Properties

Label 32-405e16-1.1-c3e16-0-1
Degree $32$
Conductor $5.239\times 10^{41}$
Sign $1$
Analytic cond. $1.13015\times 10^{22}$
Root an. cond. $4.88833$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 37·4-s + 3·5-s + 90·11-s + 613·16-s − 4·19-s + 111·20-s − 31·25-s − 516·29-s + 38·31-s + 576·41-s + 3.33e3·44-s + 2.74e3·49-s + 270·55-s − 2.20e3·59-s + 20·61-s + 6.04e3·64-s + 2.95e3·71-s − 148·76-s + 218·79-s + 1.83e3·80-s − 4.07e3·89-s − 12·95-s − 1.14e3·100-s + 7.13e3·101-s − 958·109-s − 1.90e4·116-s − 7.27e3·121-s + ⋯
L(s)  = 1  + 37/8·4-s + 0.268·5-s + 2.46·11-s + 9.57·16-s − 0.0482·19-s + 1.24·20-s − 0.247·25-s − 3.30·29-s + 0.220·31-s + 2.19·41-s + 11.4·44-s + 8.00·49-s + 0.661·55-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 + 2.57·80-s − 4.85·89-s − 0.0129·95-s − 1.14·100-s + 7.02·101-s − 0.841·109-s − 15.2·116-s − 5.46·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{64} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{64} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(32\)
Conductor: \(3^{64} \cdot 5^{16}\)
Sign: $1$
Analytic conductor: \(1.13015\times 10^{22}\)
Root analytic conductor: \(4.88833\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{405} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((32,\ 3^{64} \cdot 5^{16} ,\ ( \ : [3/2]^{16} ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(395.8557987\)
\(L(\frac12)\) \(\approx\) \(395.8557987\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 - 3 T + 8 p T^{2} + 1533 T^{3} - 5696 T^{4} - 3717 p T^{5} + 30664 p^{2} T^{6} - 141909 p^{3} T^{7} + 23902 p^{4} T^{8} - 141909 p^{6} T^{9} + 30664 p^{8} T^{10} - 3717 p^{10} T^{11} - 5696 p^{12} T^{12} + 1533 p^{15} T^{13} + 8 p^{19} T^{14} - 3 p^{21} T^{15} + p^{24} T^{16} \)
good2 \( 1 - 37 T^{2} + 189 p^{2} T^{4} - 1417 p^{3} T^{6} + 140999 T^{8} - 1566027 T^{10} + 3965011 p^{2} T^{12} - 2288161 p^{6} T^{14} + 4806027 p^{8} T^{16} - 2288161 p^{12} T^{18} + 3965011 p^{14} T^{20} - 1566027 p^{18} T^{22} + 140999 p^{24} T^{24} - 1417 p^{33} T^{26} + 189 p^{38} T^{28} - 37 p^{42} T^{30} + p^{48} T^{32} \)
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 \( 1 - 6325147 T^{2} + 19957023240705 T^{4} - 41151654463376937002 T^{6} + \)\(61\!\cdots\!84\)\( T^{8} - \)\(71\!\cdots\!96\)\( T^{10} + \)\(66\!\cdots\!35\)\( T^{12} - \)\(55\!\cdots\!75\)\( T^{14} + \)\(48\!\cdots\!30\)\( T^{16} - \)\(55\!\cdots\!75\)\( p^{6} T^{18} + \)\(66\!\cdots\!35\)\( p^{12} T^{20} - \)\(71\!\cdots\!96\)\( p^{18} T^{22} + \)\(61\!\cdots\!84\)\( p^{24} T^{24} - 41151654463376937002 p^{30} T^{26} + 19957023240705 p^{36} T^{28} - 6325147 p^{42} T^{30} + p^{48} T^{32} \)
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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

−2.64899443503262398298075914424, −2.57355310935018226107201399724, −2.50522057766394048244578201069, −2.12895245230195716271781637006, −2.04315262497542045900212827515, −2.01156321733132860449882549978, −1.97499358715995143006589285920, −1.95543857899441385843886476649, −1.87376004343772699822657295731, −1.83968987236599317078043388112, −1.79782346393968862268877824258, −1.73599365153392747175113057457, −1.68240327088307298995586195710, −1.39711037919317918263641120161, −1.18017868612074415414107841402, −1.13626210309839831616397403449, −1.12023455635093209930695127748, −0.883004334132362219806282045639, −0.861577468108321535442427938223, −0.828897949149803896013174363248, −0.60554296402804699711901727378, −0.47242124316574013651271313146, −0.46612389354710749584063572319, −0.33554698057630893631387919997, −0.16068947218620211436366387141, 0.16068947218620211436366387141, 0.33554698057630893631387919997, 0.46612389354710749584063572319, 0.47242124316574013651271313146, 0.60554296402804699711901727378, 0.828897949149803896013174363248, 0.861577468108321535442427938223, 0.883004334132362219806282045639, 1.12023455635093209930695127748, 1.13626210309839831616397403449, 1.18017868612074415414107841402, 1.39711037919317918263641120161, 1.68240327088307298995586195710, 1.73599365153392747175113057457, 1.79782346393968862268877824258, 1.83968987236599317078043388112, 1.87376004343772699822657295731, 1.95543857899441385843886476649, 1.97499358715995143006589285920, 2.01156321733132860449882549978, 2.04315262497542045900212827515, 2.12895245230195716271781637006, 2.50522057766394048244578201069, 2.57355310935018226107201399724, 2.64899443503262398298075914424

Graph of the $Z$-function along the critical line

Plot not available for L-functions of degree greater than 10.