Properties

Label 16-336e8-1.1-c6e8-0-2
Degree $16$
Conductor $1.624\times 10^{20}$
Sign $1$
Analytic cond. $1.27454\times 10^{15}$
Root an. cond. $8.79193$
Motivic weight $6$
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  − 108·3-s − 294·5-s − 232·7-s + 6.31e3·9-s − 378·11-s + 3.17e4·15-s + 852·17-s − 3.69e3·19-s + 2.50e4·21-s − 1.56e4·23-s + 1.36e4·25-s − 2.62e5·27-s − 6.86e4·29-s − 2.30e4·31-s + 4.08e4·33-s + 6.82e4·35-s + 1.59e4·37-s + 1.70e5·43-s − 1.85e6·45-s − 1.02e5·47-s + 1.05e5·49-s − 9.20e4·51-s + 1.96e5·53-s + 1.11e5·55-s + 3.98e5·57-s + 6.62e5·59-s − 2.39e4·61-s + ⋯
L(s)  = 1  − 4·3-s − 2.35·5-s − 0.676·7-s + 26/3·9-s − 0.283·11-s + 9.40·15-s + 0.173·17-s − 0.537·19-s + 2.70·21-s − 1.28·23-s + 0.874·25-s − 13.3·27-s − 2.81·29-s − 0.772·31-s + 1.13·33-s + 1.59·35-s + 0.314·37-s + 2.13·43-s − 20.3·45-s − 0.984·47-s + 0.897·49-s − 0.693·51-s + 1.31·53-s + 0.667·55-s + 2.15·57-s + 3.22·59-s − 0.105·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(7-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+3)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{32} \cdot 3^{8} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(1.27454\times 10^{15}\)
Root analytic conductor: \(8.79193\)
Motivic weight: \(6\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{32} \cdot 3^{8} \cdot 7^{8} ,\ ( \ : [3]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.03299023841\)
\(L(\frac12)\) \(\approx\) \(0.03299023841\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( ( 1 + p^{3} T + p^{5} T^{2} )^{4} \)
7 \( 1 + 232 T - 7394 p T^{2} + 56608 p^{3} T^{3} + 266611 p^{6} T^{4} + 56608 p^{9} T^{5} - 7394 p^{13} T^{6} + 232 p^{18} T^{7} + p^{24} T^{8} \)
good5 \( 1 + 294 T + 2911 p^{2} T^{2} + 12925122 T^{3} + 1936828549 T^{4} + 51436360476 p T^{5} + 1240518391354 p^{2} T^{6} + 30061064897544 p^{3} T^{7} + 738376277065834 p^{4} T^{8} + 30061064897544 p^{9} T^{9} + 1240518391354 p^{14} T^{10} + 51436360476 p^{19} T^{11} + 1936828549 p^{24} T^{12} + 12925122 p^{30} T^{13} + 2911 p^{38} T^{14} + 294 p^{42} T^{15} + p^{48} T^{16} \)
11 \( 1 + 378 T - 2135173 T^{2} + 3817304118 T^{3} + 2508469676425 T^{4} - 8435022909365340 T^{5} + 10093185308235805454 T^{6} + \)\(11\!\cdots\!76\)\( T^{7} - \)\(17\!\cdots\!94\)\( T^{8} + \)\(11\!\cdots\!76\)\( p^{6} T^{9} + 10093185308235805454 p^{12} T^{10} - 8435022909365340 p^{18} T^{11} + 2508469676425 p^{24} T^{12} + 3817304118 p^{30} T^{13} - 2135173 p^{36} T^{14} + 378 p^{42} T^{15} + p^{48} T^{16} \)
13 \( 1 - 8558882 T^{2} + 51931039845025 T^{4} - \)\(36\!\cdots\!06\)\( T^{6} + \)\(19\!\cdots\!28\)\( T^{8} - \)\(36\!\cdots\!06\)\( p^{12} T^{10} + 51931039845025 p^{24} T^{12} - 8558882 p^{36} T^{14} + p^{48} T^{16} \)
17 \( 1 - 852 T + 54742600 T^{2} - 46434538464 T^{3} + 1587802906011970 T^{4} - 5693633607771951876 T^{5} + \)\(25\!\cdots\!76\)\( T^{6} - \)\(23\!\cdots\!72\)\( T^{7} + \)\(26\!\cdots\!67\)\( T^{8} - \)\(23\!\cdots\!72\)\( p^{6} T^{9} + \)\(25\!\cdots\!76\)\( p^{12} T^{10} - 5693633607771951876 p^{18} T^{11} + 1587802906011970 p^{24} T^{12} - 46434538464 p^{30} T^{13} + 54742600 p^{36} T^{14} - 852 p^{42} T^{15} + p^{48} T^{16} \)
19 \( 1 + 3690 T + 116044543 T^{2} + 411456560670 T^{3} + 5942834641955341 T^{4} + 8841015559212216660 T^{5} + \)\(24\!\cdots\!98\)\( T^{6} - \)\(38\!\cdots\!20\)\( T^{7} + \)\(11\!\cdots\!54\)\( T^{8} - \)\(38\!\cdots\!20\)\( p^{6} T^{9} + \)\(24\!\cdots\!98\)\( p^{12} T^{10} + 8841015559212216660 p^{18} T^{11} + 5942834641955341 p^{24} T^{12} + 411456560670 p^{30} T^{13} + 116044543 p^{36} T^{14} + 3690 p^{42} T^{15} + p^{48} T^{16} \)
23 \( 1 + 15600 T - 403726396 T^{2} - 3841663026912 T^{3} + 7056563308688054 p T^{4} + \)\(93\!\cdots\!60\)\( T^{5} - \)\(34\!\cdots\!36\)\( T^{6} - \)\(33\!\cdots\!84\)\( T^{7} + \)\(65\!\cdots\!95\)\( T^{8} - \)\(33\!\cdots\!84\)\( p^{6} T^{9} - \)\(34\!\cdots\!36\)\( p^{12} T^{10} + \)\(93\!\cdots\!60\)\( p^{18} T^{11} + 7056563308688054 p^{25} T^{12} - 3841663026912 p^{30} T^{13} - 403726396 p^{36} T^{14} + 15600 p^{42} T^{15} + p^{48} T^{16} \)
29 \( ( 1 + 34302 T + 1826232133 T^{2} + 43345634488362 T^{3} + 47521785559402488 p T^{4} + 43345634488362 p^{6} T^{5} + 1826232133 p^{12} T^{6} + 34302 p^{18} T^{7} + p^{24} T^{8} )^{2} \)
31 \( 1 + 23028 T + 2354942230 T^{2} + 50159112966456 T^{3} + 2662548285992828977 T^{4} + \)\(45\!\cdots\!72\)\( T^{5} + \)\(11\!\cdots\!90\)\( T^{6} - \)\(48\!\cdots\!76\)\( T^{7} - \)\(15\!\cdots\!32\)\( T^{8} - \)\(48\!\cdots\!76\)\( p^{6} T^{9} + \)\(11\!\cdots\!90\)\( p^{12} T^{10} + \)\(45\!\cdots\!72\)\( p^{18} T^{11} + 2662548285992828977 p^{24} T^{12} + 50159112966456 p^{30} T^{13} + 2354942230 p^{36} T^{14} + 23028 p^{42} T^{15} + p^{48} T^{16} \)
37 \( 1 - 15914 T - 3568300341 T^{2} + 43959244622042 T^{3} + 7152959366788934153 T^{4} - \)\(12\!\cdots\!92\)\( T^{5} + \)\(25\!\cdots\!62\)\( T^{6} + \)\(11\!\cdots\!36\)\( T^{7} - \)\(94\!\cdots\!66\)\( T^{8} + \)\(11\!\cdots\!36\)\( p^{6} T^{9} + \)\(25\!\cdots\!62\)\( p^{12} T^{10} - \)\(12\!\cdots\!92\)\( p^{18} T^{11} + 7152959366788934153 p^{24} T^{12} + 43959244622042 p^{30} T^{13} - 3568300341 p^{36} T^{14} - 15914 p^{42} T^{15} + p^{48} T^{16} \)
41 \( 1 - 17099564888 T^{2} + \)\(12\!\cdots\!52\)\( T^{4} - \)\(62\!\cdots\!44\)\( T^{6} + \)\(26\!\cdots\!22\)\( T^{8} - \)\(62\!\cdots\!44\)\( p^{12} T^{10} + \)\(12\!\cdots\!52\)\( p^{24} T^{12} - 17099564888 p^{36} T^{14} + p^{48} T^{16} \)
43 \( ( 1 - 85022 T + 13346564329 T^{2} - 318915169525694 T^{3} + 1426277211692979340 p T^{4} - 318915169525694 p^{6} T^{5} + 13346564329 p^{12} T^{6} - 85022 p^{18} T^{7} + p^{24} T^{8} )^{2} \)
47 \( 1 + 102180 T + 27603035992 T^{2} + 2464866190918560 T^{3} + \)\(31\!\cdots\!90\)\( T^{4} + \)\(34\!\cdots\!80\)\( T^{5} + \)\(43\!\cdots\!64\)\( T^{6} + \)\(52\!\cdots\!60\)\( T^{7} + \)\(63\!\cdots\!71\)\( T^{8} + \)\(52\!\cdots\!60\)\( p^{6} T^{9} + \)\(43\!\cdots\!64\)\( p^{12} T^{10} + \)\(34\!\cdots\!80\)\( p^{18} T^{11} + \)\(31\!\cdots\!90\)\( p^{24} T^{12} + 2464866190918560 p^{30} T^{13} + 27603035992 p^{36} T^{14} + 102180 p^{42} T^{15} + p^{48} T^{16} \)
53 \( 1 - 196410 T - 38984346121 T^{2} + 6599191419883194 T^{3} + \)\(14\!\cdots\!17\)\( T^{4} - \)\(12\!\cdots\!20\)\( T^{5} - \)\(42\!\cdots\!38\)\( T^{6} + \)\(10\!\cdots\!28\)\( T^{7} + \)\(10\!\cdots\!38\)\( T^{8} + \)\(10\!\cdots\!28\)\( p^{6} T^{9} - \)\(42\!\cdots\!38\)\( p^{12} T^{10} - \)\(12\!\cdots\!20\)\( p^{18} T^{11} + \)\(14\!\cdots\!17\)\( p^{24} T^{12} + 6599191419883194 p^{30} T^{13} - 38984346121 p^{36} T^{14} - 196410 p^{42} T^{15} + p^{48} T^{16} \)
59 \( 1 - 662550 T + 297324338131 T^{2} - 100045163051569050 T^{3} + \)\(26\!\cdots\!05\)\( T^{4} - \)\(54\!\cdots\!80\)\( T^{5} + \)\(97\!\cdots\!82\)\( T^{6} - \)\(15\!\cdots\!12\)\( T^{7} + \)\(26\!\cdots\!22\)\( T^{8} - \)\(15\!\cdots\!12\)\( p^{6} T^{9} + \)\(97\!\cdots\!82\)\( p^{12} T^{10} - \)\(54\!\cdots\!80\)\( p^{18} T^{11} + \)\(26\!\cdots\!05\)\( p^{24} T^{12} - 100045163051569050 p^{30} T^{13} + 297324338131 p^{36} T^{14} - 662550 p^{42} T^{15} + p^{48} T^{16} \)
61 \( 1 + 23928 T + 126300829972 T^{2} + 3017559607278432 T^{3} + \)\(82\!\cdots\!86\)\( T^{4} - \)\(46\!\cdots\!88\)\( T^{5} + \)\(30\!\cdots\!96\)\( T^{6} - \)\(59\!\cdots\!44\)\( T^{7} + \)\(97\!\cdots\!15\)\( T^{8} - \)\(59\!\cdots\!44\)\( p^{6} T^{9} + \)\(30\!\cdots\!96\)\( p^{12} T^{10} - \)\(46\!\cdots\!88\)\( p^{18} T^{11} + \)\(82\!\cdots\!86\)\( p^{24} T^{12} + 3017559607278432 p^{30} T^{13} + 126300829972 p^{36} T^{14} + 23928 p^{42} T^{15} + p^{48} T^{16} \)
67 \( 1 + 774838 T + 57124744695 T^{2} - 24983171620492342 T^{3} + \)\(40\!\cdots\!33\)\( T^{4} + \)\(14\!\cdots\!44\)\( T^{5} - \)\(15\!\cdots\!70\)\( T^{6} + \)\(36\!\cdots\!84\)\( T^{7} + \)\(61\!\cdots\!98\)\( T^{8} + \)\(36\!\cdots\!84\)\( p^{6} T^{9} - \)\(15\!\cdots\!70\)\( p^{12} T^{10} + \)\(14\!\cdots\!44\)\( p^{18} T^{11} + \)\(40\!\cdots\!33\)\( p^{24} T^{12} - 24983171620492342 p^{30} T^{13} + 57124744695 p^{36} T^{14} + 774838 p^{42} T^{15} + p^{48} T^{16} \)
71 \( ( 1 - 360948 T - 5595070784 T^{2} - 14726539173115548 T^{3} + \)\(25\!\cdots\!14\)\( T^{4} - 14726539173115548 p^{6} T^{5} - 5595070784 p^{12} T^{6} - 360948 p^{18} T^{7} + p^{24} T^{8} )^{2} \)
73 \( 1 + 1219050 T + 810535614427 T^{2} + 384213653336359350 T^{3} + \)\(12\!\cdots\!45\)\( T^{4} + \)\(44\!\cdots\!80\)\( T^{5} + \)\(25\!\cdots\!58\)\( p T^{6} + \)\(87\!\cdots\!40\)\( T^{7} + \)\(39\!\cdots\!06\)\( T^{8} + \)\(87\!\cdots\!40\)\( p^{6} T^{9} + \)\(25\!\cdots\!58\)\( p^{13} T^{10} + \)\(44\!\cdots\!80\)\( p^{18} T^{11} + \)\(12\!\cdots\!45\)\( p^{24} T^{12} + 384213653336359350 p^{30} T^{13} + 810535614427 p^{36} T^{14} + 1219050 p^{42} T^{15} + p^{48} T^{16} \)
79 \( 1 - 493868 T - 313962316602 T^{2} + 156596837229176168 T^{3} + \)\(14\!\cdots\!09\)\( T^{4} + \)\(14\!\cdots\!52\)\( T^{5} - \)\(19\!\cdots\!66\)\( T^{6} - \)\(57\!\cdots\!04\)\( T^{7} + \)\(97\!\cdots\!76\)\( T^{8} - \)\(57\!\cdots\!04\)\( p^{6} T^{9} - \)\(19\!\cdots\!66\)\( p^{12} T^{10} + \)\(14\!\cdots\!52\)\( p^{18} T^{11} + \)\(14\!\cdots\!09\)\( p^{24} T^{12} + 156596837229176168 p^{30} T^{13} - 313962316602 p^{36} T^{14} - 493868 p^{42} T^{15} + p^{48} T^{16} \)
83 \( 1 - 905383156826 T^{2} + \)\(60\!\cdots\!65\)\( T^{4} - \)\(27\!\cdots\!66\)\( T^{6} + \)\(10\!\cdots\!40\)\( T^{8} - \)\(27\!\cdots\!66\)\( p^{12} T^{10} + \)\(60\!\cdots\!65\)\( p^{24} T^{12} - 905383156826 p^{36} T^{14} + p^{48} T^{16} \)
89 \( 1 - 604260 T + 713675215696 T^{2} - 357700871506872960 T^{3} + \)\(46\!\cdots\!90\)\( T^{4} - \)\(71\!\cdots\!80\)\( T^{5} + \)\(93\!\cdots\!92\)\( T^{6} - \)\(18\!\cdots\!88\)\( T^{7} + \)\(12\!\cdots\!27\)\( T^{8} - \)\(18\!\cdots\!88\)\( p^{6} T^{9} + \)\(93\!\cdots\!92\)\( p^{12} T^{10} - \)\(71\!\cdots\!80\)\( p^{18} T^{11} + \)\(46\!\cdots\!90\)\( p^{24} T^{12} - 357700871506872960 p^{30} T^{13} + 713675215696 p^{36} T^{14} - 604260 p^{42} T^{15} + p^{48} T^{16} \)
97 \( 1 - 3280322818322 T^{2} + \)\(56\!\cdots\!37\)\( T^{4} - \)\(70\!\cdots\!86\)\( T^{6} + \)\(68\!\cdots\!92\)\( T^{8} - \)\(70\!\cdots\!86\)\( p^{12} T^{10} + \)\(56\!\cdots\!37\)\( p^{24} T^{12} - 3280322818322 p^{36} T^{14} + p^{48} T^{16} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−4.15884467192261245327972783742, −3.98122077892947745006236634199, −3.84316381269460549152803503799, −3.74198185651049828616342340081, −3.60326499913488055069706767975, −3.59689235260945086938509851231, −3.13691847388820240238098472157, −3.03905954129942721469071476070, −2.84958621299390039535100245955, −2.81772918019281777945991637935, −2.40788040709556350662970369684, −2.16331454189434700377337234338, −2.14844314979307317915116876627, −1.86473449694360329688632801927, −1.74069838835064844463006974404, −1.69073112764812550410989528736, −1.42105078427308972461534304276, −1.08774148081665651024411130020, −1.00474660617053273415398584864, −0.76910938542099163866223704796, −0.52088961541047117603371703026, −0.50398298482539496702012137892, −0.36793496784711099151326711624, −0.29853622225695032371415622886, −0.04765513977844235000000666712, 0.04765513977844235000000666712, 0.29853622225695032371415622886, 0.36793496784711099151326711624, 0.50398298482539496702012137892, 0.52088961541047117603371703026, 0.76910938542099163866223704796, 1.00474660617053273415398584864, 1.08774148081665651024411130020, 1.42105078427308972461534304276, 1.69073112764812550410989528736, 1.74069838835064844463006974404, 1.86473449694360329688632801927, 2.14844314979307317915116876627, 2.16331454189434700377337234338, 2.40788040709556350662970369684, 2.81772918019281777945991637935, 2.84958621299390039535100245955, 3.03905954129942721469071476070, 3.13691847388820240238098472157, 3.59689235260945086938509851231, 3.60326499913488055069706767975, 3.74198185651049828616342340081, 3.84316381269460549152803503799, 3.98122077892947745006236634199, 4.15884467192261245327972783742

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

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