Dirichlet series
| L(s) = 1 | + 3·2-s + 2·4-s − 3·8-s − 12·11-s − 3·16-s − 4·19-s − 36·22-s + 13·25-s − 12·38-s + 36·41-s + 8·43-s − 24·44-s − 23·49-s + 39·50-s − 12·59-s − 3·64-s − 16·67-s − 4·73-s − 8·76-s + 108·82-s − 54·83-s + 24·86-s + 36·88-s + 8·97-s − 69·98-s + 26·100-s + 6·113-s + ⋯ |
| L(s) = 1 | + 2.12·2-s + 4-s − 1.06·8-s − 3.61·11-s − 3/4·16-s − 0.917·19-s − 7.67·22-s + 13/5·25-s − 1.94·38-s + 5.62·41-s + 1.21·43-s − 3.61·44-s − 3.28·49-s + 5.51·50-s − 1.56·59-s − 3/8·64-s − 1.95·67-s − 0.468·73-s − 0.917·76-s + 11.9·82-s − 5.92·83-s + 2.58·86-s + 3.83·88-s + 0.812·97-s − 6.97·98-s + 13/5·100-s + 0.564·113-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{48}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{48}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(2^{48} \cdot 3^{48}\) |
| Sign: | $1$ |
| Analytic conductor: | \(6133.29\) |
| Root analytic conductor: | \(1.31330\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 2^{48} \cdot 3^{48} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(2.716870171\) |
| \(L(\frac12)\) | \(\approx\) | \(2.716870171\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 - 3 T + 7 T^{2} - 3 p^{2} T^{3} + p^{4} T^{4} - 3 p^{2} T^{5} - p^{3} T^{6} + 9 p^{2} T^{7} - 17 p^{2} T^{8} + 9 p^{3} T^{9} - p^{5} T^{10} - 3 p^{5} T^{11} + p^{8} T^{12} - 3 p^{7} T^{13} + 7 p^{6} T^{14} - 3 p^{7} T^{15} + p^{8} T^{16} \) |
| 3 | \( 1 \) | |
| good | 5 | \( 1 - 13 T^{2} + 48 T^{4} + 103 T^{6} - 1099 T^{8} + 3648 T^{10} - 5222 T^{12} - 177298 T^{14} + 1667616 T^{16} - 177298 p^{2} T^{18} - 5222 p^{4} T^{20} + 3648 p^{6} T^{22} - 1099 p^{8} T^{24} + 103 p^{10} T^{26} + 48 p^{12} T^{28} - 13 p^{14} T^{30} + p^{16} T^{32} \) |
| 7 | \( 1 + 23 T^{2} + 204 T^{4} + 1399 T^{6} + 12029 T^{8} + 45000 T^{10} - 343382 T^{12} - 82438 p^{2} T^{14} - 24407256 T^{16} - 82438 p^{4} T^{18} - 343382 p^{4} T^{20} + 45000 p^{6} T^{22} + 12029 p^{8} T^{24} + 1399 p^{10} T^{26} + 204 p^{12} T^{28} + 23 p^{14} T^{30} + p^{16} T^{32} \) | |
| 11 | \( ( 1 - 3 T + 38 T^{2} - 87 T^{3} + 606 T^{4} - 87 p T^{5} + 38 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} )^{2}( 1 + 9 T + 41 T^{2} + 108 T^{3} + 276 T^{4} + 108 p T^{5} + 41 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 13 | \( 1 + 47 T^{2} + 1116 T^{4} + 17239 T^{6} + 168353 T^{8} + 463056 T^{10} - 2070506 p T^{12} - 784598686 T^{14} - 12569281176 T^{16} - 784598686 p^{2} T^{18} - 2070506 p^{5} T^{20} + 463056 p^{6} T^{22} + 168353 p^{8} T^{24} + 17239 p^{10} T^{26} + 1116 p^{12} T^{28} + 47 p^{14} T^{30} + p^{16} T^{32} \) | |
| 17 | \( ( 1 - 101 T^{2} + 4882 T^{4} - 146891 T^{6} + 2998666 T^{8} - 146891 p^{2} T^{10} + 4882 p^{4} T^{12} - 101 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 19 | \( ( 1 + T + 64 T^{2} + 49 T^{3} + 1726 T^{4} + 49 p T^{5} + 64 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} )^{4} \) | |
| 23 | \( 1 - 85 T^{2} + 3432 T^{4} - 80441 T^{6} + 1017209 T^{8} + 1764696 T^{10} - 386280290 T^{12} + 9323832998 T^{14} - 183562178736 T^{16} + 9323832998 p^{2} T^{18} - 386280290 p^{4} T^{20} + 1764696 p^{6} T^{22} + 1017209 p^{8} T^{24} - 80441 p^{10} T^{26} + 3432 p^{12} T^{28} - 85 p^{14} T^{30} + p^{16} T^{32} \) | |
| 29 | \( 1 - 97 T^{2} + 3780 T^{4} - 80561 T^{6} + 1632089 T^{8} - 54786000 T^{10} + 1038848902 T^{12} + 18974156858 T^{14} - 1329857348520 T^{16} + 18974156858 p^{2} T^{18} + 1038848902 p^{4} T^{20} - 54786000 p^{6} T^{22} + 1632089 p^{8} T^{24} - 80561 p^{10} T^{26} + 3780 p^{12} T^{28} - 97 p^{14} T^{30} + p^{16} T^{32} \) | |
| 31 | \( 1 + 131 T^{2} + 7152 T^{4} + 312151 T^{6} + 16524593 T^{8} + 706250232 T^{10} + 22761450214 T^{12} + 818587141454 T^{14} + 29243993208000 T^{16} + 818587141454 p^{2} T^{18} + 22761450214 p^{4} T^{20} + 706250232 p^{6} T^{22} + 16524593 p^{8} T^{24} + 312151 p^{10} T^{26} + 7152 p^{12} T^{28} + 131 p^{14} T^{30} + p^{16} T^{32} \) | |
| 37 | \( ( 1 - 140 T^{2} + 8596 T^{4} - 317540 T^{6} + 10470934 T^{8} - 317540 p^{2} T^{10} + 8596 p^{4} T^{12} - 140 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 41 | \( ( 1 - 18 T + 292 T^{2} - 3312 T^{3} + 34963 T^{4} - 303732 T^{5} + 2503888 T^{6} - 17908938 T^{7} + 122450608 T^{8} - 17908938 p T^{9} + 2503888 p^{2} T^{10} - 303732 p^{3} T^{11} + 34963 p^{4} T^{12} - 3312 p^{5} T^{13} + 292 p^{6} T^{14} - 18 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 43 | \( ( 1 - 4 T - 96 T^{2} + 868 T^{3} + 4061 T^{4} - 55182 T^{5} + 78652 T^{6} + 1341518 T^{7} - 8451684 T^{8} + 1341518 p T^{9} + 78652 p^{2} T^{10} - 55182 p^{3} T^{11} + 4061 p^{4} T^{12} + 868 p^{5} T^{13} - 96 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 47 | \( 1 - 265 T^{2} + 38172 T^{4} - 3663305 T^{6} + 256619789 T^{8} - 13483362840 T^{10} + 543517455226 T^{12} - 17957421673750 T^{14} + 672442507889160 T^{16} - 17957421673750 p^{2} T^{18} + 543517455226 p^{4} T^{20} - 13483362840 p^{6} T^{22} + 256619789 p^{8} T^{24} - 3663305 p^{10} T^{26} + 38172 p^{12} T^{28} - 265 p^{14} T^{30} + p^{16} T^{32} \) | |
| 53 | \( ( 1 + 196 T^{2} + 21556 T^{4} + 1665484 T^{6} + 98315734 T^{8} + 1665484 p^{2} T^{10} + 21556 p^{4} T^{12} + 196 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 59 | \( ( 1 + 6 T + 178 T^{2} + 996 T^{3} + 269 p T^{4} + 89238 T^{5} + 1194346 T^{6} + 6437052 T^{7} + 79012984 T^{8} + 6437052 p T^{9} + 1194346 p^{2} T^{10} + 89238 p^{3} T^{11} + 269 p^{5} T^{12} + 996 p^{5} T^{13} + 178 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 61 | \( 1 + 299 T^{2} + 43416 T^{4} + 4465735 T^{6} + 393470093 T^{8} + 31461221184 T^{10} + 2332722797890 T^{12} + 164022774523334 T^{14} + 10603540284680400 T^{16} + 164022774523334 p^{2} T^{18} + 2332722797890 p^{4} T^{20} + 31461221184 p^{6} T^{22} + 393470093 p^{8} T^{24} + 4465735 p^{10} T^{26} + 43416 p^{12} T^{28} + 299 p^{14} T^{30} + p^{16} T^{32} \) | |
| 67 | \( ( 1 + 8 T - 138 T^{2} - 1052 T^{3} + 11279 T^{4} + 57198 T^{5} - 930218 T^{6} - 1298482 T^{7} + 71382744 T^{8} - 1298482 p T^{9} - 930218 p^{2} T^{10} + 57198 p^{3} T^{11} + 11279 p^{4} T^{12} - 1052 p^{5} T^{13} - 138 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 71 | \( ( 1 + 400 T^{2} + 73324 T^{4} + 8374048 T^{6} + 685441990 T^{8} + 8374048 p^{2} T^{10} + 73324 p^{4} T^{12} + 400 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 73 | \( ( 1 + T + 214 T^{2} - 5 T^{3} + 20758 T^{4} - 5 p T^{5} + 214 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} )^{4} \) | |
| 79 | \( 1 + 383 T^{2} + 74724 T^{4} + 9593503 T^{6} + 916548293 T^{8} + 73505234952 T^{10} + 5745220732498 T^{12} + 475438917658202 T^{14} + 38877973133064792 T^{16} + 475438917658202 p^{2} T^{18} + 5745220732498 p^{4} T^{20} + 73505234952 p^{6} T^{22} + 916548293 p^{8} T^{24} + 9593503 p^{10} T^{26} + 74724 p^{12} T^{28} + 383 p^{14} T^{30} + p^{16} T^{32} \) | |
| 83 | \( ( 1 + 27 T + 544 T^{2} + 8127 T^{3} + 107593 T^{4} + 1304076 T^{5} + 14403022 T^{6} + 148987620 T^{7} + 1398163588 T^{8} + 148987620 p T^{9} + 14403022 p^{2} T^{10} + 1304076 p^{3} T^{11} + 107593 p^{4} T^{12} + 8127 p^{5} T^{13} + 544 p^{6} T^{14} + 27 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 89 | \( ( 1 - 440 T^{2} + 100348 T^{4} - 14946776 T^{6} + 1566592486 T^{8} - 14946776 p^{2} T^{10} + 100348 p^{4} T^{12} - 440 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 97 | \( ( 1 - 4 T - 198 T^{2} - 320 T^{3} + 21017 T^{4} + 106308 T^{5} - 1078406 T^{6} - 6931984 T^{7} + 64836612 T^{8} - 6931984 p T^{9} - 1078406 p^{2} T^{10} + 106308 p^{3} T^{11} + 21017 p^{4} T^{12} - 320 p^{5} T^{13} - 198 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
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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
−3.66530220490890198517272426881, −3.51774927401439645416663223436, −3.40934245615709631181315606349, −3.30459226541255685651313188307, −3.11087842226862977687247479677, −3.06685401357228604406068378300, −3.02683312250912332990950407405, −2.91468549325536639463487465788, −2.86650321551862434045265358158, −2.81526140886639181793701954804, −2.68666844374878534197102742140, −2.54740120379529009673904208410, −2.53306655750774436396731075108, −2.43703024952855474369778669602, −2.17788314410282264070507524753, −2.08995754414363346316245817379, −2.08784552186305106219218933182, −1.77177590401366217256348352551, −1.74108316832948756750271787310, −1.41099012573107542733979138370, −1.26528598653488102301284559293, −1.13438515222288973379094528550, −1.12096017564893920421714367734, −0.45837559524099854475959997375, −0.34582365577185995921084818390, 0.34582365577185995921084818390, 0.45837559524099854475959997375, 1.12096017564893920421714367734, 1.13438515222288973379094528550, 1.26528598653488102301284559293, 1.41099012573107542733979138370, 1.74108316832948756750271787310, 1.77177590401366217256348352551, 2.08784552186305106219218933182, 2.08995754414363346316245817379, 2.17788314410282264070507524753, 2.43703024952855474369778669602, 2.53306655750774436396731075108, 2.54740120379529009673904208410, 2.68666844374878534197102742140, 2.81526140886639181793701954804, 2.86650321551862434045265358158, 2.91468549325536639463487465788, 3.02683312250912332990950407405, 3.06685401357228604406068378300, 3.11087842226862977687247479677, 3.30459226541255685651313188307, 3.40934245615709631181315606349, 3.51774927401439645416663223436, 3.66530220490890198517272426881
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.