Dirichlet series
| L(s) = 1 | − 12·7-s + 46·9-s − 32·11-s − 12·17-s + 24·19-s + 4·23-s + 30·25-s − 176·43-s − 72·47-s − 234·49-s + 152·61-s − 552·63-s − 148·73-s + 384·77-s + 905·81-s − 208·83-s − 1.47e3·99-s − 48·101-s + 144·119-s − 356·121-s + 127-s + 131-s − 288·133-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 1.71·7-s + 46/9·9-s − 2.90·11-s − 0.705·17-s + 1.26·19-s + 4/23·23-s + 6/5·25-s − 4.09·43-s − 1.53·47-s − 4.77·49-s + 2.49·61-s − 8.76·63-s − 2.02·73-s + 4.98·77-s + 11.1·81-s − 2.50·83-s − 14.8·99-s − 0.475·101-s + 1.21·119-s − 2.94·121-s + 0.00787·127-s + 0.00763·131-s − 2.16·133-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{12} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{12} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s+1)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{24} \cdot 5^{12} \cdot 19^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.51853\times 10^{12}\) |
| Root analytic conductor: | \(3.21780\) |
| Motivic weight: | \(2\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 2^{24} \cdot 5^{12} \cdot 19^{12} ,\ ( \ : [1]^{12} ),\ 1 )\) |
Particular Values
| \(L(\frac{3}{2})\) | \(\approx\) | \(1.382301268\) |
| \(L(\frac12)\) | \(\approx\) | \(1.382301268\) |
| \(L(2)\) | 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 | \( ( 1 - p T^{2} )^{6} \) | |
| 19 | \( 1 - 24 T + 718 T^{2} - 11448 T^{3} + 16437 p T^{4} - 9072 p^{2} T^{5} + 10092 p^{3} T^{6} - 9072 p^{4} T^{7} + 16437 p^{5} T^{8} - 11448 p^{6} T^{9} + 718 p^{8} T^{10} - 24 p^{10} T^{11} + p^{12} T^{12} \) | |
| good | 3 | \( 1 - 46 T^{2} + 1211 T^{4} - 7502 p T^{6} + 108181 p T^{8} - 141176 p^{3} T^{10} + 37383802 T^{12} - 141176 p^{7} T^{14} + 108181 p^{9} T^{16} - 7502 p^{13} T^{18} + 1211 p^{16} T^{20} - 46 p^{20} T^{22} + p^{24} T^{24} \) |
| 7 | \( ( 1 + 6 T + 171 T^{2} + 146 p T^{3} + 13987 T^{4} + 1724 p^{2} T^{5} + 781538 T^{6} + 1724 p^{4} T^{7} + 13987 p^{4} T^{8} + 146 p^{7} T^{9} + 171 p^{8} T^{10} + 6 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 11 | \( ( 1 + 16 T + 562 T^{2} + 7024 T^{3} + 145599 T^{4} + 1444160 T^{5} + 22278332 T^{6} + 1444160 p^{2} T^{7} + 145599 p^{4} T^{8} + 7024 p^{6} T^{9} + 562 p^{8} T^{10} + 16 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 13 | \( 1 - 854 T^{2} + 355571 T^{4} - 100580898 T^{6} + 23037231551 T^{8} - 4679810654472 T^{10} + 845596713263626 T^{12} - 4679810654472 p^{4} T^{14} + 23037231551 p^{8} T^{16} - 100580898 p^{12} T^{18} + 355571 p^{16} T^{20} - 854 p^{20} T^{22} + p^{24} T^{24} \) | |
| 17 | \( ( 1 + 6 T + 1031 T^{2} + 4702 T^{3} + 551707 T^{4} + 2251356 T^{5} + 195760058 T^{6} + 2251356 p^{2} T^{7} + 551707 p^{4} T^{8} + 4702 p^{6} T^{9} + 1031 p^{8} T^{10} + 6 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 23 | \( ( 1 - 2 T + 1795 T^{2} + 570 p T^{3} + 1341731 T^{4} + 22855180 T^{5} + 710241586 T^{6} + 22855180 p^{2} T^{7} + 1341731 p^{4} T^{8} + 570 p^{7} T^{9} + 1795 p^{8} T^{10} - 2 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 29 | \( 1 - 3650 T^{2} + 7031031 T^{4} - 352583034 p T^{6} + 12632385843515 T^{8} - 13244160247464268 T^{10} + 11925028002350499994 T^{12} - 13244160247464268 p^{4} T^{14} + 12632385843515 p^{8} T^{16} - 352583034 p^{13} T^{18} + 7031031 p^{16} T^{20} - 3650 p^{20} T^{22} + p^{24} T^{24} \) | |
| 31 | \( 1 - 7628 T^{2} + 27743106 T^{4} - 64658387868 T^{6} + 109523415560687 T^{8} - 144392425190890648 T^{10} + \)\(15\!\cdots\!60\)\( T^{12} - 144392425190890648 p^{4} T^{14} + 109523415560687 p^{8} T^{16} - 64658387868 p^{12} T^{18} + 27743106 p^{16} T^{20} - 7628 p^{20} T^{22} + p^{24} T^{24} \) | |
| 37 | \( 1 - 7224 T^{2} + 22311206 T^{4} - 41653801768 T^{6} + 67313907267471 T^{8} - 120504737940372832 T^{10} + \)\(19\!\cdots\!16\)\( T^{12} - 120504737940372832 p^{4} T^{14} + 67313907267471 p^{8} T^{16} - 41653801768 p^{12} T^{18} + 22311206 p^{16} T^{20} - 7224 p^{20} T^{22} + p^{24} T^{24} \) | |
| 41 | \( 1 - 13052 T^{2} + 83172450 T^{4} - 343868365452 T^{6} + 1036466595892271 T^{8} - 2421432595426045432 T^{10} + \)\(45\!\cdots\!28\)\( T^{12} - 2421432595426045432 p^{4} T^{14} + 1036466595892271 p^{8} T^{16} - 343868365452 p^{12} T^{18} + 83172450 p^{16} T^{20} - 13052 p^{20} T^{22} + p^{24} T^{24} \) | |
| 43 | \( ( 1 + 88 T + 11554 T^{2} + 703000 T^{3} + 52665215 T^{4} + 2400265168 T^{5} + 128394586524 T^{6} + 2400265168 p^{2} T^{7} + 52665215 p^{4} T^{8} + 703000 p^{6} T^{9} + 11554 p^{8} T^{10} + 88 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 47 | \( ( 1 + 36 T + 5754 T^{2} + 209492 T^{3} + 22827791 T^{4} + 715807496 T^{5} + 57088342892 T^{6} + 715807496 p^{2} T^{7} + 22827791 p^{4} T^{8} + 209492 p^{6} T^{9} + 5754 p^{8} T^{10} + 36 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 53 | \( 1 - 5318 T^{2} + 31598387 T^{4} - 86434629282 T^{6} + 343060720940927 T^{8} - 795715150220512536 T^{10} + \)\(30\!\cdots\!98\)\( T^{12} - 795715150220512536 p^{4} T^{14} + 343060720940927 p^{8} T^{16} - 86434629282 p^{12} T^{18} + 31598387 p^{16} T^{20} - 5318 p^{20} T^{22} + p^{24} T^{24} \) | |
| 59 | \( 1 - 24650 T^{2} + 285179311 T^{4} - 2087819139786 T^{6} + 188485244049665 p T^{8} - 47526299717577560588 T^{10} + \)\(17\!\cdots\!14\)\( T^{12} - 47526299717577560588 p^{4} T^{14} + 188485244049665 p^{9} T^{16} - 2087819139786 p^{12} T^{18} + 285179311 p^{16} T^{20} - 24650 p^{20} T^{22} + p^{24} T^{24} \) | |
| 61 | \( ( 1 - 76 T + 19374 T^{2} - 1224700 T^{3} + 166922015 T^{4} - 8526705816 T^{5} + 807474236420 T^{6} - 8526705816 p^{2} T^{7} + 166922015 p^{4} T^{8} - 1224700 p^{6} T^{9} + 19374 p^{8} T^{10} - 76 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 67 | \( 1 - 34158 T^{2} + 557613355 T^{4} - 5911916807258 T^{6} + 46434301367345471 T^{8} - \)\(28\!\cdots\!08\)\( T^{10} + \)\(14\!\cdots\!98\)\( T^{12} - \)\(28\!\cdots\!08\)\( p^{4} T^{14} + 46434301367345471 p^{8} T^{16} - 5911916807258 p^{12} T^{18} + 557613355 p^{16} T^{20} - 34158 p^{20} T^{22} + p^{24} T^{24} \) | |
| 71 | \( 1 - 8332 T^{2} + 62516802 T^{4} - 631006741468 T^{6} + 3731375524654127 T^{8} - 21180920111267251224 T^{10} + \)\(12\!\cdots\!88\)\( T^{12} - 21180920111267251224 p^{4} T^{14} + 3731375524654127 p^{8} T^{16} - 631006741468 p^{12} T^{18} + 62516802 p^{16} T^{20} - 8332 p^{20} T^{22} + p^{24} T^{24} \) | |
| 73 | \( ( 1 + 74 T + 24871 T^{2} + 1567058 T^{3} + 280127067 T^{4} + 15035292164 T^{5} + 1881315041658 T^{6} + 15035292164 p^{2} T^{7} + 280127067 p^{4} T^{8} + 1567058 p^{6} T^{9} + 24871 p^{8} T^{10} + 74 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 79 | \( 1 - 25892 T^{2} + 436979650 T^{4} - 5345209743252 T^{6} + 52562354102437231 T^{8} - \)\(42\!\cdots\!12\)\( T^{10} + \)\(28\!\cdots\!48\)\( T^{12} - \)\(42\!\cdots\!12\)\( p^{4} T^{14} + 52562354102437231 p^{8} T^{16} - 5345209743252 p^{12} T^{18} + 436979650 p^{16} T^{20} - 25892 p^{20} T^{22} + p^{24} T^{24} \) | |
| 83 | \( ( 1 + 104 T + 29270 T^{2} + 2641096 T^{3} + 416887471 T^{4} + 31575918224 T^{5} + 3617318872852 T^{6} + 31575918224 p^{2} T^{7} + 416887471 p^{4} T^{8} + 2641096 p^{6} T^{9} + 29270 p^{8} T^{10} + 104 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 89 | \( 1 - 52276 T^{2} + 1383595746 T^{4} - 24522509395172 T^{6} + 325617115865436591 T^{8} - \)\(34\!\cdots\!68\)\( T^{10} + \)\(29\!\cdots\!36\)\( T^{12} - \)\(34\!\cdots\!68\)\( p^{4} T^{14} + 325617115865436591 p^{8} T^{16} - 24522509395172 p^{12} T^{18} + 1383595746 p^{16} T^{20} - 52276 p^{20} T^{22} + p^{24} T^{24} \) | |
| 97 | \( 1 - 57432 T^{2} + 1650944006 T^{4} - 32006220249672 T^{6} + 473513079638565487 T^{8} - \)\(57\!\cdots\!32\)\( T^{10} + \)\(58\!\cdots\!80\)\( T^{12} - \)\(57\!\cdots\!32\)\( p^{4} T^{14} + 473513079638565487 p^{8} T^{16} - 32006220249672 p^{12} T^{18} + 1650944006 p^{16} T^{20} - 57432 p^{20} T^{22} + p^{24} T^{24} \) | |
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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
−3.62800767102850370206424015137, −3.36147588083395745574807349395, −3.21741732459523443826975944264, −3.14821284056283946801554069772, −3.09029116173421565135866414185, −3.07270161922168967077122408111, −3.03241407441776207807047870855, −2.85845964860520055916338030690, −2.60756563798306434695245011606, −2.56826010512200650257849219027, −2.39713966989594173503183647104, −2.32461347737730889218970131506, −2.01969716935605406204912394824, −1.94759574677802967939741074008, −1.84621362475346705358555747328, −1.47023778932234442155924028788, −1.46694153188380593028113810142, −1.46608288577800976281416875639, −1.46531709373388902404073703841, −1.33061807840619958553860646237, −0.892487392393782729630419368534, −0.889868202891718338234430088653, −0.29616767026256703957019628561, −0.25563191469079571421240129357, −0.17409872884401585822080296139, 0.17409872884401585822080296139, 0.25563191469079571421240129357, 0.29616767026256703957019628561, 0.889868202891718338234430088653, 0.892487392393782729630419368534, 1.33061807840619958553860646237, 1.46531709373388902404073703841, 1.46608288577800976281416875639, 1.46694153188380593028113810142, 1.47023778932234442155924028788, 1.84621362475346705358555747328, 1.94759574677802967939741074008, 2.01969716935605406204912394824, 2.32461347737730889218970131506, 2.39713966989594173503183647104, 2.56826010512200650257849219027, 2.60756563798306434695245011606, 2.85845964860520055916338030690, 3.03241407441776207807047870855, 3.07270161922168967077122408111, 3.09029116173421565135866414185, 3.14821284056283946801554069772, 3.21741732459523443826975944264, 3.36147588083395745574807349395, 3.62800767102850370206424015137
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.