Dirichlet series
| L(s) = 1 | + 16·4-s + 27·9-s + 64·16-s + 112·19-s + 250·25-s + 432·36-s + 220·37-s + 286·49-s − 880·67-s − 1.07e4·73-s + 1.79e3·76-s − 9.72e3·79-s + 4.00e3·100-s + 3.64e3·103-s + 2.66e3·121-s + 127-s + 131-s + 137-s + 139-s + 1.72e3·144-s + 3.52e3·148-s + 149-s + 151-s + 157-s + 163-s + 167-s − 506·169-s + ⋯ |
| L(s) = 1 | + 2·4-s + 9-s + 16-s + 1.35·19-s + 2·25-s + 2·36-s + 0.977·37-s + 0.833·49-s − 1.60·67-s − 17.1·73-s + 2.70·76-s − 13.8·79-s + 4·100-s + 3.48·103-s + 2·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 144-s + 1.95·148-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s − 0.230·169-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{20} \cdot 67^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{20} \cdot 67^{20}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(40\) |
| Conductor: | \(3^{20} \cdot 67^{20}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.02860\times 10^{21}\) |
| Root analytic conductor: | \(3.44374\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((40,\ 3^{20} \cdot 67^{20} ,\ ( \ : [3/2]^{20} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(\approx\) | \(0.1110098972\) |
| \(L(\frac12)\) | \(\approx\) | \(0.1110098972\) |
| \(L(\frac{5}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 - p^{3} T^{2} + p^{6} T^{4} - p^{9} T^{6} + p^{12} T^{8} - p^{15} T^{10} + p^{18} T^{12} - p^{21} T^{14} + p^{24} T^{16} - p^{27} T^{18} + p^{30} T^{20} \) |
| 67 | \( 1 + 880 T + 473637 T^{2} + 152129120 T^{3} - 8578859431 T^{4} - 53304206817840 T^{5} - 44327498500653344 T^{6} - 22976265525420933120 T^{7} - \)\(68\!\cdots\!56\)\( T^{8} + \)\(84\!\cdots\!56\)\( T^{9} + \)\(28\!\cdots\!52\)\( T^{10} + \)\(84\!\cdots\!56\)\( p^{3} T^{11} - \)\(68\!\cdots\!56\)\( p^{6} T^{12} - 22976265525420933120 p^{9} T^{13} - 44327498500653344 p^{12} T^{14} - 53304206817840 p^{15} T^{15} - 8578859431 p^{18} T^{16} + 152129120 p^{21} T^{17} + 473637 p^{24} T^{18} + 880 p^{27} T^{19} + p^{30} T^{20} \) | |
| good | 2 | \( ( 1 - p^{3} T^{2} + p^{6} T^{4} - p^{9} T^{6} + p^{12} T^{8} - p^{15} T^{10} + p^{18} T^{12} - p^{21} T^{14} + p^{24} T^{16} - p^{27} T^{18} + p^{30} T^{20} )^{2} \) |
| 5 | \( ( 1 - p^{3} T^{2} + p^{6} T^{4} - p^{9} T^{6} + p^{12} T^{8} - p^{15} T^{10} + p^{18} T^{12} - p^{21} T^{14} + p^{24} T^{16} - p^{27} T^{18} + p^{30} T^{20} )^{2} \) | |
| 7 | \( ( 1 - 20 T + 57 T^{2} + 5720 T^{3} - 133951 T^{4} + 717060 T^{5} + 31603993 T^{6} - 878031440 T^{7} + 6720459201 T^{8} + 166755599900 T^{9} - 5640229503943 T^{10} + 166755599900 p^{3} T^{11} + 6720459201 p^{6} T^{12} - 878031440 p^{9} T^{13} + 31603993 p^{12} T^{14} + 717060 p^{15} T^{15} - 133951 p^{18} T^{16} + 5720 p^{21} T^{17} + 57 p^{24} T^{18} - 20 p^{27} T^{19} + p^{30} T^{20} )( 1 + 20 T + 57 T^{2} - 5720 T^{3} - 133951 T^{4} - 717060 T^{5} + 31603993 T^{6} + 878031440 T^{7} + 6720459201 T^{8} - 166755599900 T^{9} - 5640229503943 T^{10} - 166755599900 p^{3} T^{11} + 6720459201 p^{6} T^{12} + 878031440 p^{9} T^{13} + 31603993 p^{12} T^{14} - 717060 p^{15} T^{15} - 133951 p^{18} T^{16} - 5720 p^{21} T^{17} + 57 p^{24} T^{18} + 20 p^{27} T^{19} + p^{30} T^{20} ) \) | |
| 11 | \( ( 1 - p^{2} T + 5 p^{3} T^{2} - p^{5} T^{3} - p^{6} T^{4} + p^{8} T^{5} - p^{9} T^{6} - p^{11} T^{7} + 5 p^{12} T^{8} - p^{14} T^{9} + p^{15} T^{10} )^{2}( 1 + p^{2} T + 5 p^{3} T^{2} + p^{5} T^{3} - p^{6} T^{4} - p^{8} T^{5} - p^{9} T^{6} + p^{11} T^{7} + 5 p^{12} T^{8} + p^{14} T^{9} + p^{15} T^{10} )^{2} \) | |
| 13 | \( ( 1 - 70 T + 2703 T^{2} - 35420 T^{3} - 3459091 T^{4} + 319954110 T^{5} - 14797164773 T^{6} + 332862354440 T^{7} + 9209006195481 T^{8} - 1375929026388350 T^{9} + 76082845235712743 T^{10} - 1375929026388350 p^{3} T^{11} + 9209006195481 p^{6} T^{12} + 332862354440 p^{9} T^{13} - 14797164773 p^{12} T^{14} + 319954110 p^{15} T^{15} - 3459091 p^{18} T^{16} - 35420 p^{21} T^{17} + 2703 p^{24} T^{18} - 70 p^{27} T^{19} + p^{30} T^{20} )( 1 + 70 T + 2703 T^{2} + 35420 T^{3} - 3459091 T^{4} - 319954110 T^{5} - 14797164773 T^{6} - 332862354440 T^{7} + 9209006195481 T^{8} + 1375929026388350 T^{9} + 76082845235712743 T^{10} + 1375929026388350 p^{3} T^{11} + 9209006195481 p^{6} T^{12} - 332862354440 p^{9} T^{13} - 14797164773 p^{12} T^{14} - 319954110 p^{15} T^{15} - 3459091 p^{18} T^{16} + 35420 p^{21} T^{17} + 2703 p^{24} T^{18} + 70 p^{27} T^{19} + p^{30} T^{20} ) \) | |
| 17 | \( ( 1 + p^{3} T^{2} + p^{6} T^{4} + p^{9} T^{6} + p^{12} T^{8} + p^{15} T^{10} + p^{18} T^{12} + p^{21} T^{14} + p^{24} T^{16} + p^{27} T^{18} + p^{30} T^{20} )^{2} \) | |
| 19 | \( ( 1 - 56 T - 3723 T^{2} + 592592 T^{3} - 7649095 T^{4} - 3636239208 T^{5} + 256094538253 T^{6} + 10599670585504 T^{7} - 2350133990665551 T^{8} + 58904362931298920 T^{9} + 12820924717822274789 T^{10} + 58904362931298920 p^{3} T^{11} - 2350133990665551 p^{6} T^{12} + 10599670585504 p^{9} T^{13} + 256094538253 p^{12} T^{14} - 3636239208 p^{15} T^{15} - 7649095 p^{18} T^{16} + 592592 p^{21} T^{17} - 3723 p^{24} T^{18} - 56 p^{27} T^{19} + p^{30} T^{20} )^{2} \) | |
| 23 | \( ( 1 + p^{3} T^{2} + p^{6} T^{4} + p^{9} T^{6} + p^{12} T^{8} + p^{15} T^{10} + p^{18} T^{12} + p^{21} T^{14} + p^{24} T^{16} + p^{27} T^{18} + p^{30} T^{20} )^{2} \) | |
| 29 | \( ( 1 - p^{3} T^{2} )^{20} \) | |
| 31 | \( ( 1 - 308 T + 65073 T^{2} - 10866856 T^{3} + 1408401905 T^{4} - 110053279644 T^{5} - 8061291021503 T^{6} + 5761474888497328 T^{7} - 1534380344835581151 T^{8} + \)\(30\!\cdots\!60\)\( T^{9} - \)\(46\!\cdots\!39\)\( T^{10} + \)\(30\!\cdots\!60\)\( p^{3} T^{11} - 1534380344835581151 p^{6} T^{12} + 5761474888497328 p^{9} T^{13} - 8061291021503 p^{12} T^{14} - 110053279644 p^{15} T^{15} + 1408401905 p^{18} T^{16} - 10866856 p^{21} T^{17} + 65073 p^{24} T^{18} - 308 p^{27} T^{19} + p^{30} T^{20} )( 1 + 308 T + 65073 T^{2} + 10866856 T^{3} + 1408401905 T^{4} + 110053279644 T^{5} - 8061291021503 T^{6} - 5761474888497328 T^{7} - 1534380344835581151 T^{8} - \)\(30\!\cdots\!60\)\( T^{9} - \)\(46\!\cdots\!39\)\( T^{10} - \)\(30\!\cdots\!60\)\( p^{3} T^{11} - 1534380344835581151 p^{6} T^{12} - 5761474888497328 p^{9} T^{13} - 8061291021503 p^{12} T^{14} + 110053279644 p^{15} T^{15} + 1408401905 p^{18} T^{16} + 10866856 p^{21} T^{17} + 65073 p^{24} T^{18} + 308 p^{27} T^{19} + p^{30} T^{20} ) \) | |
| 37 | \( ( 1 - 110 T - 38553 T^{2} + 9812660 T^{3} + 873432509 T^{4} - 593118242970 T^{5} + 21001029848323 T^{6} + 27733105077843880 T^{7} - 4114406723469931719 T^{8} - \)\(95\!\cdots\!50\)\( T^{9} + \)\(31\!\cdots\!07\)\( T^{10} - \)\(95\!\cdots\!50\)\( p^{3} T^{11} - 4114406723469931719 p^{6} T^{12} + 27733105077843880 p^{9} T^{13} + 21001029848323 p^{12} T^{14} - 593118242970 p^{15} T^{15} + 873432509 p^{18} T^{16} + 9812660 p^{21} T^{17} - 38553 p^{24} T^{18} - 110 p^{27} T^{19} + p^{30} T^{20} )^{2} \) | |
| 41 | \( ( 1 - p^{3} T^{2} + p^{6} T^{4} - p^{9} T^{6} + p^{12} T^{8} - p^{15} T^{10} + p^{18} T^{12} - p^{21} T^{14} + p^{24} T^{16} - p^{27} T^{18} + p^{30} T^{20} )^{2} \) | |
| 43 | \( ( 1 - 520 T + 190893 T^{2} - 57920720 T^{3} + 14941444649 T^{4} - 3164448532440 T^{5} + 457563797160757 T^{6} + 13662634945113440 T^{7} - 43484094992319295599 T^{8} + \)\(21\!\cdots\!00\)\( T^{9} - \)\(77\!\cdots\!07\)\( T^{10} + \)\(21\!\cdots\!00\)\( p^{3} T^{11} - 43484094992319295599 p^{6} T^{12} + 13662634945113440 p^{9} T^{13} + 457563797160757 p^{12} T^{14} - 3164448532440 p^{15} T^{15} + 14941444649 p^{18} T^{16} - 57920720 p^{21} T^{17} + 190893 p^{24} T^{18} - 520 p^{27} T^{19} + p^{30} T^{20} )( 1 + 520 T + 190893 T^{2} + 57920720 T^{3} + 14941444649 T^{4} + 3164448532440 T^{5} + 457563797160757 T^{6} - 13662634945113440 T^{7} - 43484094992319295599 T^{8} - \)\(21\!\cdots\!00\)\( T^{9} - \)\(77\!\cdots\!07\)\( T^{10} - \)\(21\!\cdots\!00\)\( p^{3} T^{11} - 43484094992319295599 p^{6} T^{12} - 13662634945113440 p^{9} T^{13} + 457563797160757 p^{12} T^{14} + 3164448532440 p^{15} T^{15} + 14941444649 p^{18} T^{16} + 57920720 p^{21} T^{17} + 190893 p^{24} T^{18} + 520 p^{27} T^{19} + p^{30} T^{20} ) \) | |
| 47 | \( ( 1 + p^{3} T^{2} + p^{6} T^{4} + p^{9} T^{6} + p^{12} T^{8} + p^{15} T^{10} + p^{18} T^{12} + p^{21} T^{14} + p^{24} T^{16} + p^{27} T^{18} + p^{30} T^{20} )^{2} \) | |
| 53 | \( ( 1 - p^{3} T^{2} + p^{6} T^{4} - p^{9} T^{6} + p^{12} T^{8} - p^{15} T^{10} + p^{18} T^{12} - p^{21} T^{14} + p^{24} T^{16} - p^{27} T^{18} + p^{30} T^{20} )^{2} \) | |
| 59 | \( ( 1 + p^{3} T^{2} + p^{6} T^{4} + p^{9} T^{6} + p^{12} T^{8} + p^{15} T^{10} + p^{18} T^{12} + p^{21} T^{14} + p^{24} T^{16} + p^{27} T^{18} + p^{30} T^{20} )^{2} \) | |
| 61 | \( ( 1 - 182 T - 193857 T^{2} + 76592516 T^{3} + 30062017805 T^{4} - 22856333114706 T^{5} - 2663654236520213 T^{6} + 5672738417755761352 T^{7} - \)\(42\!\cdots\!11\)\( T^{8} - \)\(12\!\cdots\!10\)\( T^{9} + \)\(31\!\cdots\!11\)\( T^{10} - \)\(12\!\cdots\!10\)\( p^{3} T^{11} - \)\(42\!\cdots\!11\)\( p^{6} T^{12} + 5672738417755761352 p^{9} T^{13} - 2663654236520213 p^{12} T^{14} - 22856333114706 p^{15} T^{15} + 30062017805 p^{18} T^{16} + 76592516 p^{21} T^{17} - 193857 p^{24} T^{18} - 182 p^{27} T^{19} + p^{30} T^{20} )( 1 + 182 T - 193857 T^{2} - 76592516 T^{3} + 30062017805 T^{4} + 22856333114706 T^{5} - 2663654236520213 T^{6} - 5672738417755761352 T^{7} - \)\(42\!\cdots\!11\)\( T^{8} + \)\(12\!\cdots\!10\)\( T^{9} + \)\(31\!\cdots\!11\)\( T^{10} + \)\(12\!\cdots\!10\)\( p^{3} T^{11} - \)\(42\!\cdots\!11\)\( p^{6} T^{12} - 5672738417755761352 p^{9} T^{13} - 2663654236520213 p^{12} T^{14} + 22856333114706 p^{15} T^{15} + 30062017805 p^{18} T^{16} - 76592516 p^{21} T^{17} - 193857 p^{24} T^{18} + 182 p^{27} T^{19} + p^{30} T^{20} ) \) | |
| 71 | \( ( 1 + p^{3} T^{2} + p^{6} T^{4} + p^{9} T^{6} + p^{12} T^{8} + p^{15} T^{10} + p^{18} T^{12} + p^{21} T^{14} + p^{24} T^{16} + p^{27} T^{18} + p^{30} T^{20} )^{2} \) | |
| 73 | \( ( 1 + 1190 T + p^{3} T^{2} )^{10}( 1 - 1190 T + 1027083 T^{2} - 759298540 T^{3} + 504012515189 T^{4} - 304394852939730 T^{5} + 166160438376999487 T^{6} - 79316149162574444120 T^{7} + \)\(29\!\cdots\!21\)\( T^{8} - \)\(45\!\cdots\!50\)\( T^{9} - \)\(61\!\cdots\!57\)\( T^{10} - \)\(45\!\cdots\!50\)\( p^{3} T^{11} + \)\(29\!\cdots\!21\)\( p^{6} T^{12} - 79316149162574444120 p^{9} T^{13} + 166160438376999487 p^{12} T^{14} - 304394852939730 p^{15} T^{15} + 504012515189 p^{18} T^{16} - 759298540 p^{21} T^{17} + 1027083 p^{24} T^{18} - 1190 p^{27} T^{19} + p^{30} T^{20} ) \) | |
| 79 | \( ( 1 + 884 T + p^{3} T^{2} )^{10}( 1 + 884 T + 288417 T^{2} - 180885848 T^{3} - 302103918895 T^{4} - 177876086691108 T^{5} - 8293446566867567 T^{6} + 80368441140986267984 T^{7} + \)\(75\!\cdots\!69\)\( T^{8} + \)\(26\!\cdots\!20\)\( T^{9} - \)\(13\!\cdots\!11\)\( T^{10} + \)\(26\!\cdots\!20\)\( p^{3} T^{11} + \)\(75\!\cdots\!69\)\( p^{6} T^{12} + 80368441140986267984 p^{9} T^{13} - 8293446566867567 p^{12} T^{14} - 177876086691108 p^{15} T^{15} - 302103918895 p^{18} T^{16} - 180885848 p^{21} T^{17} + 288417 p^{24} T^{18} + 884 p^{27} T^{19} + p^{30} T^{20} ) \) | |
| 83 | \( ( 1 + p^{3} T^{2} + p^{6} T^{4} + p^{9} T^{6} + p^{12} T^{8} + p^{15} T^{10} + p^{18} T^{12} + p^{21} T^{14} + p^{24} T^{16} + p^{27} T^{18} + p^{30} T^{20} )^{2} \) | |
| 89 | \( ( 1 + p^{3} T^{2} + p^{6} T^{4} + p^{9} T^{6} + p^{12} T^{8} + p^{15} T^{10} + p^{18} T^{12} + p^{21} T^{14} + p^{24} T^{16} + p^{27} T^{18} + p^{30} T^{20} )^{2} \) | |
| 97 | \( ( 1 - 1330 T + 856227 T^{2} + 75073180 T^{3} - 881302594171 T^{4} + 1103615185837290 T^{5} - 663467114633766617 T^{6} - \)\(12\!\cdots\!60\)\( T^{7} + \)\(77\!\cdots\!41\)\( T^{8} - \)\(91\!\cdots\!50\)\( T^{9} + \)\(50\!\cdots\!07\)\( T^{10} - \)\(91\!\cdots\!50\)\( p^{3} T^{11} + \)\(77\!\cdots\!41\)\( p^{6} T^{12} - \)\(12\!\cdots\!60\)\( p^{9} T^{13} - 663467114633766617 p^{12} T^{14} + 1103615185837290 p^{15} T^{15} - 881302594171 p^{18} T^{16} + 75073180 p^{21} T^{17} + 856227 p^{24} T^{18} - 1330 p^{27} T^{19} + p^{30} T^{20} )( 1 + 1330 T + 856227 T^{2} - 75073180 T^{3} - 881302594171 T^{4} - 1103615185837290 T^{5} - 663467114633766617 T^{6} + \)\(12\!\cdots\!60\)\( T^{7} + \)\(77\!\cdots\!41\)\( T^{8} + \)\(91\!\cdots\!50\)\( T^{9} + \)\(50\!\cdots\!07\)\( T^{10} + \)\(91\!\cdots\!50\)\( p^{3} T^{11} + \)\(77\!\cdots\!41\)\( p^{6} T^{12} + \)\(12\!\cdots\!60\)\( p^{9} T^{13} - 663467114633766617 p^{12} T^{14} - 1103615185837290 p^{15} T^{15} - 881302594171 p^{18} T^{16} - 75073180 p^{21} T^{17} + 856227 p^{24} T^{18} + 1330 p^{27} T^{19} + p^{30} T^{20} ) \) | |
| show more | ||
| show less | ||
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{40} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.59395385249878428307768240487, −2.54577392593775490111731448181, −2.41824532651759082946038680030, −2.37316305895919780205707680650, −2.27909549878923813820335644152, −1.94110494646768899267088129419, −1.77620702362142540897702498093, −1.77195057412478957422992056795, −1.77087436685935864981232633078, −1.71841465154480833360097771709, −1.70237081698586924321910898859, −1.68866184438205825505810645228, −1.55409744144064674399809167321, −1.46121951314796661695581790896, −1.28896032737097182130050409148, −1.17950517494264521580499045623, −1.01883740211686542060530639828, −1.00607380950368537451596985180, −0.942727064246679946033507193319, −0.73296003576768113360046697963, −0.61066264076637889210987216359, −0.49224247725420449901742509842, −0.12866323792005689870661749180, −0.11274515509818615284182103754, −0.03476846108436118780732045018, 0.03476846108436118780732045018, 0.11274515509818615284182103754, 0.12866323792005689870661749180, 0.49224247725420449901742509842, 0.61066264076637889210987216359, 0.73296003576768113360046697963, 0.942727064246679946033507193319, 1.00607380950368537451596985180, 1.01883740211686542060530639828, 1.17950517494264521580499045623, 1.28896032737097182130050409148, 1.46121951314796661695581790896, 1.55409744144064674399809167321, 1.68866184438205825505810645228, 1.70237081698586924321910898859, 1.71841465154480833360097771709, 1.77087436685935864981232633078, 1.77195057412478957422992056795, 1.77620702362142540897702498093, 1.94110494646768899267088129419, 2.27909549878923813820335644152, 2.37316305895919780205707680650, 2.41824532651759082946038680030, 2.54577392593775490111731448181, 2.59395385249878428307768240487
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.