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
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 3^{32} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 3^{32} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(2^{64} \cdot 3^{32} \cdot 7^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.45033\times 10^{28}\) |
| Root analytic conductor: | \(7.71193\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 2^{64} \cdot 3^{32} \cdot 7^{16} ,\ ( \ : [3/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(\approx\) | \(0.1853438180\) |
| \(L(\frac12)\) | \(\approx\) | \(0.1853438180\) |
| \(L(\frac{5}{2})\) | 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
−2.11883087752529687761457754884, −1.96726813337646930494692631449, −1.90698099821409430155158002506, −1.88872945978739345178632672030, −1.74431603761100430087874516399, −1.73371332019896879629343539303, −1.66603984071872707379475154358, −1.61279334616616587279668463224, −1.57785035943779076673226656169, −1.34033405742634834525018539231, −1.27177480811660621477425077068, −1.08748203715038715409827658256, −1.07007521477820572056678683223, −1.05774236902615291970067287810, −1.03693477737668297453739621535, −0.916477429235834045490268798112, −0.906576007330222998678357179156, −0.870653395969605574673208149724, −0.77111966981038671271788010639, −0.46113121890979819766727837279, −0.38480929888588060042711246001, −0.34300000304872457911850922919, −0.13454894418261246178983100699, −0.099513709393977380031489422703, −0.03287082706919651521109232894, 0.03287082706919651521109232894, 0.099513709393977380031489422703, 0.13454894418261246178983100699, 0.34300000304872457911850922919, 0.38480929888588060042711246001, 0.46113121890979819766727837279, 0.77111966981038671271788010639, 0.870653395969605574673208149724, 0.906576007330222998678357179156, 0.916477429235834045490268798112, 1.03693477737668297453739621535, 1.05774236902615291970067287810, 1.07007521477820572056678683223, 1.08748203715038715409827658256, 1.27177480811660621477425077068, 1.34033405742634834525018539231, 1.57785035943779076673226656169, 1.61279334616616587279668463224, 1.66603984071872707379475154358, 1.73371332019896879629343539303, 1.74431603761100430087874516399, 1.88872945978739345178632672030, 1.90698099821409430155158002506, 1.96726813337646930494692631449, 2.11883087752529687761457754884
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.