 2539 T^{5} + 13237 T^{6} + 2539 p T^{7} + 972 p^{2} T^{8} + 144 p^{3} T^{9} + 45 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 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 | \( ( 1 - T + 50 T^{2} + 16 T^{3} + 1180 T^{4} + 1175 T^{5} + 23331 T^{6} + 1175 p T^{7} + 1180 p^{2} T^{8} + 16 p^{3} T^{9} + 50 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 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 | \( 1 - 16 T + 20 T^{2} + 594 T^{3} + 2163 T^{4} - 43649 T^{5} - 56125 T^{6} + 1282696 T^{7} + 2984747 T^{8} - 22743273 T^{9} - 180497697 T^{10} + 655302586 T^{11} + 2182678017 T^{12} + 655302586 p T^{13} - 180497697 p^{2} T^{14} - 22743273 p^{3} T^{15} + 2984747 p^{4} T^{16} + 1282696 p^{5} T^{17} - 56125 p^{6} T^{18} - 43649 p^{7} T^{19} + 2163 p^{8} T^{20} + 594 p^{9} T^{21} + 20 p^{10} T^{22} - 16 p^{11} T^{23} + p^{12} T^{24} \) | |
| 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 | \( 1 + T - 104 T^{2} - 189 T^{3} + 5335 T^{4} + 164 p T^{5} - 69863 T^{6} + 514255 T^{7} - 7627520 T^{8} - 55687467 T^{9} + 662939941 T^{10} + 1686387922 T^{11} - 35399065407 T^{12} + 1686387922 p T^{13} + 662939941 p^{2} T^{14} - 55687467 p^{3} T^{15} - 7627520 p^{4} T^{16} + 514255 p^{5} T^{17} - 69863 p^{6} T^{18} + 164 p^{8} T^{19} + 5335 p^{8} T^{20} - 189 p^{9} T^{21} - 104 p^{10} T^{22} + p^{11} T^{23} + p^{12} T^{24} \) | |
| 53 | \( 1 + 2 T - 214 T^{2} - 252 T^{3} + 24796 T^{4} + 13772 T^{5} - 1921862 T^{6} + 82142 T^{7} + 113089342 T^{8} - 43114584 T^{9} - 5653831794 T^{10} + 1443208718 T^{11} + 285781391787 T^{12} + 1443208718 p T^{13} - 5653831794 p^{2} T^{14} - 43114584 p^{3} T^{15} + 113089342 p^{4} T^{16} + 82142 p^{5} T^{17} - 1921862 p^{6} T^{18} + 13772 p^{7} T^{19} + 24796 p^{8} T^{20} - 252 p^{9} T^{21} - 214 p^{10} T^{22} + 2 p^{11} T^{23} + p^{12} T^{24} \) | |
| 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 | \( ( 1 - 5 T + 165 T^{2} - 599 T^{3} + 15743 T^{4} - 53393 T^{5} + 1179159 T^{6} - 53393 p T^{7} + 15743 p^{2} T^{8} - 599 p^{3} T^{9} + 165 p^{4} T^{10} - 5 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 67 | \( ( 1 - 11 T + 296 T^{2} - 2796 T^{3} + 41197 T^{4} - 326168 T^{5} + 3447813 T^{6} - 326168 p T^{7} + 41197 p^{2} T^{8} - 2796 p^{3} T^{9} + 296 p^{4} T^{10} - 11 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 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 | \( 1 + 30 T + 224 T^{2} - 1118 T^{3} - 5021 T^{4} + 290169 T^{5} + 1854677 T^{6} - 9817892 T^{7} - 27971653 T^{8} + 688598777 T^{9} - 1819010273 T^{10} - 20701972840 T^{11} + 235631264151 T^{12} - 20701972840 p T^{13} - 1819010273 p^{2} T^{14} + 688598777 p^{3} T^{15} - 27971653 p^{4} T^{16} - 9817892 p^{5} T^{17} + 1854677 p^{6} T^{18} + 290169 p^{7} T^{19} - 5021 p^{8} T^{20} - 1118 p^{9} T^{21} + 224 p^{10} T^{22} + 30 p^{11} T^{23} + p^{12} T^{24} \) | |
| 79 | \( 1 - 7 T - 277 T^{2} + 2628 T^{3} + 34995 T^{4} - 387429 T^{5} - 3070086 T^{6} + 26237658 T^{7} + 339376855 T^{8} - 565746882 T^{9} - 45365142063 T^{10} - 8895648284 T^{11} + 4474615429807 T^{12} - 8895648284 p T^{13} - 45365142063 p^{2} T^{14} - 565746882 p^{3} T^{15} + 339376855 p^{4} T^{16} + 26237658 p^{5} T^{17} - 3070086 p^{6} T^{18} - 387429 p^{7} T^{19} + 34995 p^{8} T^{20} + 2628 p^{9} T^{21} - 277 p^{10} T^{22} - 7 p^{11} T^{23} + p^{12} T^{24} \) | |
| 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
−5.31339248127785475489640391337, −5.28910117630892829738891696677, −5.24979365640528738254496654193, −5.07076087141044920224967328556, −4.89389388606952577196881471963, −4.86070233394752450882833618282, −4.59994424967326469857119611745, −4.36439657250420829042629980134, −4.35703602722561885347661362994, −4.13412758953910039249696777954, −3.96115763005030510270935471274, −3.49160821166836662000830581435, −3.47056810512680249336675206530, −3.24472151618467011588974359923, −3.11928989381995533090059555544, −3.03989103486242207375010228791, −2.81605100482040806701727653512, −2.80729778271699423842019959130, −2.66162183724471948812018366829, −2.57410502882543459336409725158, −2.37414593614358078020455315650, −2.19310979310703892056007369531, −1.83464653610401243271167157870, −1.34663010083685641620956609761, −1.27123782086708242007570856389, 1.27123782086708242007570856389, 1.34663010083685641620956609761, 1.83464653610401243271167157870, 2.19310979310703892056007369531, 2.37414593614358078020455315650, 2.57410502882543459336409725158, 2.66162183724471948812018366829, 2.80729778271699423842019959130, 2.81605100482040806701727653512, 3.03989103486242207375010228791, 3.11928989381995533090059555544, 3.24472151618467011588974359923, 3.47056810512680249336675206530, 3.49160821166836662000830581435, 3.96115763005030510270935471274, 4.13412758953910039249696777954, 4.35703602722561885347661362994, 4.36439657250420829042629980134, 4.59994424967326469857119611745, 4.86070233394752450882833618282, 4.89389388606952577196881471963, 5.07076087141044920224967328556, 5.24979365640528738254496654193, 5.28910117630892829738891696677, 5.31339248127785475489640391337
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.