Properties

Degree 20
Conductor $ 2^{30} $
Sign $1$
Motivic weight 13
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 110·2-s + 3.69e3·4-s + 5.86e5·7-s − 1.27e5·8-s + 8.97e6·9-s + 6.45e7·14-s − 1.15e6·16-s + 2.17e8·17-s + 9.87e8·18-s − 7.86e7·23-s + 4.56e9·25-s + 2.16e9·28-s + 6.48e8·31-s − 1.11e9·32-s + 2.39e10·34-s + 3.31e10·36-s + 5.93e10·41-s − 8.65e9·46-s − 1.01e10·47-s − 2.20e11·49-s + 5.02e11·50-s − 7.50e10·56-s + 7.13e10·62-s + 5.27e12·63-s − 5.49e11·64-s + 8.02e11·68-s + 7.26e11·71-s + ⋯
L(s)  = 1  + 1.21·2-s + 0.450·4-s + 1.88·7-s − 0.172·8-s + 5.63·9-s + 2.29·14-s − 0.0172·16-s + 2.18·17-s + 6.84·18-s − 0.110·23-s + 3.73·25-s + 0.849·28-s + 0.131·31-s − 0.184·32-s + 2.65·34-s + 2.53·36-s + 1.95·41-s − 0.134·46-s − 0.137·47-s − 2.27·49-s + 4.54·50-s − 0.324·56-s + 0.159·62-s + 10.6·63-s − 0.999·64-s + 0.984·68-s + 0.672·71-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{30}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr =\mathstrut & \,\Lambda(14-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{30}\right)^{s/2} \, \Gamma_{\C}(s+13/2)^{10} \, L(s)\cr =\mathstrut & \,\Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(20\)
\( N \)  =  \(2^{30}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(13\)
character  :  induced by $\chi_{8} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(20,\ 2^{30} ,\ ( \ : [13/2]^{10} ),\ 1 )$
$L(7)$  $\approx$  $137.509$
$L(\frac12)$  $\approx$  $137.509$
$L(\frac{15}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 2$, \(F_p\) is a polynomial of degree 20. If $p = 2$, then $F_p$ is a polynomial of degree at most 19.
$p$$F_p$
bad2 \( 1 - 55 p T + 1051 p^{3} T^{2} - 6109 p^{6} T^{3} - 457 p^{11} T^{4} + 27561 p^{17} T^{5} - 457 p^{24} T^{6} - 6109 p^{32} T^{7} + 1051 p^{42} T^{8} - 55 p^{53} T^{9} + p^{65} T^{10} \)
good3 \( 1 - 8978642 T^{2} + 4658209641485 p^{2} T^{4} - 181750892080261208 p^{6} T^{6} + 64862993246405867138 p^{14} T^{8} - \)\(14\!\cdots\!28\)\( p^{18} T^{10} + 64862993246405867138 p^{40} T^{12} - 181750892080261208 p^{58} T^{14} + 4658209641485 p^{80} T^{16} - 8978642 p^{104} T^{18} + p^{130} T^{20} \)
5 \( 1 - 4565314338 T^{2} + 10147469641592691461 T^{4} - \)\(57\!\cdots\!32\)\( p^{2} T^{6} + \)\(10\!\cdots\!74\)\( p^{6} T^{8} - \)\(18\!\cdots\!44\)\( p^{10} T^{10} + \)\(10\!\cdots\!74\)\( p^{32} T^{12} - \)\(57\!\cdots\!32\)\( p^{54} T^{14} + 10147469641592691461 p^{78} T^{16} - 4565314338 p^{104} T^{18} + p^{130} T^{20} \)
7 \( ( 1 - 293480 T + 239610643203 T^{2} - 1947286877479264 p^{2} T^{3} + \)\(65\!\cdots\!26\)\( p^{2} T^{4} - \)\(39\!\cdots\!96\)\( p^{3} T^{5} + \)\(65\!\cdots\!26\)\( p^{15} T^{6} - 1947286877479264 p^{28} T^{7} + 239610643203 p^{39} T^{8} - 293480 p^{52} T^{9} + p^{65} T^{10} )^{2} \)
11 \( 1 - 1482155238098 p^{2} T^{2} + \)\(11\!\cdots\!01\)\( p^{4} T^{4} - \)\(59\!\cdots\!00\)\( p^{6} T^{6} + \)\(23\!\cdots\!38\)\( p^{8} T^{8} - \)\(74\!\cdots\!64\)\( p^{10} T^{10} + \)\(23\!\cdots\!38\)\( p^{34} T^{12} - \)\(59\!\cdots\!00\)\( p^{58} T^{14} + \)\(11\!\cdots\!01\)\( p^{82} T^{16} - 1482155238098 p^{106} T^{18} + p^{130} T^{20} \)
13 \( 1 - 1881700477395890 T^{2} + \)\(17\!\cdots\!13\)\( T^{4} - \)\(10\!\cdots\!00\)\( T^{6} + \)\(48\!\cdots\!02\)\( T^{8} - \)\(16\!\cdots\!20\)\( T^{10} + \)\(48\!\cdots\!02\)\( p^{26} T^{12} - \)\(10\!\cdots\!00\)\( p^{52} T^{14} + \)\(17\!\cdots\!13\)\( p^{78} T^{16} - 1881700477395890 p^{104} T^{18} + p^{130} T^{20} \)
17 \( ( 1 - 108663002 T + 32276400633367229 T^{2} - \)\(23\!\cdots\!84\)\( T^{3} + \)\(50\!\cdots\!42\)\( T^{4} - \)\(29\!\cdots\!76\)\( T^{5} + \)\(50\!\cdots\!42\)\( p^{13} T^{6} - \)\(23\!\cdots\!84\)\( p^{26} T^{7} + 32276400633367229 p^{39} T^{8} - 108663002 p^{52} T^{9} + p^{65} T^{10} )^{2} \)
19 \( 1 - 8567413845488966 p T^{2} + \)\(13\!\cdots\!97\)\( T^{4} - \)\(80\!\cdots\!36\)\( T^{6} + \)\(39\!\cdots\!66\)\( T^{8} - \)\(17\!\cdots\!68\)\( T^{10} + \)\(39\!\cdots\!66\)\( p^{26} T^{12} - \)\(80\!\cdots\!36\)\( p^{52} T^{14} + \)\(13\!\cdots\!97\)\( p^{78} T^{16} - 8567413845488966 p^{105} T^{18} + p^{130} T^{20} \)
23 \( ( 1 + 39339976 T + 1242751119340586771 T^{2} + \)\(42\!\cdots\!92\)\( T^{3} + \)\(76\!\cdots\!02\)\( T^{4} + \)\(38\!\cdots\!08\)\( T^{5} + \)\(76\!\cdots\!02\)\( p^{13} T^{6} + \)\(42\!\cdots\!92\)\( p^{26} T^{7} + 1242751119340586771 p^{39} T^{8} + 39339976 p^{52} T^{9} + p^{65} T^{10} )^{2} \)
29 \( 1 - 60047671171855484498 T^{2} + \)\(18\!\cdots\!41\)\( T^{4} - \)\(38\!\cdots\!60\)\( T^{6} + \)\(58\!\cdots\!78\)\( T^{8} - \)\(67\!\cdots\!84\)\( T^{10} + \)\(58\!\cdots\!78\)\( p^{26} T^{12} - \)\(38\!\cdots\!60\)\( p^{52} T^{14} + \)\(18\!\cdots\!41\)\( p^{78} T^{16} - 60047671171855484498 p^{104} T^{18} + p^{130} T^{20} \)
31 \( ( 1 - 324116896 T + 94759416869201467419 T^{2} - \)\(76\!\cdots\!28\)\( T^{3} + \)\(40\!\cdots\!58\)\( T^{4} - \)\(32\!\cdots\!48\)\( T^{5} + \)\(40\!\cdots\!58\)\( p^{13} T^{6} - \)\(76\!\cdots\!28\)\( p^{26} T^{7} + 94759416869201467419 p^{39} T^{8} - 324116896 p^{52} T^{9} + p^{65} T^{10} )^{2} \)
37 \( 1 - \)\(85\!\cdots\!14\)\( T^{2} + \)\(35\!\cdots\!29\)\( T^{4} - \)\(83\!\cdots\!24\)\( T^{6} + \)\(11\!\cdots\!78\)\( T^{8} - \)\(14\!\cdots\!64\)\( T^{10} + \)\(11\!\cdots\!78\)\( p^{26} T^{12} - \)\(83\!\cdots\!24\)\( p^{52} T^{14} + \)\(35\!\cdots\!29\)\( p^{78} T^{16} - \)\(85\!\cdots\!14\)\( p^{104} T^{18} + p^{130} T^{20} \)
41 \( ( 1 - 29662320178 T + \)\(14\!\cdots\!89\)\( T^{2} - \)\(15\!\cdots\!16\)\( T^{3} + \)\(10\!\cdots\!62\)\( T^{4} - \)\(29\!\cdots\!76\)\( T^{5} + \)\(10\!\cdots\!62\)\( p^{13} T^{6} - \)\(15\!\cdots\!16\)\( p^{26} T^{7} + \)\(14\!\cdots\!89\)\( p^{39} T^{8} - 29662320178 p^{52} T^{9} + p^{65} T^{10} )^{2} \)
43 \( 1 - \)\(61\!\cdots\!14\)\( T^{2} + \)\(15\!\cdots\!05\)\( T^{4} - \)\(25\!\cdots\!04\)\( T^{6} + \)\(52\!\cdots\!62\)\( T^{8} - \)\(10\!\cdots\!24\)\( T^{10} + \)\(52\!\cdots\!62\)\( p^{26} T^{12} - \)\(25\!\cdots\!04\)\( p^{52} T^{14} + \)\(15\!\cdots\!05\)\( p^{78} T^{16} - \)\(61\!\cdots\!14\)\( p^{104} T^{18} + p^{130} T^{20} \)
47 \( ( 1 + 5088267408 T + \)\(10\!\cdots\!43\)\( T^{2} + \)\(16\!\cdots\!28\)\( T^{3} + \)\(79\!\cdots\!70\)\( T^{4} + \)\(90\!\cdots\!04\)\( T^{5} + \)\(79\!\cdots\!70\)\( p^{13} T^{6} + \)\(16\!\cdots\!28\)\( p^{26} T^{7} + \)\(10\!\cdots\!43\)\( p^{39} T^{8} + 5088267408 p^{52} T^{9} + p^{65} T^{10} )^{2} \)
53 \( 1 - \)\(15\!\cdots\!98\)\( T^{2} + \)\(12\!\cdots\!49\)\( T^{4} - \)\(63\!\cdots\!52\)\( T^{6} + \)\(24\!\cdots\!06\)\( T^{8} - \)\(72\!\cdots\!20\)\( T^{10} + \)\(24\!\cdots\!06\)\( p^{26} T^{12} - \)\(63\!\cdots\!52\)\( p^{52} T^{14} + \)\(12\!\cdots\!49\)\( p^{78} T^{16} - \)\(15\!\cdots\!98\)\( p^{104} T^{18} + p^{130} T^{20} \)
59 \( 1 - \)\(54\!\cdots\!82\)\( T^{2} + \)\(16\!\cdots\!77\)\( T^{4} - \)\(32\!\cdots\!52\)\( T^{6} + \)\(49\!\cdots\!86\)\( T^{8} - \)\(58\!\cdots\!80\)\( T^{10} + \)\(49\!\cdots\!86\)\( p^{26} T^{12} - \)\(32\!\cdots\!52\)\( p^{52} T^{14} + \)\(16\!\cdots\!77\)\( p^{78} T^{16} - \)\(54\!\cdots\!82\)\( p^{104} T^{18} + p^{130} T^{20} \)
61 \( 1 - \)\(10\!\cdots\!62\)\( T^{2} + \)\(47\!\cdots\!37\)\( T^{4} - \)\(13\!\cdots\!12\)\( T^{6} + \)\(29\!\cdots\!66\)\( T^{8} - \)\(51\!\cdots\!60\)\( T^{10} + \)\(29\!\cdots\!66\)\( p^{26} T^{12} - \)\(13\!\cdots\!12\)\( p^{52} T^{14} + \)\(47\!\cdots\!37\)\( p^{78} T^{16} - \)\(10\!\cdots\!62\)\( p^{104} T^{18} + p^{130} T^{20} \)
67 \( 1 - \)\(19\!\cdots\!42\)\( T^{2} + \)\(23\!\cdots\!49\)\( T^{4} - \)\(18\!\cdots\!88\)\( T^{6} + \)\(12\!\cdots\!86\)\( T^{8} - \)\(71\!\cdots\!60\)\( T^{10} + \)\(12\!\cdots\!86\)\( p^{26} T^{12} - \)\(18\!\cdots\!88\)\( p^{52} T^{14} + \)\(23\!\cdots\!49\)\( p^{78} T^{16} - \)\(19\!\cdots\!42\)\( p^{104} T^{18} + p^{130} T^{20} \)
71 \( ( 1 - 363180589992 T + \)\(34\!\cdots\!95\)\( T^{2} - \)\(14\!\cdots\!16\)\( T^{3} + \)\(61\!\cdots\!94\)\( T^{4} - \)\(22\!\cdots\!64\)\( T^{5} + \)\(61\!\cdots\!94\)\( p^{13} T^{6} - \)\(14\!\cdots\!16\)\( p^{26} T^{7} + \)\(34\!\cdots\!95\)\( p^{39} T^{8} - 363180589992 p^{52} T^{9} + p^{65} T^{10} )^{2} \)
73 \( ( 1 + 316620182766 T + \)\(64\!\cdots\!53\)\( T^{2} + \)\(24\!\cdots\!84\)\( T^{3} + \)\(18\!\cdots\!54\)\( T^{4} + \)\(60\!\cdots\!52\)\( T^{5} + \)\(18\!\cdots\!54\)\( p^{13} T^{6} + \)\(24\!\cdots\!84\)\( p^{26} T^{7} + \)\(64\!\cdots\!53\)\( p^{39} T^{8} + 316620182766 p^{52} T^{9} + p^{65} T^{10} )^{2} \)
79 \( ( 1 - 2722551782672 T + \)\(12\!\cdots\!19\)\( T^{2} - \)\(22\!\cdots\!64\)\( T^{3} + \)\(55\!\cdots\!14\)\( T^{4} - \)\(94\!\cdots\!68\)\( T^{5} + \)\(55\!\cdots\!14\)\( p^{13} T^{6} - \)\(22\!\cdots\!64\)\( p^{26} T^{7} + \)\(12\!\cdots\!19\)\( p^{39} T^{8} - 2722551782672 p^{52} T^{9} + p^{65} T^{10} )^{2} \)
83 \( 1 - \)\(43\!\cdots\!74\)\( T^{2} + \)\(87\!\cdots\!65\)\( T^{4} - \)\(12\!\cdots\!64\)\( T^{6} + \)\(14\!\cdots\!62\)\( T^{8} - \)\(14\!\cdots\!44\)\( T^{10} + \)\(14\!\cdots\!62\)\( p^{26} T^{12} - \)\(12\!\cdots\!64\)\( p^{52} T^{14} + \)\(87\!\cdots\!65\)\( p^{78} T^{16} - \)\(43\!\cdots\!74\)\( p^{104} T^{18} + p^{130} T^{20} \)
89 \( ( 1 - 2753172404002 T + \)\(62\!\cdots\!29\)\( T^{2} - \)\(14\!\cdots\!84\)\( T^{3} + \)\(21\!\cdots\!54\)\( T^{4} - \)\(47\!\cdots\!48\)\( T^{5} + \)\(21\!\cdots\!54\)\( p^{13} T^{6} - \)\(14\!\cdots\!84\)\( p^{26} T^{7} + \)\(62\!\cdots\!29\)\( p^{39} T^{8} - 2753172404002 p^{52} T^{9} + p^{65} T^{10} )^{2} \)
97 \( ( 1 - 680566660394 T + \)\(17\!\cdots\!53\)\( T^{2} + \)\(11\!\cdots\!08\)\( T^{3} + \)\(15\!\cdots\!70\)\( T^{4} + \)\(20\!\cdots\!28\)\( T^{5} + \)\(15\!\cdots\!70\)\( p^{13} T^{6} + \)\(11\!\cdots\!08\)\( p^{26} T^{7} + \)\(17\!\cdots\!53\)\( p^{39} T^{8} - 680566660394 p^{52} T^{9} + p^{65} T^{10} )^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{20} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−6.66886947381929636598797455366, −6.08447339184230149496102052983, −5.80726223661423727537792140543, −5.72427749505962053177887307845, −5.23353068686638011330789676137, −4.84951984004623786832158653480, −4.82721055450893994686983548483, −4.79021458568723709447934504244, −4.75350634760065164536498711230, −4.25622886693741870371577012967, −4.13847733670970732735505770859, −4.04426730356893955415865051031, −3.63292485695676890412364927162, −3.21749486277340490413237687312, −3.15722795232523809558507010116, −3.00011105243157225660093515947, −2.23587952828443494914035394069, −2.05597772219470668917871605546, −1.74098504658635278114485834352, −1.47206363951694865860601046652, −1.46182439319606293728102147730, −1.13182754764996151619051846223, −0.916042999419600349163614330458, −0.78765903334435241606060204547, −0.49486991567735139869259407329, 0.49486991567735139869259407329, 0.78765903334435241606060204547, 0.916042999419600349163614330458, 1.13182754764996151619051846223, 1.46182439319606293728102147730, 1.47206363951694865860601046652, 1.74098504658635278114485834352, 2.05597772219470668917871605546, 2.23587952828443494914035394069, 3.00011105243157225660093515947, 3.15722795232523809558507010116, 3.21749486277340490413237687312, 3.63292485695676890412364927162, 4.04426730356893955415865051031, 4.13847733670970732735505770859, 4.25622886693741870371577012967, 4.75350634760065164536498711230, 4.79021458568723709447934504244, 4.82721055450893994686983548483, 4.84951984004623786832158653480, 5.23353068686638011330789676137, 5.72427749505962053177887307845, 5.80726223661423727537792140543, 6.08447339184230149496102052983, 6.66886947381929636598797455366

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

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