Dirichlet series
| L(s) = 1 | − 37·4-s + 90·11-s + 613·16-s + 4·19-s + 516·29-s + 38·31-s + 576·41-s − 3.33e3·44-s − 2.74e3·49-s + 2.20e3·59-s + 20·61-s − 6.04e3·64-s + 2.95e3·71-s − 148·76-s − 218·79-s + 4.07e3·89-s + 7.13e3·101-s + 958·109-s − 1.90e4·116-s − 7.27e3·121-s − 1.40e3·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 4.62·4-s + 2.46·11-s + 9.57·16-s + 0.0482·19-s + 3.30·29-s + 0.220·31-s + 2.19·41-s − 11.4·44-s − 8.00·49-s + 4.85·59-s + 0.0419·61-s − 11.8·64-s + 4.93·71-s − 0.223·76-s − 0.310·79-s + 4.85·89-s + 7.02·101-s + 0.841·109-s − 15.2·116-s − 5.46·121-s − 1.01·124-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{64} \cdot 5^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{64} \cdot 5^{32}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(3^{64} \cdot 5^{32}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.72447\times 10^{33}\) |
| Root analytic conductor: | \(10.9306\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 3^{64} \cdot 5^{32} ,\ ( \ : [3/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(\approx\) | \(23.73992672\) |
| \(L(\frac12)\) | \(\approx\) | \(23.73992672\) |
| \(L(\frac{5}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) | |
| good | 2 | \( 1 + 37 T^{2} + 189 p^{2} T^{4} + 1417 p^{3} T^{6} + 140999 T^{8} + 1566027 T^{10} + 3965011 p^{2} T^{12} + 2288161 p^{6} T^{14} + 4806027 p^{8} T^{16} + 2288161 p^{12} T^{18} + 3965011 p^{14} T^{20} + 1566027 p^{18} T^{22} + 140999 p^{24} T^{24} + 1417 p^{33} T^{26} + 189 p^{38} T^{28} + 37 p^{42} T^{30} + p^{48} T^{32} \) |
| 7 | \( 1 + 2746 T^{2} + 3920535 T^{4} + 3799787198 T^{6} + 2786049777317 T^{8} + 1640205583267248 T^{10} + 115126196667169306 p T^{12} + \)\(33\!\cdots\!92\)\( T^{14} + \)\(12\!\cdots\!90\)\( T^{16} + \)\(33\!\cdots\!92\)\( p^{6} T^{18} + 115126196667169306 p^{13} T^{20} + 1640205583267248 p^{18} T^{22} + 2786049777317 p^{24} T^{24} + 3799787198 p^{30} T^{26} + 3920535 p^{36} T^{28} + 2746 p^{42} T^{30} + p^{48} T^{32} \) | |
| 11 | \( ( 1 - 45 T + 6673 T^{2} - 258714 T^{3} + 22094686 T^{4} - 764410356 T^{5} + 4377741821 p T^{6} - 1476648574551 T^{7} + 75234818756890 T^{8} - 1476648574551 p^{3} T^{9} + 4377741821 p^{7} T^{10} - 764410356 p^{9} T^{11} + 22094686 p^{12} T^{12} - 258714 p^{15} T^{13} + 6673 p^{18} T^{14} - 45 p^{21} T^{15} + p^{24} T^{16} )^{2} \) | |
| 13 | \( 1 + 16669 T^{2} + 131836626 T^{4} + 664100547179 T^{6} + 2449602067304840 T^{8} + 7361875938870845289 T^{10} + \)\(19\!\cdots\!74\)\( T^{12} + \)\(49\!\cdots\!35\)\( T^{14} + \)\(11\!\cdots\!38\)\( T^{16} + \)\(49\!\cdots\!35\)\( p^{6} T^{18} + \)\(19\!\cdots\!74\)\( p^{12} T^{20} + 7361875938870845289 p^{18} T^{22} + 2449602067304840 p^{24} T^{24} + 664100547179 p^{30} T^{26} + 131836626 p^{36} T^{28} + 16669 p^{42} T^{30} + p^{48} T^{32} \) | |
| 17 | \( 1 + 2450 p T^{2} + 902277741 T^{4} + 13395318881222 T^{6} + 151327515243116846 T^{8} + \)\(13\!\cdots\!06\)\( T^{10} + \)\(10\!\cdots\!23\)\( T^{12} + \)\(64\!\cdots\!62\)\( T^{14} + \)\(11\!\cdots\!22\)\( p^{2} T^{16} + \)\(64\!\cdots\!62\)\( p^{6} T^{18} + \)\(10\!\cdots\!23\)\( p^{12} T^{20} + \)\(13\!\cdots\!06\)\( p^{18} T^{22} + 151327515243116846 p^{24} T^{24} + 13395318881222 p^{30} T^{26} + 902277741 p^{36} T^{28} + 2450 p^{43} T^{30} + p^{48} T^{32} \) | |
| 19 | \( ( 1 - 2 T + 30531 T^{2} - 541942 T^{3} + 465624254 T^{4} - 13840800234 T^{5} + 4769220969385 T^{6} - 167543587475774 T^{7} + 36901908228236058 T^{8} - 167543587475774 p^{3} T^{9} + 4769220969385 p^{6} T^{10} - 13840800234 p^{9} T^{11} + 465624254 p^{12} T^{12} - 541942 p^{15} T^{13} + 30531 p^{18} T^{14} - 2 p^{21} T^{15} + p^{24} T^{16} )^{2} \) | |
| 23 | \( 1 + 71542 T^{2} + 2977162083 T^{4} + 90987915764606 T^{6} + 2211781935528224105 T^{8} + \)\(44\!\cdots\!08\)\( T^{10} + \)\(77\!\cdots\!42\)\( T^{12} + \)\(11\!\cdots\!84\)\( T^{14} + \)\(15\!\cdots\!22\)\( T^{16} + \)\(11\!\cdots\!84\)\( p^{6} T^{18} + \)\(77\!\cdots\!42\)\( p^{12} T^{20} + \)\(44\!\cdots\!08\)\( p^{18} T^{22} + 2211781935528224105 p^{24} T^{24} + 90987915764606 p^{30} T^{26} + 2977162083 p^{36} T^{28} + 71542 p^{42} T^{30} + p^{48} T^{32} \) | |
| 29 | \( ( 1 - 258 T + 114367 T^{2} - 26481270 T^{3} + 7156276849 T^{4} - 1361396125908 T^{5} + 293841763913878 T^{6} - 47439998673210312 T^{7} + 8428524038498605474 T^{8} - 47439998673210312 p^{3} T^{9} + 293841763913878 p^{6} T^{10} - 1361396125908 p^{9} T^{11} + 7156276849 p^{12} T^{12} - 26481270 p^{15} T^{13} + 114367 p^{18} T^{14} - 258 p^{21} T^{15} + p^{24} T^{16} )^{2} \) | |
| 31 | \( ( 1 - 19 T + 100674 T^{2} + 1895179 T^{3} + 5575730000 T^{4} + 206828855631 T^{5} + 229438602271414 T^{6} + 11942468233871849 T^{7} + 7393722570659007774 T^{8} + 11942468233871849 p^{3} T^{9} + 229438602271414 p^{6} T^{10} + 206828855631 p^{9} T^{11} + 5575730000 p^{12} T^{12} + 1895179 p^{15} T^{13} + 100674 p^{18} T^{14} - 19 p^{21} T^{15} + p^{24} T^{16} )^{2} \) | |
| 37 | \( 1 + 440548 T^{2} + 93516072384 T^{4} + 12630824927332076 T^{6} + \)\(12\!\cdots\!32\)\( T^{8} + \)\(89\!\cdots\!76\)\( T^{10} + \)\(52\!\cdots\!88\)\( T^{12} + \)\(27\!\cdots\!16\)\( T^{14} + \)\(13\!\cdots\!50\)\( T^{16} + \)\(27\!\cdots\!16\)\( p^{6} T^{18} + \)\(52\!\cdots\!88\)\( p^{12} T^{20} + \)\(89\!\cdots\!76\)\( p^{18} T^{22} + \)\(12\!\cdots\!32\)\( p^{24} T^{24} + 12630824927332076 p^{30} T^{26} + 93516072384 p^{36} T^{28} + 440548 p^{42} T^{30} + p^{48} T^{32} \) | |
| 41 | \( ( 1 - 288 T + 280552 T^{2} - 73221588 T^{3} + 40717856062 T^{4} - 9351582084468 T^{5} + 4102782756503920 T^{6} - 863614938864737844 T^{7} + \)\(31\!\cdots\!03\)\( T^{8} - 863614938864737844 p^{3} T^{9} + 4102782756503920 p^{6} T^{10} - 9351582084468 p^{9} T^{11} + 40717856062 p^{12} T^{12} - 73221588 p^{15} T^{13} + 280552 p^{18} T^{14} - 288 p^{21} T^{15} + p^{24} T^{16} )^{2} \) | |
| 43 | \( 1 + 950095 T^{2} + 437299902657 T^{4} + 129824459037449726 T^{6} + \)\(27\!\cdots\!76\)\( T^{8} + \)\(46\!\cdots\!08\)\( T^{10} + \)\(60\!\cdots\!71\)\( T^{12} + \)\(64\!\cdots\!11\)\( T^{14} + \)\(56\!\cdots\!34\)\( T^{16} + \)\(64\!\cdots\!11\)\( p^{6} T^{18} + \)\(60\!\cdots\!71\)\( p^{12} T^{20} + \)\(46\!\cdots\!08\)\( p^{18} T^{22} + \)\(27\!\cdots\!76\)\( p^{24} T^{24} + 129824459037449726 p^{30} T^{26} + 437299902657 p^{36} T^{28} + 950095 p^{42} T^{30} + p^{48} T^{32} \) | |
| 47 | \( 1 + 1066654 T^{2} + 532421607303 T^{4} + 164828062009030154 T^{6} + \)\(35\!\cdots\!61\)\( T^{8} + \)\(56\!\cdots\!48\)\( T^{10} + \)\(70\!\cdots\!38\)\( T^{12} + \)\(74\!\cdots\!20\)\( T^{14} + \)\(76\!\cdots\!14\)\( T^{16} + \)\(74\!\cdots\!20\)\( p^{6} T^{18} + \)\(70\!\cdots\!38\)\( p^{12} T^{20} + \)\(56\!\cdots\!48\)\( p^{18} T^{22} + \)\(35\!\cdots\!61\)\( p^{24} T^{24} + 164828062009030154 p^{30} T^{26} + 532421607303 p^{36} T^{28} + 1066654 p^{42} T^{30} + p^{48} T^{32} \) | |
| 53 | \( 1 + 1817908 T^{2} + 1611006757428 T^{4} + 922312347044964140 T^{6} + \)\(38\!\cdots\!04\)\( T^{8} + \)\(12\!\cdots\!20\)\( T^{10} + \)\(30\!\cdots\!20\)\( T^{12} + \)\(61\!\cdots\!04\)\( T^{14} + \)\(10\!\cdots\!54\)\( T^{16} + \)\(61\!\cdots\!04\)\( p^{6} T^{18} + \)\(30\!\cdots\!20\)\( p^{12} T^{20} + \)\(12\!\cdots\!20\)\( p^{18} T^{22} + \)\(38\!\cdots\!04\)\( p^{24} T^{24} + 922312347044964140 p^{30} T^{26} + 1611006757428 p^{36} T^{28} + 1817908 p^{42} T^{30} + p^{48} T^{32} \) | |
| 59 | \( ( 1 - 1101 T + 1632325 T^{2} - 1246231302 T^{3} + 1091627498134 T^{4} - 645042288302388 T^{5} + 420293547846430531 T^{6} - \)\(20\!\cdots\!11\)\( T^{7} + \)\(10\!\cdots\!66\)\( T^{8} - \)\(20\!\cdots\!11\)\( p^{3} T^{9} + 420293547846430531 p^{6} T^{10} - 645042288302388 p^{9} T^{11} + 1091627498134 p^{12} T^{12} - 1246231302 p^{15} T^{13} + 1632325 p^{18} T^{14} - 1101 p^{21} T^{15} + p^{24} T^{16} )^{2} \) | |
| 61 | \( ( 1 - 10 T + 1065681 T^{2} - 77244494 T^{3} + 572514786419 T^{4} - 60269722986600 T^{5} + 206056746724365154 T^{6} - 23165304849108125200 T^{7} + \)\(54\!\cdots\!08\)\( T^{8} - 23165304849108125200 p^{3} T^{9} + 206056746724365154 p^{6} T^{10} - 60269722986600 p^{9} T^{11} + 572514786419 p^{12} T^{12} - 77244494 p^{15} T^{13} + 1065681 p^{18} T^{14} - 10 p^{21} T^{15} + p^{24} T^{16} )^{2} \) | |
| 67 | \( 1 + 2656108 T^{2} + 3532638993804 T^{4} + 3143729963654001860 T^{6} + \)\(21\!\cdots\!78\)\( T^{8} + \)\(11\!\cdots\!88\)\( T^{10} + \)\(50\!\cdots\!76\)\( T^{12} + \)\(19\!\cdots\!72\)\( T^{14} + \)\(62\!\cdots\!47\)\( T^{16} + \)\(19\!\cdots\!72\)\( p^{6} T^{18} + \)\(50\!\cdots\!76\)\( p^{12} T^{20} + \)\(11\!\cdots\!88\)\( p^{18} T^{22} + \)\(21\!\cdots\!78\)\( p^{24} T^{24} + 3143729963654001860 p^{30} T^{26} + 3532638993804 p^{36} T^{28} + 2656108 p^{42} T^{30} + p^{48} T^{32} \) | |
| 71 | \( ( 1 - 1476 T + 3187450 T^{2} - 3360993624 T^{3} + 4161371889196 T^{4} - 3386869531237728 T^{5} + 3025874711031382510 T^{6} - \)\(19\!\cdots\!64\)\( T^{7} + \)\(13\!\cdots\!82\)\( T^{8} - \)\(19\!\cdots\!64\)\( p^{3} T^{9} + 3025874711031382510 p^{6} T^{10} - 3386869531237728 p^{9} T^{11} + 4161371889196 p^{12} T^{12} - 3360993624 p^{15} T^{13} + 3187450 p^{18} T^{14} - 1476 p^{21} T^{15} + p^{24} T^{16} )^{2} \) | |
| 73 | \( 1 + 3404446 T^{2} + 6126306632589 T^{4} + 7521328741695950774 T^{6} + \)\(69\!\cdots\!86\)\( T^{8} + \)\(51\!\cdots\!42\)\( T^{10} + \)\(31\!\cdots\!39\)\( T^{12} + \)\(15\!\cdots\!62\)\( T^{14} + \)\(66\!\cdots\!30\)\( T^{16} + \)\(15\!\cdots\!62\)\( p^{6} T^{18} + \)\(31\!\cdots\!39\)\( p^{12} T^{20} + \)\(51\!\cdots\!42\)\( p^{18} T^{22} + \)\(69\!\cdots\!86\)\( p^{24} T^{24} + 7521328741695950774 p^{30} T^{26} + 6126306632589 p^{36} T^{28} + 3404446 p^{42} T^{30} + p^{48} T^{32} \) | |
| 79 | \( ( 1 + 109 T + 2801466 T^{2} + 331495223 T^{3} + 3830980169408 T^{4} + 446725089992151 T^{5} + 3301239660918451750 T^{6} + \)\(34\!\cdots\!69\)\( T^{7} + \)\(19\!\cdots\!22\)\( T^{8} + \)\(34\!\cdots\!69\)\( p^{3} T^{9} + 3301239660918451750 p^{6} T^{10} + 446725089992151 p^{9} T^{11} + 3830980169408 p^{12} T^{12} + 331495223 p^{15} T^{13} + 2801466 p^{18} T^{14} + 109 p^{21} T^{15} + p^{24} T^{16} )^{2} \) | |
| 83 | \( 1 + 6737518 T^{2} + 256659614721 p T^{4} + 42075161735524670174 T^{6} + \)\(58\!\cdots\!81\)\( T^{8} + \)\(61\!\cdots\!72\)\( T^{10} + \)\(51\!\cdots\!38\)\( T^{12} + \)\(35\!\cdots\!76\)\( T^{14} + \)\(21\!\cdots\!14\)\( T^{16} + \)\(35\!\cdots\!76\)\( p^{6} T^{18} + \)\(51\!\cdots\!38\)\( p^{12} T^{20} + \)\(61\!\cdots\!72\)\( p^{18} T^{22} + \)\(58\!\cdots\!81\)\( p^{24} T^{24} + 42075161735524670174 p^{30} T^{26} + 256659614721 p^{37} T^{28} + 6737518 p^{42} T^{30} + p^{48} T^{32} \) | |
| 89 | \( ( 1 - 2037 T + 5372956 T^{2} - 7622227437 T^{3} + 11878232772349 T^{4} - 13022692821688500 T^{5} + 15169726987524966634 T^{6} - \)\(13\!\cdots\!46\)\( T^{7} + \)\(12\!\cdots\!80\)\( T^{8} - \)\(13\!\cdots\!46\)\( p^{3} T^{9} + 15169726987524966634 p^{6} T^{10} - 13022692821688500 p^{9} T^{11} + 11878232772349 p^{12} T^{12} - 7622227437 p^{15} T^{13} + 5372956 p^{18} T^{14} - 2037 p^{21} T^{15} + p^{24} T^{16} )^{2} \) | |
| 97 | \( 1 + 6325147 T^{2} + 19957023240705 T^{4} + 41151654463376937002 T^{6} + \)\(61\!\cdots\!84\)\( T^{8} + \)\(71\!\cdots\!96\)\( T^{10} + \)\(66\!\cdots\!35\)\( T^{12} + \)\(55\!\cdots\!75\)\( T^{14} + \)\(48\!\cdots\!30\)\( T^{16} + \)\(55\!\cdots\!75\)\( p^{6} T^{18} + \)\(66\!\cdots\!35\)\( p^{12} T^{20} + \)\(71\!\cdots\!96\)\( p^{18} T^{22} + \)\(61\!\cdots\!84\)\( p^{24} T^{24} + 41151654463376937002 p^{30} T^{26} + 19957023240705 p^{36} T^{28} + 6325147 p^{42} T^{30} + p^{48} T^{32} \) | |
| show more | ||
| show less | ||
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−1.69058239451996903867769637607, −1.68923441526931022837143467594, −1.66535710138387711667396144140, −1.63248483165862694921208007499, −1.57768007589075281822795368850, −1.47274203793044631473458190187, −1.45767078405961214393846690918, −1.44568713540376133810607024517, −1.40082793166661906214909051039, −1.18645673954448865670781870491, −1.09605918350408668000802006064, −0.829637153159304817262722857767, −0.795459719974522991118828433693, −0.76292138916473603447540538663, −0.69861553313859516375260535644, −0.69687126605436575228234959587, −0.68289884495884877015638892587, −0.60610105440904540375054898335, −0.59396362714844643970435840556, −0.49772264570663878215299649845, −0.44924185018653150220653678230, −0.44074670752905130658629967767, −0.35588066229596096729088267515, −0.14127159633678638782669511899, −0.084441934440631649648233859862, 0.084441934440631649648233859862, 0.14127159633678638782669511899, 0.35588066229596096729088267515, 0.44074670752905130658629967767, 0.44924185018653150220653678230, 0.49772264570663878215299649845, 0.59396362714844643970435840556, 0.60610105440904540375054898335, 0.68289884495884877015638892587, 0.69687126605436575228234959587, 0.69861553313859516375260535644, 0.76292138916473603447540538663, 0.795459719974522991118828433693, 0.829637153159304817262722857767, 1.09605918350408668000802006064, 1.18645673954448865670781870491, 1.40082793166661906214909051039, 1.44568713540376133810607024517, 1.45767078405961214393846690918, 1.47274203793044631473458190187, 1.57768007589075281822795368850, 1.63248483165862694921208007499, 1.66535710138387711667396144140, 1.68923441526931022837143467594, 1.69058239451996903867769637607
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.