Dirichlet series
| L(s) = 1 | − 28·4-s − 152·9-s − 124·11-s + 302·16-s − 228·19-s − 1.03e3·29-s − 592·31-s + 4.25e3·36-s − 712·41-s + 3.47e3·44-s − 2.78e3·49-s − 2.74e3·59-s − 28·61-s − 1.46e3·64-s − 1.61e3·71-s + 6.38e3·76-s − 976·79-s + 1.05e4·81-s − 5.60e3·89-s + 1.88e4·99-s − 6.47e3·101-s − 5.34e3·109-s + 2.90e4·116-s − 1.62e3·121-s + 1.65e4·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 7/2·4-s − 5.62·9-s − 3.39·11-s + 4.71·16-s − 2.75·19-s − 6.63·29-s − 3.42·31-s + 19.7·36-s − 2.71·41-s + 11.8·44-s − 8.11·49-s − 6.04·59-s − 0.0587·61-s − 2.86·64-s − 2.70·71-s + 9.63·76-s − 1.38·79-s + 14.4·81-s − 6.67·89-s + 19.1·99-s − 6.38·101-s − 4.69·109-s + 23.2·116-s − 1.22·121-s + 12.0·124-s + 0.000698·127-s + 0.000666·131-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{24} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{24} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(5^{24} \cdot 19^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.34809\times 10^{17}\) |
| Root analytic conductor: | \(5.29395\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(12\) |
| Selberg data: | \((24,\ 5^{24} \cdot 19^{12} ,\ ( \ : [3/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{5}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 5 | \( 1 \) |
| 19 | \( ( 1 + p T )^{12} \) | |
| good | 2 | \( 1 + 7 p^{2} T^{2} + 241 p T^{4} + 1627 p^{2} T^{6} + 69773 T^{8} + 81241 p^{3} T^{10} + 86017 p^{6} T^{12} + 81241 p^{9} T^{14} + 69773 p^{12} T^{16} + 1627 p^{20} T^{18} + 241 p^{25} T^{20} + 7 p^{32} T^{22} + p^{36} T^{24} \) |
| 3 | \( 1 + 152 T^{2} + 4201 p T^{4} + 737578 T^{6} + 33408739 T^{8} + 44947706 p^{3} T^{10} + 36093753634 T^{12} + 44947706 p^{9} T^{14} + 33408739 p^{12} T^{16} + 737578 p^{18} T^{18} + 4201 p^{25} T^{20} + 152 p^{30} T^{22} + p^{36} T^{24} \) | |
| 7 | \( 1 + 2782 T^{2} + 3485607 T^{4} + 2612456934 T^{6} + 1334832787003 T^{8} + 523973795384092 T^{10} + 182874689287329978 T^{12} + 523973795384092 p^{6} T^{14} + 1334832787003 p^{12} T^{16} + 2612456934 p^{18} T^{18} + 3485607 p^{24} T^{20} + 2782 p^{30} T^{22} + p^{36} T^{24} \) | |
| 11 | \( ( 1 + 62 T + 6580 T^{2} + 279470 T^{3} + 18159895 T^{4} + 613838232 T^{5} + 30323939160 T^{6} + 613838232 p^{3} T^{7} + 18159895 p^{6} T^{8} + 279470 p^{9} T^{9} + 6580 p^{12} T^{10} + 62 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 13 | \( 1 + 5612 T^{2} + 22575795 T^{4} + 79257761118 T^{6} + 227375072069499 T^{8} + 574124821353766742 T^{10} + \)\(13\!\cdots\!70\)\( T^{12} + 574124821353766742 p^{6} T^{14} + 227375072069499 p^{12} T^{16} + 79257761118 p^{18} T^{18} + 22575795 p^{24} T^{20} + 5612 p^{30} T^{22} + p^{36} T^{24} \) | |
| 17 | \( 1 + 29586 T^{2} + 412979983 T^{4} + 3684916493242 T^{6} + 24304598483080395 T^{8} + \)\(13\!\cdots\!28\)\( T^{10} + \)\(66\!\cdots\!02\)\( T^{12} + \)\(13\!\cdots\!28\)\( p^{6} T^{14} + 24304598483080395 p^{12} T^{16} + 3684916493242 p^{18} T^{18} + 412979983 p^{24} T^{20} + 29586 p^{30} T^{22} + p^{36} T^{24} \) | |
| 23 | \( 1 + 94182 T^{2} + 4344004791 T^{4} + 130407416441190 T^{6} + 2857011095832392555 T^{8} + \)\(48\!\cdots\!56\)\( T^{10} + \)\(65\!\cdots\!86\)\( T^{12} + \)\(48\!\cdots\!56\)\( p^{6} T^{14} + 2857011095832392555 p^{12} T^{16} + 130407416441190 p^{18} T^{18} + 4344004791 p^{24} T^{20} + 94182 p^{30} T^{22} + p^{36} T^{24} \) | |
| 29 | \( ( 1 + 518 T + 222799 T^{2} + 65095602 T^{3} + 16196706251 T^{4} + 111108190384 p T^{5} + 554054293296298 T^{6} + 111108190384 p^{4} T^{7} + 16196706251 p^{6} T^{8} + 65095602 p^{9} T^{9} + 222799 p^{12} T^{10} + 518 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 31 | \( ( 1 + 296 T + 60572 T^{2} + 9415656 T^{3} + 2659473895 T^{4} + 432612007040 T^{5} + 80928011847800 T^{6} + 432612007040 p^{3} T^{7} + 2659473895 p^{6} T^{8} + 9415656 p^{9} T^{9} + 60572 p^{12} T^{10} + 296 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 37 | \( 1 + 257254 T^{2} + 32881632666 T^{4} + 2858586787547134 T^{6} + \)\(19\!\cdots\!99\)\( T^{8} + \)\(11\!\cdots\!44\)\( T^{10} + \)\(59\!\cdots\!88\)\( T^{12} + \)\(11\!\cdots\!44\)\( p^{6} T^{14} + \)\(19\!\cdots\!99\)\( p^{12} T^{16} + 2858586787547134 p^{18} T^{18} + 32881632666 p^{24} T^{20} + 257254 p^{30} T^{22} + p^{36} T^{24} \) | |
| 41 | \( ( 1 + 356 T + 297044 T^{2} + 67986500 T^{3} + 36915882167 T^{4} + 6449806138680 T^{5} + 2976796357109384 T^{6} + 6449806138680 p^{3} T^{7} + 36915882167 p^{6} T^{8} + 67986500 p^{9} T^{9} + 297044 p^{12} T^{10} + 356 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 43 | \( 1 + 559152 T^{2} + 154868418926 T^{4} + 28225816016931824 T^{6} + \)\(38\!\cdots\!03\)\( T^{8} + \)\(40\!\cdots\!24\)\( T^{10} + \)\(35\!\cdots\!40\)\( T^{12} + \)\(40\!\cdots\!24\)\( p^{6} T^{14} + \)\(38\!\cdots\!03\)\( p^{12} T^{16} + 28225816016931824 p^{18} T^{18} + 154868418926 p^{24} T^{20} + 559152 p^{30} T^{22} + p^{36} T^{24} \) | |
| 47 | \( 1 + 544220 T^{2} + 177405577218 T^{4} + 40572151454383596 T^{6} + \)\(71\!\cdots\!67\)\( T^{8} + \)\(10\!\cdots\!80\)\( T^{10} + \)\(11\!\cdots\!88\)\( T^{12} + \)\(10\!\cdots\!80\)\( p^{6} T^{14} + \)\(71\!\cdots\!67\)\( p^{12} T^{16} + 40572151454383596 p^{18} T^{18} + 177405577218 p^{24} T^{20} + 544220 p^{30} T^{22} + p^{36} T^{24} \) | |
| 53 | \( 1 + 714884 T^{2} + 243694635571 T^{4} + 54745292756146766 T^{6} + \)\(10\!\cdots\!59\)\( T^{8} + \)\(34\!\cdots\!06\)\( p T^{10} + \)\(29\!\cdots\!38\)\( T^{12} + \)\(34\!\cdots\!06\)\( p^{7} T^{14} + \)\(10\!\cdots\!59\)\( p^{12} T^{16} + 54745292756146766 p^{18} T^{18} + 243694635571 p^{24} T^{20} + 714884 p^{30} T^{22} + p^{36} T^{24} \) | |
| 59 | \( ( 1 + 1370 T + 1334749 T^{2} + 1088793082 T^{3} + 714448941611 T^{4} + 400648972161208 T^{5} + 196670469700493998 T^{6} + 400648972161208 p^{3} T^{7} + 714448941611 p^{6} T^{8} + 1088793082 p^{9} T^{9} + 1334749 p^{12} T^{10} + 1370 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 61 | \( ( 1 + 14 T + 651032 T^{2} - 109635070 T^{3} + 241412848127 T^{4} - 42477997355144 T^{5} + 68563037856130720 T^{6} - 42477997355144 p^{3} T^{7} + 241412848127 p^{6} T^{8} - 109635070 p^{9} T^{9} + 651032 p^{12} T^{10} + 14 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 67 | \( 1 + 1882032 T^{2} + 1901065949867 T^{4} + 1318077207189216170 T^{6} + \)\(69\!\cdots\!23\)\( T^{8} + \)\(28\!\cdots\!66\)\( T^{10} + \)\(95\!\cdots\!38\)\( T^{12} + \)\(28\!\cdots\!66\)\( p^{6} T^{14} + \)\(69\!\cdots\!23\)\( p^{12} T^{16} + 1318077207189216170 p^{18} T^{18} + 1901065949867 p^{24} T^{20} + 1882032 p^{30} T^{22} + p^{36} T^{24} \) | |
| 71 | \( ( 1 + 808 T + 987392 T^{2} + 583513584 T^{3} + 620310785415 T^{4} + 293824601495912 T^{5} + 236511794090688336 T^{6} + 293824601495912 p^{3} T^{7} + 620310785415 p^{6} T^{8} + 583513584 p^{9} T^{9} + 987392 p^{12} T^{10} + 808 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 73 | \( 1 + 2399554 T^{2} + 3062285951711 T^{4} + 2673641094139452586 T^{6} + \)\(17\!\cdots\!19\)\( T^{8} + \)\(93\!\cdots\!88\)\( T^{10} + \)\(39\!\cdots\!98\)\( T^{12} + \)\(93\!\cdots\!88\)\( p^{6} T^{14} + \)\(17\!\cdots\!19\)\( p^{12} T^{16} + 2673641094139452586 p^{18} T^{18} + 3062285951711 p^{24} T^{20} + 2399554 p^{30} T^{22} + p^{36} T^{24} \) | |
| 79 | \( ( 1 + 488 T + 1927814 T^{2} + 983829416 T^{3} + 1922642704511 T^{4} + 885012893225728 T^{5} + 1170719795948197428 T^{6} + 885012893225728 p^{3} T^{7} + 1922642704511 p^{6} T^{8} + 983829416 p^{9} T^{9} + 1927814 p^{12} T^{10} + 488 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 83 | \( 1 + 5543340 T^{2} + 14626779373122 T^{4} + 24314947992479211420 T^{6} + \)\(28\!\cdots\!23\)\( T^{8} + \)\(24\!\cdots\!40\)\( T^{10} + \)\(16\!\cdots\!08\)\( T^{12} + \)\(24\!\cdots\!40\)\( p^{6} T^{14} + \)\(28\!\cdots\!23\)\( p^{12} T^{16} + 24314947992479211420 p^{18} T^{18} + 14626779373122 p^{24} T^{20} + 5543340 p^{30} T^{22} + p^{36} T^{24} \) | |
| 89 | \( ( 1 + 2804 T + 4771572 T^{2} + 6309927972 T^{3} + 7225629724919 T^{4} + 7326024850162696 T^{5} + 6525277049522332296 T^{6} + 7326024850162696 p^{3} T^{7} + 7225629724919 p^{6} T^{8} + 6309927972 p^{9} T^{9} + 4771572 p^{12} T^{10} + 2804 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 97 | \( 1 + 7241934 T^{2} + 25787394237290 T^{4} + 59760729384247583894 T^{6} + \)\(10\!\cdots\!79\)\( T^{8} + \)\(13\!\cdots\!16\)\( T^{10} + \)\(13\!\cdots\!80\)\( T^{12} + \)\(13\!\cdots\!16\)\( p^{6} T^{14} + \)\(10\!\cdots\!79\)\( p^{12} T^{16} + 59760729384247583894 p^{18} T^{18} + 25787394237290 p^{24} T^{20} + 7241934 p^{30} T^{22} + p^{36} T^{24} \) | |
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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
−3.75617374476556042989071934839, −3.63672399568653210585836797872, −3.38950163306814623598565189820, −3.37325287004279962784774830735, −3.32668345085185914450063554478, −3.30478783989741412212792098395, −3.23162895289540028789051502591, −3.12558884821172284772839932742, −3.00676543378691775499885290338, −2.86442024617978160193288240676, −2.75607913316559697298687914163, −2.56548721045061460450304206882, −2.51817401938462246802370332410, −2.50530221543413593643689283893, −2.48028551464549649692694879659, −2.13503272822294266081561151108, −2.02053237465976973813466812157, −1.85604864798487855958539687426, −1.71718035997069763015625609429, −1.71034187740750230240358829007, −1.65534439536014920915323103209, −1.31942172527313957903874924436, −1.24992480209233941420098872649, −1.21139803813977552569464266411, −1.11939735880213571589229845783, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1.11939735880213571589229845783, 1.21139803813977552569464266411, 1.24992480209233941420098872649, 1.31942172527313957903874924436, 1.65534439536014920915323103209, 1.71034187740750230240358829007, 1.71718035997069763015625609429, 1.85604864798487855958539687426, 2.02053237465976973813466812157, 2.13503272822294266081561151108, 2.48028551464549649692694879659, 2.50530221543413593643689283893, 2.51817401938462246802370332410, 2.56548721045061460450304206882, 2.75607913316559697298687914163, 2.86442024617978160193288240676, 3.00676543378691775499885290338, 3.12558884821172284772839932742, 3.23162895289540028789051502591, 3.30478783989741412212792098395, 3.32668345085185914450063554478, 3.37325287004279962784774830735, 3.38950163306814623598565189820, 3.63672399568653210585836797872, 3.75617374476556042989071934839
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.