Properties

Degree 32
Conductor $ 2^{80} \cdot 7^{16} $
Sign $1$
Motivic weight 2
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 16·9-s − 48·17-s − 60·25-s − 16·29-s + 72·37-s + 64·49-s + 40·53-s + 216·61-s − 312·73-s + 122·81-s + 24·89-s − 480·101-s + 112·109-s + 1.32e3·113-s + 432·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 768·153-s + 157-s + 163-s + 167-s + 2.11e3·169-s + 173-s + ⋯
L(s)  = 1  − 1.77·9-s − 2.82·17-s − 2.39·25-s − 0.551·29-s + 1.94·37-s + 1.30·49-s + 0.754·53-s + 3.54·61-s − 4.27·73-s + 1.50·81-s + 0.269·89-s − 4.75·101-s + 1.02·109-s + 11.7·113-s + 3.57·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 5.01·153-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 12.4·169-s + 0.00578·173-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{80} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{80} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(32\)
\( N \)  =  \(2^{80} \cdot 7^{16}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(2\)
character  :  induced by $\chi_{224} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((32,\ 2^{80} \cdot 7^{16} ,\ ( \ : [1]^{16} ),\ 1 )\)
\(L(\frac{3}{2})\)  \(\approx\)  \(6.72481\)
\(L(\frac12)\)  \(\approx\)  \(6.72481\)
\(L(2)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;7\}$,\(F_p(T)\) is a polynomial of degree 32. If $p \in \{2,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 31.
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 - 64 T^{2} + 324 p T^{4} + 64 p^{2} T^{6} - 2938 p^{4} T^{8} + 64 p^{6} T^{10} + 324 p^{9} T^{12} - 64 p^{12} T^{14} + p^{16} T^{16} \)
good3 \( 1 + 16 T^{2} + 134 T^{4} + 1120 T^{6} + 7897 T^{8} + 73856 T^{10} + 1194422 T^{12} + 13226000 T^{14} + 117489172 T^{16} + 13226000 p^{4} T^{18} + 1194422 p^{8} T^{20} + 73856 p^{12} T^{22} + 7897 p^{16} T^{24} + 1120 p^{20} T^{26} + 134 p^{24} T^{28} + 16 p^{28} T^{30} + p^{32} T^{32} \)
5 \( ( 1 + 6 p T^{2} + 321 T^{4} + 48 p^{2} T^{5} - 186 p T^{6} + 27792 p T^{7} - 137884 T^{8} + 27792 p^{3} T^{9} - 186 p^{5} T^{10} + 48 p^{8} T^{11} + 321 p^{8} T^{12} + 6 p^{13} T^{14} + p^{16} T^{16} )^{2} \)
11 \( 1 - 432 T^{2} + 7714 p T^{4} - 7780896 T^{6} + 22311049 T^{8} + 70438127616 T^{10} + 13160037970 p T^{12} - 2441540782042416 T^{14} + 459119675281580212 T^{16} - 2441540782042416 p^{4} T^{18} + 13160037970 p^{9} T^{20} + 70438127616 p^{12} T^{22} + 22311049 p^{16} T^{24} - 7780896 p^{20} T^{26} + 7714 p^{25} T^{28} - 432 p^{28} T^{30} + p^{32} T^{32} \)
13 \( ( 1 - 1056 T^{2} + 510492 T^{4} - 150812640 T^{6} + 30377120006 T^{8} - 150812640 p^{4} T^{10} + 510492 p^{8} T^{12} - 1056 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
17 \( ( 1 + 24 T + 1022 T^{2} + 19920 T^{3} + 479985 T^{4} + 519360 p T^{5} + 194168350 T^{6} + 193952472 p T^{7} + 67118725988 T^{8} + 193952472 p^{3} T^{9} + 194168350 p^{4} T^{10} + 519360 p^{7} T^{11} + 479985 p^{8} T^{12} + 19920 p^{10} T^{13} + 1022 p^{12} T^{14} + 24 p^{14} T^{15} + p^{16} T^{16} )^{2} \)
19 \( 1 + 1440 T^{2} + 935910 T^{4} + 23102400 p T^{6} + 186482707321 T^{8} + 56682302220480 T^{10} + 7723885307690070 T^{12} - 285644590063174560 T^{14} - \)\(25\!\cdots\!40\)\( T^{16} - 285644590063174560 p^{4} T^{18} + 7723885307690070 p^{8} T^{20} + 56682302220480 p^{12} T^{22} + 186482707321 p^{16} T^{24} + 23102400 p^{21} T^{26} + 935910 p^{24} T^{28} + 1440 p^{28} T^{30} + p^{32} T^{32} \)
23 \( 1 - 1648 T^{2} + 1017702 T^{4} - 249312032 T^{6} + 42919392953 T^{8} - 81219467468736 T^{10} + 45669118532829142 T^{12} + 12560582613209435984 T^{14} - \)\(19\!\cdots\!76\)\( T^{16} + 12560582613209435984 p^{4} T^{18} + 45669118532829142 p^{8} T^{20} - 81219467468736 p^{12} T^{22} + 42919392953 p^{16} T^{24} - 249312032 p^{20} T^{26} + 1017702 p^{24} T^{28} - 1648 p^{28} T^{30} + p^{32} T^{32} \)
29 \( ( 1 + 4 T + 2608 T^{2} + 8572 T^{3} + 2977102 T^{4} + 8572 p^{2} T^{5} + 2608 p^{4} T^{6} + 4 p^{6} T^{7} + p^{8} T^{8} )^{4} \)
31 \( 1 + 4944 T^{2} + 12401142 T^{4} + 21706728288 T^{6} + 30278528813833 T^{8} + 35431200182034624 T^{10} + 36428088221729894790 T^{12} + \)\(35\!\cdots\!36\)\( T^{14} + \)\(33\!\cdots\!48\)\( T^{16} + \)\(35\!\cdots\!36\)\( p^{4} T^{18} + 36428088221729894790 p^{8} T^{20} + 35431200182034624 p^{12} T^{22} + 30278528813833 p^{16} T^{24} + 21706728288 p^{20} T^{26} + 12401142 p^{24} T^{28} + 4944 p^{28} T^{30} + p^{32} T^{32} \)
37 \( ( 1 - 36 T - 2658 T^{2} + 179832 T^{3} + 2782297 T^{4} - 349680552 T^{5} + 4944965886 T^{6} + 248707896180 T^{7} - 11947732536204 T^{8} + 248707896180 p^{2} T^{9} + 4944965886 p^{4} T^{10} - 349680552 p^{6} T^{11} + 2782297 p^{8} T^{12} + 179832 p^{10} T^{13} - 2658 p^{12} T^{14} - 36 p^{14} T^{15} + p^{16} T^{16} )^{2} \)
41 \( ( 1 - 10816 T^{2} + 53826588 T^{4} - 162742362560 T^{6} + 329722113042758 T^{8} - 162742362560 p^{4} T^{10} + 53826588 p^{8} T^{12} - 10816 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
43 \( ( 1 + 5992 T^{2} + 25433596 T^{4} + 70773684952 T^{6} + 153071479203142 T^{8} + 70773684952 p^{4} T^{10} + 25433596 p^{8} T^{12} + 5992 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
47 \( 1 + 7200 T^{2} + 21086790 T^{4} + 446537280 p T^{6} - 28679941197479 T^{8} - 70284530432138880 T^{10} + \)\(21\!\cdots\!30\)\( T^{12} + \)\(13\!\cdots\!80\)\( T^{14} + \)\(38\!\cdots\!20\)\( T^{16} + \)\(13\!\cdots\!80\)\( p^{4} T^{18} + \)\(21\!\cdots\!30\)\( p^{8} T^{20} - 70284530432138880 p^{12} T^{22} - 28679941197479 p^{16} T^{24} + 446537280 p^{21} T^{26} + 21086790 p^{24} T^{28} + 7200 p^{28} T^{30} + p^{32} T^{32} \)
53 \( ( 1 - 20 T - 6098 T^{2} + 355352 T^{3} + 299365 p T^{4} - 1398217832 T^{5} + 8057716910 T^{6} + 2171403120260 T^{7} - 80650058761388 T^{8} + 2171403120260 p^{2} T^{9} + 8057716910 p^{4} T^{10} - 1398217832 p^{6} T^{11} + 299365 p^{9} T^{12} + 355352 p^{10} T^{13} - 6098 p^{12} T^{14} - 20 p^{14} T^{15} + p^{16} T^{16} )^{2} \)
59 \( 1 + 13056 T^{2} + 66899190 T^{4} + 216438091392 T^{6} + 1041428149652041 T^{8} + 6415617182341109568 T^{10} + \)\(27\!\cdots\!66\)\( T^{12} + \)\(85\!\cdots\!44\)\( T^{14} + \)\(25\!\cdots\!80\)\( T^{16} + \)\(85\!\cdots\!44\)\( p^{4} T^{18} + \)\(27\!\cdots\!66\)\( p^{8} T^{20} + 6415617182341109568 p^{12} T^{22} + 1041428149652041 p^{16} T^{24} + 216438091392 p^{20} T^{26} + 66899190 p^{24} T^{28} + 13056 p^{28} T^{30} + p^{32} T^{32} \)
61 \( ( 1 - 108 T + 15678 T^{2} - 1273320 T^{3} + 112514745 T^{4} - 8281698840 T^{5} + 635104006110 T^{6} - 41916162290292 T^{7} + 2814046950849908 T^{8} - 41916162290292 p^{2} T^{9} + 635104006110 p^{4} T^{10} - 8281698840 p^{6} T^{11} + 112514745 p^{8} T^{12} - 1273320 p^{10} T^{13} + 15678 p^{12} T^{14} - 108 p^{14} T^{15} + p^{16} T^{16} )^{2} \)
67 \( 1 - 19248 T^{2} + 171825062 T^{4} - 1105279125792 T^{6} + 6322631625605113 T^{8} - 28716244420307792256 T^{10} + \)\(95\!\cdots\!42\)\( T^{12} - \)\(31\!\cdots\!16\)\( T^{14} + \)\(13\!\cdots\!84\)\( T^{16} - \)\(31\!\cdots\!16\)\( p^{4} T^{18} + \)\(95\!\cdots\!42\)\( p^{8} T^{20} - 28716244420307792256 p^{12} T^{22} + 6322631625605113 p^{16} T^{24} - 1105279125792 p^{20} T^{26} + 171825062 p^{24} T^{28} - 19248 p^{28} T^{30} + p^{32} T^{32} \)
71 \( ( 1 + 24232 T^{2} + 314425660 T^{4} + 2642217676312 T^{6} + 15735155452974982 T^{8} + 2642217676312 p^{4} T^{10} + 314425660 p^{8} T^{12} + 24232 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
73 \( ( 1 + 156 T + 22270 T^{2} + 2208648 T^{3} + 203639977 T^{4} + 17676576216 T^{5} + 1406019151294 T^{6} + 115719264813828 T^{7} + 8350708700464372 T^{8} + 115719264813828 p^{2} T^{9} + 1406019151294 p^{4} T^{10} + 17676576216 p^{6} T^{11} + 203639977 p^{8} T^{12} + 2208648 p^{10} T^{13} + 22270 p^{12} T^{14} + 156 p^{14} T^{15} + p^{16} T^{16} )^{2} \)
79 \( 1 - 34640 T^{2} + 602988550 T^{4} - 7689396100960 T^{6} + 82922855327705561 T^{8} - \)\(77\!\cdots\!00\)\( T^{10} + \)\(63\!\cdots\!50\)\( T^{12} - \)\(46\!\cdots\!20\)\( T^{14} + \)\(30\!\cdots\!00\)\( T^{16} - \)\(46\!\cdots\!20\)\( p^{4} T^{18} + \)\(63\!\cdots\!50\)\( p^{8} T^{20} - \)\(77\!\cdots\!00\)\( p^{12} T^{22} + 82922855327705561 p^{16} T^{24} - 7689396100960 p^{20} T^{26} + 602988550 p^{24} T^{28} - 34640 p^{28} T^{30} + p^{32} T^{32} \)
83 \( ( 1 - 35368 T^{2} + 588987420 T^{4} - 6301011254168 T^{6} + 49538235115611782 T^{8} - 6301011254168 p^{4} T^{10} + 588987420 p^{8} T^{12} - 35368 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
89 \( ( 1 - 12 T + 8678 T^{2} - 103560 T^{3} + 5715441 T^{4} + 150778776 T^{5} - 475047477962 T^{6} + 26903544978684 T^{7} - 3925857799931356 T^{8} + 26903544978684 p^{2} T^{9} - 475047477962 p^{4} T^{10} + 150778776 p^{6} T^{11} + 5715441 p^{8} T^{12} - 103560 p^{10} T^{13} + 8678 p^{12} T^{14} - 12 p^{14} T^{15} + p^{16} T^{16} )^{2} \)
97 \( ( 1 - 39520 T^{2} + 789394140 T^{4} - 11378905321376 T^{6} + 124539045704549702 T^{8} - 11378905321376 p^{4} T^{10} + 789394140 p^{8} T^{12} - 39520 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−3.16335981417903371727982824187, −3.11765020079369687459135643074, −3.10491990161699864803769981103, −2.82235296501782338640749084961, −2.81175518254212283457069922435, −2.76405253086093310099125156390, −2.75337086249543600104361649701, −2.73718027712871353251950662767, −2.39464673530070294548699442503, −2.23581547545644182610997290427, −2.15218306056580456671119052096, −1.99909742582060575444615696951, −1.93325317674117615646919904034, −1.89868251426243492415123981123, −1.85880789534000425918911589658, −1.73311892041356998062349990124, −1.69371738830042619451989810749, −1.60236020686703517486543008118, −0.967093651953591820472313406476, −0.804959339867129721082139697475, −0.76857155342408799668170881777, −0.69667925966916715460957959241, −0.47921383347889656192385013247, −0.47547215199922233592959120398, −0.20429724425088123869977150741, 0.20429724425088123869977150741, 0.47547215199922233592959120398, 0.47921383347889656192385013247, 0.69667925966916715460957959241, 0.76857155342408799668170881777, 0.804959339867129721082139697475, 0.967093651953591820472313406476, 1.60236020686703517486543008118, 1.69371738830042619451989810749, 1.73311892041356998062349990124, 1.85880789534000425918911589658, 1.89868251426243492415123981123, 1.93325317674117615646919904034, 1.99909742582060575444615696951, 2.15218306056580456671119052096, 2.23581547545644182610997290427, 2.39464673530070294548699442503, 2.73718027712871353251950662767, 2.75337086249543600104361649701, 2.76405253086093310099125156390, 2.81175518254212283457069922435, 2.82235296501782338640749084961, 3.10491990161699864803769981103, 3.11765020079369687459135643074, 3.16335981417903371727982824187

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

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