Dirichlet series
| L(s) = 1 | + 12·2-s + 85·4-s + 504·8-s − 216·9-s + 352·13-s + 2.55e3·16-s − 2.59e3·18-s + 4.22e3·26-s − 3.45e3·29-s + 9.82e3·32-s − 1.83e4·36-s − 9.37e3·37-s + 1.24e3·41-s + 1.72e4·49-s + 2.99e4·52-s + 5.18e3·53-s − 4.14e4·58-s − 3.80e3·61-s + 2.58e4·64-s − 1.08e5·72-s − 1.10e4·73-s − 1.12e5·74-s + 2.62e4·81-s + 1.49e4·82-s + 7.58e3·89-s + 1.44e4·97-s + 2.06e5·98-s + ⋯ |
| L(s) = 1 | + 3·2-s + 5.31·4-s + 63/8·8-s − 8/3·9-s + 2.08·13-s + 9.98·16-s − 8·18-s + 6.24·26-s − 4.10·29-s + 9.59·32-s − 14.1·36-s − 6.84·37-s + 0.742·41-s + 7.17·49-s + 11.0·52-s + 1.84·53-s − 12.3·58-s − 1.02·61-s + 6.30·64-s − 21·72-s − 2.07·73-s − 20.5·74-s + 4·81-s + 2.22·82-s + 0.957·89-s + 1.54·97-s + 21.5·98-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{16} \cdot 5^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{16} \cdot 5^{32}\right)^{s/2} \, \Gamma_{\C}(s+2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(2^{32} \cdot 3^{16} \cdot 5^{32}\) |
| Sign: | $1$ |
| Analytic conductor: | \(7.31559\times 10^{23}\) |
| Root analytic conductor: | \(5.56875\) |
| Motivic weight: | \(4\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 2^{32} \cdot 3^{16} \cdot 5^{32} ,\ ( \ : [2]^{16} ),\ 1 )\) |
Particular Values
| \(L(\frac{5}{2})\) | \(\approx\) | \(0.1594173127\) |
| \(L(\frac12)\) | \(\approx\) | \(0.1594173127\) |
| \(L(3)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 - 3 p^{2} T + 59 T^{2} - 3 p^{6} T^{3} + 195 p^{2} T^{4} - 15 p^{7} T^{5} - 329 p^{4} T^{6} + 9 p^{12} T^{7} - 203 p^{9} T^{8} + 9 p^{16} T^{9} - 329 p^{12} T^{10} - 15 p^{19} T^{11} + 195 p^{18} T^{12} - 3 p^{26} T^{13} + 59 p^{24} T^{14} - 3 p^{30} T^{15} + p^{32} T^{16} \) |
| 3 | \( ( 1 + p^{3} T^{2} )^{8} \) | |
| 5 | \( 1 \) | |
| good | 7 | \( 1 - 17232 T^{2} + 136818168 T^{4} - 680657380848 T^{6} + 2517512063806492 T^{8} - 8029440611585623632 T^{10} + \)\(24\!\cdots\!72\)\( T^{12} - \)\(69\!\cdots\!80\)\( T^{14} + \)\(17\!\cdots\!94\)\( T^{16} - \)\(69\!\cdots\!80\)\( p^{8} T^{18} + \)\(24\!\cdots\!72\)\( p^{16} T^{20} - 8029440611585623632 p^{24} T^{22} + 2517512063806492 p^{32} T^{24} - 680657380848 p^{40} T^{26} + 136818168 p^{48} T^{28} - 17232 p^{56} T^{30} + p^{64} T^{32} \) |
| 11 | \( 1 - 62928 T^{2} + 2074869240 T^{4} - 50681547786096 T^{6} + 1061223539946340636 T^{8} - \)\(18\!\cdots\!08\)\( p T^{10} + \)\(34\!\cdots\!08\)\( T^{12} - \)\(55\!\cdots\!64\)\( T^{14} + \)\(84\!\cdots\!18\)\( T^{16} - \)\(55\!\cdots\!64\)\( p^{8} T^{18} + \)\(34\!\cdots\!08\)\( p^{16} T^{20} - \)\(18\!\cdots\!08\)\( p^{25} T^{22} + 1061223539946340636 p^{32} T^{24} - 50681547786096 p^{40} T^{26} + 2074869240 p^{48} T^{28} - 62928 p^{56} T^{30} + p^{64} T^{32} \) | |
| 13 | \( ( 1 - 176 T + 120088 T^{2} - 15535504 T^{3} + 6148186076 T^{4} - 666071960624 T^{5} + 211894024736168 T^{6} - 22608154112614288 T^{7} + 6337852768553827654 T^{8} - 22608154112614288 p^{4} T^{9} + 211894024736168 p^{8} T^{10} - 666071960624 p^{12} T^{11} + 6148186076 p^{16} T^{12} - 15535504 p^{20} T^{13} + 120088 p^{24} T^{14} - 176 p^{28} T^{15} + p^{32} T^{16} )^{2} \) | |
| 17 | \( ( 1 + 292152 T^{2} - 10007040 T^{3} + 33408102556 T^{4} - 3077630814720 T^{5} + 2039887696288392 T^{6} - 425509686457820160 T^{7} + \)\(11\!\cdots\!54\)\( T^{8} - 425509686457820160 p^{4} T^{9} + 2039887696288392 p^{8} T^{10} - 3077630814720 p^{12} T^{11} + 33408102556 p^{16} T^{12} - 10007040 p^{20} T^{13} + 292152 p^{24} T^{14} + p^{32} T^{16} )^{2} \) | |
| 19 | \( 1 - 776912 T^{2} + 326220320248 T^{4} - 94716496394482288 T^{6} + \)\(21\!\cdots\!32\)\( T^{8} - \)\(37\!\cdots\!52\)\( T^{10} + \)\(57\!\cdots\!92\)\( T^{12} - \)\(80\!\cdots\!20\)\( T^{14} + \)\(10\!\cdots\!54\)\( T^{16} - \)\(80\!\cdots\!20\)\( p^{8} T^{18} + \)\(57\!\cdots\!92\)\( p^{16} T^{20} - \)\(37\!\cdots\!52\)\( p^{24} T^{22} + \)\(21\!\cdots\!32\)\( p^{32} T^{24} - 94716496394482288 p^{40} T^{26} + 326220320248 p^{48} T^{28} - 776912 p^{56} T^{30} + p^{64} T^{32} \) | |
| 23 | \( 1 - 2546384 T^{2} + 3106897075960 T^{4} - 2431209279725770864 T^{6} + \)\(13\!\cdots\!36\)\( T^{8} - \)\(61\!\cdots\!64\)\( T^{10} + \)\(22\!\cdots\!80\)\( T^{12} - \)\(74\!\cdots\!40\)\( T^{14} + \)\(21\!\cdots\!46\)\( T^{16} - \)\(74\!\cdots\!40\)\( p^{8} T^{18} + \)\(22\!\cdots\!80\)\( p^{16} T^{20} - \)\(61\!\cdots\!64\)\( p^{24} T^{22} + \)\(13\!\cdots\!36\)\( p^{32} T^{24} - 2431209279725770864 p^{40} T^{26} + 3106897075960 p^{48} T^{28} - 2546384 p^{56} T^{30} + p^{64} T^{32} \) | |
| 29 | \( ( 1 + 1728 T + 2988344 T^{2} + 4323706944 T^{3} + 5340269169948 T^{4} + 6037406711468736 T^{5} + 6196853389876661128 T^{6} + \)\(58\!\cdots\!72\)\( T^{7} + \)\(51\!\cdots\!82\)\( T^{8} + \)\(58\!\cdots\!72\)\( p^{4} T^{9} + 6196853389876661128 p^{8} T^{10} + 6037406711468736 p^{12} T^{11} + 5340269169948 p^{16} T^{12} + 4323706944 p^{20} T^{13} + 2988344 p^{24} T^{14} + 1728 p^{28} T^{15} + p^{32} T^{16} )^{2} \) | |
| 31 | \( 1 - 9087696 T^{2} + 41491468948728 T^{4} - \)\(12\!\cdots\!92\)\( T^{6} + \)\(28\!\cdots\!92\)\( T^{8} - \)\(50\!\cdots\!44\)\( T^{10} + \)\(73\!\cdots\!80\)\( T^{12} - \)\(88\!\cdots\!08\)\( T^{14} + \)\(88\!\cdots\!14\)\( T^{16} - \)\(88\!\cdots\!08\)\( p^{8} T^{18} + \)\(73\!\cdots\!80\)\( p^{16} T^{20} - \)\(50\!\cdots\!44\)\( p^{24} T^{22} + \)\(28\!\cdots\!92\)\( p^{32} T^{24} - \)\(12\!\cdots\!92\)\( p^{40} T^{26} + 41491468948728 p^{48} T^{28} - 9087696 p^{56} T^{30} + p^{64} T^{32} \) | |
| 37 | \( ( 1 + 4688 T + 19328728 T^{2} + 53195113840 T^{3} + 132027938064476 T^{4} + 265802490052125392 T^{5} + 13257236948539071944 p T^{6} + \)\(77\!\cdots\!20\)\( T^{7} + \)\(11\!\cdots\!74\)\( T^{8} + \)\(77\!\cdots\!20\)\( p^{4} T^{9} + 13257236948539071944 p^{9} T^{10} + 265802490052125392 p^{12} T^{11} + 132027938064476 p^{16} T^{12} + 53195113840 p^{20} T^{13} + 19328728 p^{24} T^{14} + 4688 p^{28} T^{15} + p^{32} T^{16} )^{2} \) | |
| 41 | \( ( 1 - 624 T + 8875576 T^{2} - 3753492048 T^{3} + 30295021037980 T^{4} - 5394202699532400 T^{5} + 28464857682185758600 T^{6} + \)\(17\!\cdots\!76\)\( T^{7} - \)\(37\!\cdots\!66\)\( T^{8} + \)\(17\!\cdots\!76\)\( p^{4} T^{9} + 28464857682185758600 p^{8} T^{10} - 5394202699532400 p^{12} T^{11} + 30295021037980 p^{16} T^{12} - 3753492048 p^{20} T^{13} + 8875576 p^{24} T^{14} - 624 p^{28} T^{15} + p^{32} T^{16} )^{2} \) | |
| 43 | \( 1 - 31341200 T^{2} + 487359340065400 T^{4} - \)\(49\!\cdots\!88\)\( T^{6} + \)\(37\!\cdots\!84\)\( T^{8} - \)\(22\!\cdots\!88\)\( T^{10} + \)\(11\!\cdots\!96\)\( T^{12} - \)\(47\!\cdots\!76\)\( T^{14} + \)\(17\!\cdots\!18\)\( T^{16} - \)\(47\!\cdots\!76\)\( p^{8} T^{18} + \)\(11\!\cdots\!96\)\( p^{16} T^{20} - \)\(22\!\cdots\!88\)\( p^{24} T^{22} + \)\(37\!\cdots\!84\)\( p^{32} T^{24} - \)\(49\!\cdots\!88\)\( p^{40} T^{26} + 487359340065400 p^{48} T^{28} - 31341200 p^{56} T^{30} + p^{64} T^{32} \) | |
| 47 | \( 1 - 46791248 T^{2} + 1043859960850936 T^{4} - \)\(14\!\cdots\!80\)\( T^{6} + \)\(15\!\cdots\!32\)\( T^{8} - \)\(12\!\cdots\!68\)\( T^{10} + \)\(80\!\cdots\!20\)\( T^{12} - \)\(45\!\cdots\!28\)\( T^{14} + \)\(23\!\cdots\!30\)\( T^{16} - \)\(45\!\cdots\!28\)\( p^{8} T^{18} + \)\(80\!\cdots\!20\)\( p^{16} T^{20} - \)\(12\!\cdots\!68\)\( p^{24} T^{22} + \)\(15\!\cdots\!32\)\( p^{32} T^{24} - \)\(14\!\cdots\!80\)\( p^{40} T^{26} + 1043859960850936 p^{48} T^{28} - 46791248 p^{56} T^{30} + p^{64} T^{32} \) | |
| 53 | \( ( 1 - 2592 T + 16188728 T^{2} - 35131993440 T^{3} + 231133127641116 T^{4} - 362403670187658528 T^{5} + \)\(18\!\cdots\!28\)\( T^{6} - \)\(26\!\cdots\!20\)\( T^{7} + \)\(17\!\cdots\!74\)\( T^{8} - \)\(26\!\cdots\!20\)\( p^{4} T^{9} + \)\(18\!\cdots\!28\)\( p^{8} T^{10} - 362403670187658528 p^{12} T^{11} + 231133127641116 p^{16} T^{12} - 35131993440 p^{20} T^{13} + 16188728 p^{24} T^{14} - 2592 p^{28} T^{15} + p^{32} T^{16} )^{2} \) | |
| 59 | \( 1 - 65171152 T^{2} + 1981452915163128 T^{4} - \)\(38\!\cdots\!68\)\( T^{6} + \)\(58\!\cdots\!52\)\( T^{8} - \)\(81\!\cdots\!12\)\( T^{10} + \)\(10\!\cdots\!12\)\( T^{12} - \)\(12\!\cdots\!20\)\( T^{14} + \)\(15\!\cdots\!94\)\( T^{16} - \)\(12\!\cdots\!20\)\( p^{8} T^{18} + \)\(10\!\cdots\!12\)\( p^{16} T^{20} - \)\(81\!\cdots\!12\)\( p^{24} T^{22} + \)\(58\!\cdots\!52\)\( p^{32} T^{24} - \)\(38\!\cdots\!68\)\( p^{40} T^{26} + 1981452915163128 p^{48} T^{28} - 65171152 p^{56} T^{30} + p^{64} T^{32} \) | |
| 61 | \( ( 1 + 1904 T + 55588216 T^{2} + 110588795728 T^{3} + 1721588177636252 T^{4} + 3470109544215419504 T^{5} + \)\(37\!\cdots\!52\)\( T^{6} + \)\(67\!\cdots\!28\)\( T^{7} + \)\(60\!\cdots\!66\)\( T^{8} + \)\(67\!\cdots\!28\)\( p^{4} T^{9} + \)\(37\!\cdots\!52\)\( p^{8} T^{10} + 3470109544215419504 p^{12} T^{11} + 1721588177636252 p^{16} T^{12} + 110588795728 p^{20} T^{13} + 55588216 p^{24} T^{14} + 1904 p^{28} T^{15} + p^{32} T^{16} )^{2} \) | |
| 67 | \( 1 - 157133712 T^{2} + 13346244284840568 T^{4} - \)\(78\!\cdots\!08\)\( T^{6} + \)\(35\!\cdots\!92\)\( T^{8} - \)\(12\!\cdots\!12\)\( T^{10} + \)\(39\!\cdots\!52\)\( T^{12} - \)\(10\!\cdots\!00\)\( T^{14} + \)\(21\!\cdots\!94\)\( T^{16} - \)\(10\!\cdots\!00\)\( p^{8} T^{18} + \)\(39\!\cdots\!52\)\( p^{16} T^{20} - \)\(12\!\cdots\!12\)\( p^{24} T^{22} + \)\(35\!\cdots\!92\)\( p^{32} T^{24} - \)\(78\!\cdots\!08\)\( p^{40} T^{26} + 13346244284840568 p^{48} T^{28} - 157133712 p^{56} T^{30} + p^{64} T^{32} \) | |
| 71 | \( 1 - 178757904 T^{2} + 15919921832924280 T^{4} - \)\(96\!\cdots\!92\)\( T^{6} + \)\(46\!\cdots\!36\)\( T^{8} - \)\(18\!\cdots\!44\)\( T^{10} + \)\(64\!\cdots\!32\)\( T^{12} - \)\(19\!\cdots\!36\)\( T^{14} + \)\(52\!\cdots\!54\)\( T^{16} - \)\(19\!\cdots\!36\)\( p^{8} T^{18} + \)\(64\!\cdots\!32\)\( p^{16} T^{20} - \)\(18\!\cdots\!44\)\( p^{24} T^{22} + \)\(46\!\cdots\!36\)\( p^{32} T^{24} - \)\(96\!\cdots\!92\)\( p^{40} T^{26} + 15919921832924280 p^{48} T^{28} - 178757904 p^{56} T^{30} + p^{64} T^{32} \) | |
| 73 | \( ( 1 + 5520 T + 123421176 T^{2} + 381355216560 T^{3} + 6457821303709468 T^{4} + 12407075531934222480 T^{5} + \)\(25\!\cdots\!08\)\( T^{6} + \)\(43\!\cdots\!40\)\( T^{7} + \)\(82\!\cdots\!14\)\( T^{8} + \)\(43\!\cdots\!40\)\( p^{4} T^{9} + \)\(25\!\cdots\!08\)\( p^{8} T^{10} + 12407075531934222480 p^{12} T^{11} + 6457821303709468 p^{16} T^{12} + 381355216560 p^{20} T^{13} + 123421176 p^{24} T^{14} + 5520 p^{28} T^{15} + p^{32} T^{16} )^{2} \) | |
| 79 | \( 1 - 295546320 T^{2} + 44889007478055672 T^{4} - \)\(46\!\cdots\!60\)\( T^{6} + \)\(37\!\cdots\!56\)\( T^{8} - \)\(24\!\cdots\!40\)\( T^{10} + \)\(13\!\cdots\!52\)\( T^{12} - \)\(66\!\cdots\!60\)\( T^{14} + \)\(27\!\cdots\!74\)\( T^{16} - \)\(66\!\cdots\!60\)\( p^{8} T^{18} + \)\(13\!\cdots\!52\)\( p^{16} T^{20} - \)\(24\!\cdots\!40\)\( p^{24} T^{22} + \)\(37\!\cdots\!56\)\( p^{32} T^{24} - \)\(46\!\cdots\!60\)\( p^{40} T^{26} + 44889007478055672 p^{48} T^{28} - 295546320 p^{56} T^{30} + p^{64} T^{32} \) | |
| 83 | \( 1 - 214350992 T^{2} + 28796953652331640 T^{4} - \)\(26\!\cdots\!04\)\( T^{6} + \)\(19\!\cdots\!04\)\( T^{8} - \)\(11\!\cdots\!00\)\( T^{10} + \)\(60\!\cdots\!64\)\( T^{12} - \)\(28\!\cdots\!56\)\( T^{14} + \)\(13\!\cdots\!02\)\( T^{16} - \)\(28\!\cdots\!56\)\( p^{8} T^{18} + \)\(60\!\cdots\!64\)\( p^{16} T^{20} - \)\(11\!\cdots\!00\)\( p^{24} T^{22} + \)\(19\!\cdots\!04\)\( p^{32} T^{24} - \)\(26\!\cdots\!04\)\( p^{40} T^{26} + 28796953652331640 p^{48} T^{28} - 214350992 p^{56} T^{30} + p^{64} T^{32} \) | |
| 89 | \( ( 1 - 3792 T + 177375608 T^{2} - 1023293054064 T^{3} + 21218942601872412 T^{4} - \)\(11\!\cdots\!88\)\( T^{5} + \)\(18\!\cdots\!44\)\( T^{6} - \)\(94\!\cdots\!40\)\( T^{7} + \)\(12\!\cdots\!02\)\( T^{8} - \)\(94\!\cdots\!40\)\( p^{4} T^{9} + \)\(18\!\cdots\!44\)\( p^{8} T^{10} - \)\(11\!\cdots\!88\)\( p^{12} T^{11} + 21218942601872412 p^{16} T^{12} - 1023293054064 p^{20} T^{13} + 177375608 p^{24} T^{14} - 3792 p^{28} T^{15} + p^{32} T^{16} )^{2} \) | |
| 97 | \( ( 1 - 7248 T + 474839096 T^{2} - 3706917179376 T^{3} + 113797092743603356 T^{4} - \)\(84\!\cdots\!44\)\( T^{5} + \)\(17\!\cdots\!68\)\( T^{6} - \)\(11\!\cdots\!24\)\( T^{7} + \)\(18\!\cdots\!98\)\( T^{8} - \)\(11\!\cdots\!24\)\( p^{4} T^{9} + \)\(17\!\cdots\!68\)\( p^{8} T^{10} - \)\(84\!\cdots\!44\)\( p^{12} T^{11} + 113797092743603356 p^{16} T^{12} - 3706917179376 p^{20} T^{13} + 474839096 p^{24} T^{14} - 7248 p^{28} T^{15} + p^{32} T^{16} )^{2} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.64477781017275777970719183618, −2.41287170802147012057905306750, −2.33084303044278753274853907928, −2.25194567940267359118506960782, −2.20657685003600808325649628184, −2.15047264522489798768502482571, −2.11251130524515090331436818432, −2.02680086276322784296045507191, −1.77950649512195422734803180314, −1.77769162527465717728223018604, −1.69814727461648609327160061324, −1.54434478830045962051254438918, −1.42348068707786779467077011466, −1.41860028081020085546024184171, −1.13769131781389968622586701635, −1.11938681543897745432313218844, −1.11219033850203479403140953174, −0.975521603394746865250463712732, −0.69902616495119175225193258680, −0.69105831825206306782304215804, −0.51709815663732345673192687127, −0.30299510680052331234129333340, −0.24373607291584639192899211900, −0.20255138533635836647853347664, −0.008096590414680710627243170113, 0.008096590414680710627243170113, 0.20255138533635836647853347664, 0.24373607291584639192899211900, 0.30299510680052331234129333340, 0.51709815663732345673192687127, 0.69105831825206306782304215804, 0.69902616495119175225193258680, 0.975521603394746865250463712732, 1.11219033850203479403140953174, 1.11938681543897745432313218844, 1.13769131781389968622586701635, 1.41860028081020085546024184171, 1.42348068707786779467077011466, 1.54434478830045962051254438918, 1.69814727461648609327160061324, 1.77769162527465717728223018604, 1.77950649512195422734803180314, 2.02680086276322784296045507191, 2.11251130524515090331436818432, 2.15047264522489798768502482571, 2.20657685003600808325649628184, 2.25194567940267359118506960782, 2.33084303044278753274853907928, 2.41287170802147012057905306750, 2.64477781017275777970719183618
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.