Dirichlet series
| L(s) = 1 | + 8·7-s − 8·9-s − 8·11-s + 8·13-s − 12·23-s − 8·25-s − 4·29-s + 4·41-s − 14·49-s − 64·63-s + 16·67-s + 40·73-s − 64·77-s − 32·79-s + 36·81-s + 64·91-s + 64·99-s − 12·101-s − 64·103-s + 8·107-s − 64·117-s − 48·121-s + 127-s + 131-s + 137-s + 139-s − 64·143-s + ⋯ |
| L(s) = 1 | + 3.02·7-s − 8/3·9-s − 2.41·11-s + 2.21·13-s − 2.50·23-s − 8/5·25-s − 0.742·29-s + 0.624·41-s − 2·49-s − 8.06·63-s + 1.95·67-s + 4.68·73-s − 7.29·77-s − 3.60·79-s + 4·81-s + 6.70·91-s + 6.43·99-s − 1.19·101-s − 6.30·103-s + 0.773·107-s − 5.91·117-s − 4.36·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 5.35·143-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 3^{16} \cdot 5^{16} \cdot 23^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 3^{16} \cdot 5^{16} \cdot 23^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(2^{64} \cdot 3^{16} \cdot 5^{16} \cdot 23^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.02982\times 10^{26}\) |
| Root analytic conductor: | \(6.63908\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 2^{64} \cdot 3^{16} \cdot 5^{16} \cdot 23^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(3.876080716\) |
| \(L(\frac12)\) | \(\approx\) | \(3.876080716\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 + T^{2} )^{8} \) | |
| 5 | \( ( 1 + T^{2} )^{8} \) | |
| 23 | \( 1 + 12 T + 72 T^{2} + 540 T^{3} + 2844 T^{4} + 6828 T^{5} + 21880 T^{6} + 8284 T^{7} - 600954 T^{8} + 8284 p T^{9} + 21880 p^{2} T^{10} + 6828 p^{3} T^{11} + 2844 p^{4} T^{12} + 540 p^{5} T^{13} + 72 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} \) | |
| good | 7 | \( ( 1 - 4 T + 31 T^{2} - 88 T^{3} + 361 T^{4} - 652 T^{5} + 1962 T^{6} - 1572 T^{7} + 8590 T^{8} - 1572 p T^{9} + 1962 p^{2} T^{10} - 652 p^{3} T^{11} + 361 p^{4} T^{12} - 88 p^{5} T^{13} + 31 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} )^{2} \) |
| 11 | \( ( 1 + 4 T + 48 T^{2} + 92 T^{3} + 808 T^{4} + 268 T^{5} + 8564 T^{6} - 6508 T^{7} + 92166 T^{8} - 6508 p T^{9} + 8564 p^{2} T^{10} + 268 p^{3} T^{11} + 808 p^{4} T^{12} + 92 p^{5} T^{13} + 48 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 13 | \( ( 1 - 4 T + 64 T^{2} - 272 T^{3} + 1964 T^{4} - 680 p T^{5} + 3012 p T^{6} - 175620 T^{7} + 578910 T^{8} - 175620 p T^{9} + 3012 p^{3} T^{10} - 680 p^{4} T^{11} + 1964 p^{4} T^{12} - 272 p^{5} T^{13} + 64 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 17 | \( 1 - 178 T^{2} + 915 p T^{4} - 889798 T^{6} + 37439985 T^{8} - 1232479556 T^{10} + 32905027030 T^{12} - 42814485776 p T^{14} + 13491943771858 T^{16} - 42814485776 p^{3} T^{18} + 32905027030 p^{4} T^{20} - 1232479556 p^{6} T^{22} + 37439985 p^{8} T^{24} - 889798 p^{10} T^{26} + 915 p^{13} T^{28} - 178 p^{14} T^{30} + p^{16} T^{32} \) | |
| 19 | \( ( 1 + 50 T^{2} + 64 T^{3} + 1176 T^{4} + 2552 T^{5} + 22866 T^{6} + 61704 T^{7} + 427422 T^{8} + 61704 p T^{9} + 22866 p^{2} T^{10} + 2552 p^{3} T^{11} + 1176 p^{4} T^{12} + 64 p^{5} T^{13} + 50 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 29 | \( ( 1 + 2 T + 119 T^{2} + 342 T^{3} + 7449 T^{4} + 788 p T^{5} + 333194 T^{6} + 905480 T^{7} + 11186594 T^{8} + 905480 p T^{9} + 333194 p^{2} T^{10} + 788 p^{4} T^{11} + 7449 p^{4} T^{12} + 342 p^{5} T^{13} + 119 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 31 | \( 1 - 238 T^{2} + 30043 T^{4} - 2614554 T^{6} + 174358425 T^{8} - 9419039108 T^{10} + 425918878462 T^{12} - 16442868416736 T^{14} + 547778433766914 T^{16} - 16442868416736 p^{2} T^{18} + 425918878462 p^{4} T^{20} - 9419039108 p^{6} T^{22} + 174358425 p^{8} T^{24} - 2614554 p^{10} T^{26} + 30043 p^{12} T^{28} - 238 p^{14} T^{30} + p^{16} T^{32} \) | |
| 37 | \( 1 - 342 T^{2} + 58815 T^{4} - 6694806 T^{6} + 564290469 T^{8} - 37494470992 T^{10} + 2046807727462 T^{12} - 94483573507460 T^{14} + 3755544774871262 T^{16} - 94483573507460 p^{2} T^{18} + 2046807727462 p^{4} T^{20} - 37494470992 p^{6} T^{22} + 564290469 p^{8} T^{24} - 6694806 p^{10} T^{26} + 58815 p^{12} T^{28} - 342 p^{14} T^{30} + p^{16} T^{32} \) | |
| 41 | \( ( 1 - 2 T + 187 T^{2} + 34 T^{3} + 16757 T^{4} + 24948 T^{5} + 1044142 T^{6} + 1926280 T^{7} + 49280146 T^{8} + 1926280 p T^{9} + 1044142 p^{2} T^{10} + 24948 p^{3} T^{11} + 16757 p^{4} T^{12} + 34 p^{5} T^{13} + 187 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 43 | \( ( 1 + 136 T^{2} + 48 T^{3} + 12108 T^{4} + 3760 T^{5} + 761736 T^{6} + 251776 T^{7} + 37544806 T^{8} + 251776 p T^{9} + 761736 p^{2} T^{10} + 3760 p^{3} T^{11} + 12108 p^{4} T^{12} + 48 p^{5} T^{13} + 136 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 47 | \( 1 - 368 T^{2} + 70336 T^{4} - 9185464 T^{6} + 917173292 T^{8} - 74253599256 T^{10} + 5044209707344 T^{12} - 293477181573760 T^{14} + 14792178435806470 T^{16} - 293477181573760 p^{2} T^{18} + 5044209707344 p^{4} T^{20} - 74253599256 p^{6} T^{22} + 917173292 p^{8} T^{24} - 9185464 p^{10} T^{26} + 70336 p^{12} T^{28} - 368 p^{14} T^{30} + p^{16} T^{32} \) | |
| 53 | \( 1 - 346 T^{2} + 67699 T^{4} - 9492686 T^{6} + 1043869505 T^{8} - 94527170244 T^{10} + 7236369960390 T^{12} - 475628040447216 T^{14} + 27063529258860962 T^{16} - 475628040447216 p^{2} T^{18} + 7236369960390 p^{4} T^{20} - 94527170244 p^{6} T^{22} + 1043869505 p^{8} T^{24} - 9492686 p^{10} T^{26} + 67699 p^{12} T^{28} - 346 p^{14} T^{30} + p^{16} T^{32} \) | |
| 59 | \( 1 - 602 T^{2} + 180215 T^{4} - 35538826 T^{6} + 5167349197 T^{8} - 588150123560 T^{10} + 54313192383838 T^{12} - 4159267695735644 T^{14} + 267338025341042334 T^{16} - 4159267695735644 p^{2} T^{18} + 54313192383838 p^{4} T^{20} - 588150123560 p^{6} T^{22} + 5167349197 p^{8} T^{24} - 35538826 p^{10} T^{26} + 180215 p^{12} T^{28} - 602 p^{14} T^{30} + p^{16} T^{32} \) | |
| 61 | \( 1 - 8 p T^{2} + 123368 T^{4} - 21466608 T^{6} + 2867544780 T^{8} - 310328261312 T^{10} + 28038976323880 T^{12} - 2152174590605032 T^{14} + 141602545675915782 T^{16} - 2152174590605032 p^{2} T^{18} + 28038976323880 p^{4} T^{20} - 310328261312 p^{6} T^{22} + 2867544780 p^{8} T^{24} - 21466608 p^{10} T^{26} + 123368 p^{12} T^{28} - 8 p^{15} T^{30} + p^{16} T^{32} \) | |
| 67 | \( ( 1 - 8 T + 311 T^{2} - 2492 T^{3} + 51633 T^{4} - 383796 T^{5} + 5670586 T^{6} - 37872228 T^{7} + 445033726 T^{8} - 37872228 p T^{9} + 5670586 p^{2} T^{10} - 383796 p^{3} T^{11} + 51633 p^{4} T^{12} - 2492 p^{5} T^{13} + 311 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 71 | \( 1 - 306 T^{2} + 49671 T^{4} - 6317362 T^{6} + 681045469 T^{8} - 63072265400 T^{10} + 5274953776718 T^{12} - 408381941088396 T^{14} + 29689479595513390 T^{16} - 408381941088396 p^{2} T^{18} + 5274953776718 p^{4} T^{20} - 63072265400 p^{6} T^{22} + 681045469 p^{8} T^{24} - 6317362 p^{10} T^{26} + 49671 p^{12} T^{28} - 306 p^{14} T^{30} + p^{16} T^{32} \) | |
| 73 | \( ( 1 - 20 T + 496 T^{2} - 7816 T^{3} + 113244 T^{4} - 1389584 T^{5} + 15491236 T^{6} - 150450916 T^{7} + 1383911070 T^{8} - 150450916 p T^{9} + 15491236 p^{2} T^{10} - 1389584 p^{3} T^{11} + 113244 p^{4} T^{12} - 7816 p^{5} T^{13} + 496 p^{6} T^{14} - 20 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 79 | \( ( 1 + 16 T + 350 T^{2} + 4544 T^{3} + 52760 T^{4} + 533600 T^{5} + 4867330 T^{6} + 38802608 T^{7} + 376756494 T^{8} + 38802608 p T^{9} + 4867330 p^{2} T^{10} + 533600 p^{3} T^{11} + 52760 p^{4} T^{12} + 4544 p^{5} T^{13} + 350 p^{6} T^{14} + 16 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 83 | \( ( 1 + 347 T^{2} - 412 T^{3} + 56257 T^{4} - 186772 T^{5} + 5857638 T^{6} - 31016492 T^{7} + 498461926 T^{8} - 31016492 p T^{9} + 5857638 p^{2} T^{10} - 186772 p^{3} T^{11} + 56257 p^{4} T^{12} - 412 p^{5} T^{13} + 347 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 89 | \( 1 - 936 T^{2} + 431992 T^{4} - 130870264 T^{6} + 29190909724 T^{8} - 5090492345000 T^{10} + 718445537995080 T^{12} - 83730895659802296 T^{14} + 8143876361971114438 T^{16} - 83730895659802296 p^{2} T^{18} + 718445537995080 p^{4} T^{20} - 5090492345000 p^{6} T^{22} + 29190909724 p^{8} T^{24} - 130870264 p^{10} T^{26} + 431992 p^{12} T^{28} - 936 p^{14} T^{30} + p^{16} T^{32} \) | |
| 97 | \( 1 - 780 T^{2} + 283156 T^{4} - 62701364 T^{6} + 9165153620 T^{8} - 854978615964 T^{10} + 35142036221420 T^{12} + 2830546579047132 T^{14} - 557041678088950954 T^{16} + 2830546579047132 p^{2} T^{18} + 35142036221420 p^{4} T^{20} - 854978615964 p^{6} T^{22} + 9165153620 p^{8} T^{24} - 62701364 p^{10} T^{26} + 283156 p^{12} T^{28} - 780 p^{14} T^{30} + p^{16} T^{32} \) | |
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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
−1.91430361330510138117738358568, −1.85558284288887408858241288639, −1.81804114457543391763559866390, −1.75423763922538200921090383859, −1.72527649368548237948332824704, −1.67088766410399717856783944619, −1.46934993074984020614483183621, −1.46242632379710099817267222908, −1.37922795294303896356385013591, −1.30447436318360766801696418597, −1.30346048427862691993128791674, −1.28918832308562768114402928183, −1.13791351877730029212899168700, −1.09814245840356466900920876992, −1.09227620943827397466019202696, −0.909025478744700049156887050374, −0.77582384956138551952940319761, −0.60107779288901273858120843300, −0.51363191093722892102691548385, −0.43770958148564349140230039713, −0.42317292055213164144178485279, −0.33103828136453240524683841560, −0.28886490044777906332376277259, −0.19028270674210585963516531558, −0.080967666142078345130102281535, 0.080967666142078345130102281535, 0.19028270674210585963516531558, 0.28886490044777906332376277259, 0.33103828136453240524683841560, 0.42317292055213164144178485279, 0.43770958148564349140230039713, 0.51363191093722892102691548385, 0.60107779288901273858120843300, 0.77582384956138551952940319761, 0.909025478744700049156887050374, 1.09227620943827397466019202696, 1.09814245840356466900920876992, 1.13791351877730029212899168700, 1.28918832308562768114402928183, 1.30346048427862691993128791674, 1.30447436318360766801696418597, 1.37922795294303896356385013591, 1.46242632379710099817267222908, 1.46934993074984020614483183621, 1.67088766410399717856783944619, 1.72527649368548237948332824704, 1.75423763922538200921090383859, 1.81804114457543391763559866390, 1.85558284288887408858241288639, 1.91430361330510138117738358568
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.