Dirichlet series
| L(s) = 1 | − 4·7-s − 72·19-s + 394·25-s − 708·31-s + 76·37-s + 1.40e3·43-s + 208·49-s − 1.63e3·61-s − 1.52e3·67-s − 2.70e3·73-s − 364·79-s − 4.99e3·103-s + 772·109-s − 5.84e3·121-s + 127-s + 131-s + 288·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2.95e4·169-s + 173-s − 1.57e3·175-s + ⋯ |
| L(s) = 1 | − 0.215·7-s − 0.869·19-s + 3.15·25-s − 4.10·31-s + 0.337·37-s + 4.99·43-s + 0.606·49-s − 3.42·61-s − 2.78·67-s − 4.32·73-s − 0.518·79-s − 4.77·103-s + 0.678·109-s − 4.38·121-s + 0.000698·127-s + 0.000666·131-s + 0.187·133-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 13.4·169-s + 0.000439·173-s − 0.680·175-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{32} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{32} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(2^{32} \cdot 3^{32} \cdot 7^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(5.70513\times 10^{18}\) |
| Root analytic conductor: | \(3.85596\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 2^{32} \cdot 3^{32} \cdot 7^{16} ,\ ( \ : [3/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(\approx\) | \(0.4029865434\) |
| \(L(\frac12)\) | \(\approx\) | \(0.4029865434\) |
| \(L(\frac{5}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) | |
| 7 | \( ( 1 + 2 T - 2 p^{2} T^{2} + 80 p T^{3} + 4519 p^{2} T^{4} + 80 p^{4} T^{5} - 2 p^{8} T^{6} + 2 p^{9} T^{7} + p^{12} T^{8} )^{2} \) | |
| good | 5 | \( 1 - 394 T^{2} + 79383 T^{4} - 13135862 T^{6} + 1573353173 T^{8} - 72279808932 T^{10} - 11170130994038 T^{12} + 735694168015888 p T^{14} - 600595446685632534 T^{16} + 735694168015888 p^{7} T^{18} - 11170130994038 p^{12} T^{20} - 72279808932 p^{18} T^{22} + 1573353173 p^{24} T^{24} - 13135862 p^{30} T^{26} + 79383 p^{36} T^{28} - 394 p^{42} T^{30} + p^{48} T^{32} \) |
| 11 | \( 1 + 5842 T^{2} + 17711847 T^{4} + 34861793198 T^{6} + 46760130210965 T^{8} + 35906625944396532 T^{10} - 10800538135038597014 T^{12} - \)\(85\!\cdots\!92\)\( T^{14} - \)\(15\!\cdots\!02\)\( T^{16} - \)\(85\!\cdots\!92\)\( p^{6} T^{18} - 10800538135038597014 p^{12} T^{20} + 35906625944396532 p^{18} T^{22} + 46760130210965 p^{24} T^{24} + 34861793198 p^{30} T^{26} + 17711847 p^{36} T^{28} + 5842 p^{42} T^{30} + p^{48} T^{32} \) | |
| 13 | \( ( 1 - 14750 T^{2} + 100165933 T^{4} - 31441942490 p T^{6} + 1095322120237240 T^{8} - 31441942490 p^{7} T^{10} + 100165933 p^{12} T^{12} - 14750 p^{18} T^{14} + p^{24} T^{16} )^{2} \) | |
| 17 | \( 1 - 17908 T^{2} + 183721800 T^{4} - 1337837599208 T^{6} + 6758666240315906 T^{8} - 17525131900492504332 T^{10} - \)\(47\!\cdots\!28\)\( T^{12} + \)\(88\!\cdots\!12\)\( T^{14} - \)\(56\!\cdots\!25\)\( T^{16} + \)\(88\!\cdots\!12\)\( p^{6} T^{18} - \)\(47\!\cdots\!28\)\( p^{12} T^{20} - 17525131900492504332 p^{18} T^{22} + 6758666240315906 p^{24} T^{24} - 1337837599208 p^{30} T^{26} + 183721800 p^{36} T^{28} - 17908 p^{42} T^{30} + p^{48} T^{32} \) | |
| 19 | \( ( 1 + 36 T + 12655 T^{2} + 440028 T^{3} + 69273163 T^{4} + 128221056 T^{5} - 8891738156 p T^{6} - 30396011988456 T^{7} - 3052907831237222 T^{8} - 30396011988456 p^{3} T^{9} - 8891738156 p^{7} T^{10} + 128221056 p^{9} T^{11} + 69273163 p^{12} T^{12} + 440028 p^{15} T^{13} + 12655 p^{18} T^{14} + 36 p^{21} T^{15} + p^{24} T^{16} )^{2} \) | |
| 23 | \( 1 + 25876 T^{2} + 243513432 T^{4} + 1681286205992 T^{6} + 1505945652861218 T^{8} - \)\(37\!\cdots\!76\)\( T^{10} - \)\(25\!\cdots\!88\)\( T^{12} + \)\(23\!\cdots\!32\)\( p^{2} T^{14} + \)\(15\!\cdots\!91\)\( T^{16} + \)\(23\!\cdots\!32\)\( p^{8} T^{18} - \)\(25\!\cdots\!88\)\( p^{12} T^{20} - \)\(37\!\cdots\!76\)\( p^{18} T^{22} + 1505945652861218 p^{24} T^{24} + 1681286205992 p^{30} T^{26} + 243513432 p^{36} T^{28} + 25876 p^{42} T^{30} + p^{48} T^{32} \) | |
| 29 | \( ( 1 - 2138 p T^{2} + 2900130409 T^{4} - 97031213045314 T^{6} + 2790721647797586244 T^{8} - 97031213045314 p^{6} T^{10} + 2900130409 p^{12} T^{12} - 2138 p^{19} T^{14} + p^{24} T^{16} )^{2} \) | |
| 31 | \( ( 1 + 354 T + 75094 T^{2} + 11795988 T^{3} + 883174573 T^{4} + 3505223700 T^{5} - 50834565509222 T^{6} - 19486802838799662 T^{7} - 3853637391311667068 T^{8} - 19486802838799662 p^{3} T^{9} - 50834565509222 p^{6} T^{10} + 3505223700 p^{9} T^{11} + 883174573 p^{12} T^{12} + 11795988 p^{15} T^{13} + 75094 p^{18} T^{14} + 354 p^{21} T^{15} + p^{24} T^{16} )^{2} \) | |
| 37 | \( ( 1 - 38 T - 91851 T^{2} - 20169358 T^{3} + 5124013139 T^{4} + 1492587433200 T^{5} + 200369748531112 T^{6} - 60212649379130144 T^{7} - 16861961095425650754 T^{8} - 60212649379130144 p^{3} T^{9} + 200369748531112 p^{6} T^{10} + 1492587433200 p^{9} T^{11} + 5124013139 p^{12} T^{12} - 20169358 p^{15} T^{13} - 91851 p^{18} T^{14} - 38 p^{21} T^{15} + p^{24} T^{16} )^{2} \) | |
| 41 | \( ( 1 + 311308 T^{2} + 49778800408 T^{4} + 5249339160024580 T^{6} + \)\(41\!\cdots\!42\)\( T^{8} + 5249339160024580 p^{6} T^{10} + 49778800408 p^{12} T^{12} + 311308 p^{18} T^{14} + p^{24} T^{16} )^{2} \) | |
| 43 | \( ( 1 - 352 T + 171577 T^{2} - 15193816 T^{3} + 8385609238 T^{4} - 15193816 p^{3} T^{5} + 171577 p^{6} T^{6} - 352 p^{9} T^{7} + p^{12} T^{8} )^{4} \) | |
| 47 | \( 1 - 307672 T^{2} + 35469631884 T^{4} - 1804225864402640 T^{6} + 27431766123470856938 T^{8} + \)\(11\!\cdots\!68\)\( T^{10} - \)\(25\!\cdots\!40\)\( T^{12} + \)\(15\!\cdots\!32\)\( T^{14} - \)\(22\!\cdots\!25\)\( T^{16} + \)\(15\!\cdots\!32\)\( p^{6} T^{18} - \)\(25\!\cdots\!40\)\( p^{12} T^{20} + \)\(11\!\cdots\!68\)\( p^{18} T^{22} + 27431766123470856938 p^{24} T^{24} - 1804225864402640 p^{30} T^{26} + 35469631884 p^{36} T^{28} - 307672 p^{42} T^{30} + p^{48} T^{32} \) | |
| 53 | \( 1 + 938962 T^{2} + 467576763291 T^{4} + 165402918375811934 T^{6} + \)\(46\!\cdots\!33\)\( T^{8} + \)\(10\!\cdots\!96\)\( T^{10} + \)\(22\!\cdots\!82\)\( T^{12} + \)\(39\!\cdots\!84\)\( T^{14} + \)\(62\!\cdots\!74\)\( T^{16} + \)\(39\!\cdots\!84\)\( p^{6} T^{18} + \)\(22\!\cdots\!82\)\( p^{12} T^{20} + \)\(10\!\cdots\!96\)\( p^{18} T^{22} + \)\(46\!\cdots\!33\)\( p^{24} T^{24} + 165402918375811934 p^{30} T^{26} + 467576763291 p^{36} T^{28} + 938962 p^{42} T^{30} + p^{48} T^{32} \) | |
| 59 | \( 1 - 937618 T^{2} + 432339337947 T^{4} - 128957067202564334 T^{6} + \)\(29\!\cdots\!01\)\( T^{8} - \)\(61\!\cdots\!08\)\( T^{10} + \)\(14\!\cdots\!42\)\( T^{12} - \)\(10\!\cdots\!40\)\( p^{2} T^{14} + \)\(66\!\cdots\!62\)\( p^{4} T^{16} - \)\(10\!\cdots\!40\)\( p^{8} T^{18} + \)\(14\!\cdots\!42\)\( p^{12} T^{20} - \)\(61\!\cdots\!08\)\( p^{18} T^{22} + \)\(29\!\cdots\!01\)\( p^{24} T^{24} - 128957067202564334 p^{30} T^{26} + 432339337947 p^{36} T^{28} - 937618 p^{42} T^{30} + p^{48} T^{32} \) | |
| 61 | \( ( 1 + 816 T + 1193926 T^{2} + 13002144 p T^{3} + 758515460050 T^{4} + 6819529429512 p T^{5} + 297926214028110592 T^{6} + \)\(13\!\cdots\!16\)\( T^{7} + \)\(81\!\cdots\!47\)\( T^{8} + \)\(13\!\cdots\!16\)\( p^{3} T^{9} + 297926214028110592 p^{6} T^{10} + 6819529429512 p^{10} T^{11} + 758515460050 p^{12} T^{12} + 13002144 p^{16} T^{13} + 1193926 p^{18} T^{14} + 816 p^{21} T^{15} + p^{24} T^{16} )^{2} \) | |
| 67 | \( ( 1 + 764 T - 273663 T^{2} - 344243228 T^{3} - 11559707059 T^{4} + 21061240190208 T^{5} - 22917938128164926 T^{6} + 4683940411980034808 T^{7} + \)\(15\!\cdots\!66\)\( T^{8} + 4683940411980034808 p^{3} T^{9} - 22917938128164926 p^{6} T^{10} + 21061240190208 p^{9} T^{11} - 11559707059 p^{12} T^{12} - 344243228 p^{15} T^{13} - 273663 p^{18} T^{14} + 764 p^{21} T^{15} + p^{24} T^{16} )^{2} \) | |
| 71 | \( ( 1 - 1856440 T^{2} + 1761671174068 T^{4} - 1073069689503907240 T^{6} + \)\(45\!\cdots\!66\)\( T^{8} - 1073069689503907240 p^{6} T^{10} + 1761671174068 p^{12} T^{12} - 1856440 p^{18} T^{14} + p^{24} T^{16} )^{2} \) | |
| 73 | \( ( 1 + 1350 T + 1649863 T^{2} + 1407190050 T^{3} + 1214635307077 T^{4} + 919047647125260 T^{5} + 624120328609647082 T^{6} + \)\(41\!\cdots\!20\)\( T^{7} + \)\(25\!\cdots\!02\)\( T^{8} + \)\(41\!\cdots\!20\)\( p^{3} T^{9} + 624120328609647082 p^{6} T^{10} + 919047647125260 p^{9} T^{11} + 1214635307077 p^{12} T^{12} + 1407190050 p^{15} T^{13} + 1649863 p^{18} T^{14} + 1350 p^{21} T^{15} + p^{24} T^{16} )^{2} \) | |
| 79 | \( ( 1 + 182 T - 1303458 T^{2} + 51410212 T^{3} + 935956948277 T^{4} - 127720931027844 T^{5} - 463014917975063918 T^{6} + 34428841086426043166 T^{7} + \)\(20\!\cdots\!88\)\( T^{8} + 34428841086426043166 p^{3} T^{9} - 463014917975063918 p^{6} T^{10} - 127720931027844 p^{9} T^{11} + 935956948277 p^{12} T^{12} + 51410212 p^{15} T^{13} - 1303458 p^{18} T^{14} + 182 p^{21} T^{15} + p^{24} T^{16} )^{2} \) | |
| 83 | \( ( 1 + 2984326 T^{2} + 4434375721357 T^{4} + 4272111895669322434 T^{6} + \)\(28\!\cdots\!68\)\( T^{8} + 4272111895669322434 p^{6} T^{10} + 4434375721357 p^{12} T^{12} + 2984326 p^{18} T^{14} + p^{24} T^{16} )^{2} \) | |
| 89 | \( 1 - 2563264 T^{2} + 3275474914692 T^{4} - 2650752142645956224 T^{6} + \)\(12\!\cdots\!98\)\( T^{8} + \)\(65\!\cdots\!92\)\( T^{10} - \)\(73\!\cdots\!16\)\( T^{12} + \)\(87\!\cdots\!44\)\( T^{14} - \)\(70\!\cdots\!05\)\( T^{16} + \)\(87\!\cdots\!44\)\( p^{6} T^{18} - \)\(73\!\cdots\!16\)\( p^{12} T^{20} + \)\(65\!\cdots\!92\)\( p^{18} T^{22} + \)\(12\!\cdots\!98\)\( p^{24} T^{24} - 2650752142645956224 p^{30} T^{26} + 3275474914692 p^{36} T^{28} - 2563264 p^{42} T^{30} + p^{48} T^{32} \) | |
| 97 | \( ( 1 - 2165930 T^{2} + 2100306931633 T^{4} - 316975242262618658 T^{6} - \)\(53\!\cdots\!72\)\( T^{8} - 316975242262618658 p^{6} T^{10} + 2100306931633 p^{12} T^{12} - 2165930 p^{18} T^{14} + p^{24} T^{16} )^{2} \) | |
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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
−2.95840187237013479615099002153, −2.78238634345708709309443282419, −2.61533623480734461307672275936, −2.53148572080518187227480101440, −2.52887062492109936393277588509, −2.38266346542616148351791433626, −2.34852543885013009298244581673, −2.26707478775007110286628792219, −2.22996470999556180095862059994, −1.72448720054161169681059898312, −1.71473084293138441761314249989, −1.67796320238435111741742480310, −1.67234977862149698056638382783, −1.64304524735341496611219640428, −1.62825289044305193895389147982, −1.19641289905962759398958801183, −1.09139915643883468909744306563, −0.986610211091218122616689936139, −0.955814519366399086550285932077, −0.889311145770136357043106808631, −0.75326318466503477988626633132, −0.40910187322158419476926970646, −0.22680378513484566205493684715, −0.19151335620659905818457719205, −0.05843175486299573575556118256, 0.05843175486299573575556118256, 0.19151335620659905818457719205, 0.22680378513484566205493684715, 0.40910187322158419476926970646, 0.75326318466503477988626633132, 0.889311145770136357043106808631, 0.955814519366399086550285932077, 0.986610211091218122616689936139, 1.09139915643883468909744306563, 1.19641289905962759398958801183, 1.62825289044305193895389147982, 1.64304524735341496611219640428, 1.67234977862149698056638382783, 1.67796320238435111741742480310, 1.71473084293138441761314249989, 1.72448720054161169681059898312, 2.22996470999556180095862059994, 2.26707478775007110286628792219, 2.34852543885013009298244581673, 2.38266346542616148351791433626, 2.52887062492109936393277588509, 2.53148572080518187227480101440, 2.61533623480734461307672275936, 2.78238634345708709309443282419, 2.95840187237013479615099002153
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.