Properties

Label 32-147e16-1.1-c5e16-0-0
Degree $32$
Conductor $4.754\times 10^{34}$
Sign $1$
Analytic cond. $9.11268\times 10^{21}$
Root an. cond. $4.85555$
Motivic weight $5$
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  + 170·4-s + 606·9-s + 1.45e4·16-s − 1.84e4·25-s + 1.03e5·36-s + 2.09e4·37-s + 1.11e5·43-s + 8.31e5·64-s + 2.04e5·67-s + 5.02e5·79-s + 9.39e4·81-s − 3.13e6·100-s − 8.90e3·109-s + 2.10e6·121-s + 127-s + 131-s + 137-s + 139-s + 8.78e6·144-s + 3.55e6·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4.28e6·169-s + 1.88e7·172-s + ⋯
L(s)  = 1  + 5.31·4-s + 2.49·9-s + 14.1·16-s − 5.89·25-s + 13.2·36-s + 2.51·37-s + 9.15·43-s + 25.3·64-s + 5.56·67-s + 9.06·79-s + 1.59·81-s − 31.3·100-s − 0.0717·109-s + 13.1·121-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + 35.3·144-s + 13.3·148-s + 3.69e−6·149-s + 3.56e−6·151-s + 3.23e−6·157-s + 2.94e−6·163-s + 2.77e−6·167-s + 11.5·169-s + 48.6·172-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{32}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(32\)
Conductor: \(3^{16} \cdot 7^{32}\)
Sign: $1$
Analytic conductor: \(9.11268\times 10^{21}\)
Root analytic conductor: \(4.85555\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{147} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((32,\ 3^{16} \cdot 7^{32} ,\ ( \ : [5/2]^{16} ),\ 1 )\)

Particular Values

\(L(3)\) \(\approx\) \(357.7512266\)
\(L(\frac12)\) \(\approx\) \(357.7512266\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 202 p T^{2} + 30361 p^{2} T^{4} - 128578 p^{6} T^{6} + 405484 p^{10} T^{8} - 128578 p^{16} T^{10} + 30361 p^{22} T^{12} - 202 p^{31} T^{14} + p^{40} T^{16} \)
7 \( 1 \)
good2 \( ( 1 - 85 T^{2} + 1793 p T^{4} - 12763 p^{3} T^{6} + 92635 p^{5} T^{8} - 12763 p^{13} T^{10} + 1793 p^{21} T^{12} - 85 p^{30} T^{14} + p^{40} T^{16} )^{2} \)
5 \( ( 1 + 9211 T^{2} + 45434461 T^{4} + 182003669566 T^{6} + 632775698529886 T^{8} + 182003669566 p^{10} T^{10} + 45434461 p^{20} T^{12} + 9211 p^{30} T^{14} + p^{40} T^{16} )^{2} \)
11 \( ( 1 - 1054897 T^{2} + 46048407287 p T^{4} - 146853363030047186 T^{6} + \)\(28\!\cdots\!94\)\( T^{8} - 146853363030047186 p^{10} T^{10} + 46048407287 p^{21} T^{12} - 1054897 p^{30} T^{14} + p^{40} T^{16} )^{2} \)
13 \( ( 1 - 2144633 T^{2} + 2239709913742 T^{4} - 1466006418442869023 T^{6} + \)\(65\!\cdots\!38\)\( T^{8} - 1466006418442869023 p^{10} T^{10} + 2239709913742 p^{20} T^{12} - 2144633 p^{30} T^{14} + p^{40} T^{16} )^{2} \)
17 \( ( 1 + 9084106 T^{2} + 38356486297777 T^{4} + 98586445151688130222 T^{6} + \)\(16\!\cdots\!20\)\( T^{8} + 98586445151688130222 p^{10} T^{10} + 38356486297777 p^{20} T^{12} + 9084106 p^{30} T^{14} + p^{40} T^{16} )^{2} \)
19 \( ( 1 - 2955779 T^{2} + 7463522925721 T^{4} - 1535726214901369406 p T^{6} + \)\(97\!\cdots\!94\)\( T^{8} - 1535726214901369406 p^{11} T^{10} + 7463522925721 p^{20} T^{12} - 2955779 p^{30} T^{14} + p^{40} T^{16} )^{2} \)
23 \( ( 1 - 17175838 T^{2} + 204555149435065 T^{4} - \)\(18\!\cdots\!54\)\( T^{6} + \)\(12\!\cdots\!00\)\( T^{8} - \)\(18\!\cdots\!54\)\( p^{10} T^{10} + 204555149435065 p^{20} T^{12} - 17175838 p^{30} T^{14} + p^{40} T^{16} )^{2} \)
29 \( ( 1 - 68423503 T^{2} + 2743707440153470 T^{4} - \)\(78\!\cdots\!21\)\( T^{6} + \)\(17\!\cdots\!14\)\( T^{8} - \)\(78\!\cdots\!21\)\( p^{10} T^{10} + 2743707440153470 p^{20} T^{12} - 68423503 p^{30} T^{14} + p^{40} T^{16} )^{2} \)
31 \( ( 1 - 144062468 T^{2} + 10475805095963554 T^{4} - \)\(49\!\cdots\!04\)\( T^{6} + \)\(16\!\cdots\!95\)\( T^{8} - \)\(49\!\cdots\!04\)\( p^{10} T^{10} + 10475805095963554 p^{20} T^{12} - 144062468 p^{30} T^{14} + p^{40} T^{16} )^{2} \)
37 \( ( 1 - 5233 T + 242181853 T^{2} - 1004596508078 T^{3} + 24215098505313878 T^{4} - 1004596508078 p^{5} T^{5} + 242181853 p^{10} T^{6} - 5233 p^{15} T^{7} + p^{20} T^{8} )^{4} \)
41 \( ( 1 + 417110896 T^{2} + 104784626889300316 T^{4} + \)\(18\!\cdots\!16\)\( T^{6} + \)\(24\!\cdots\!34\)\( T^{8} + \)\(18\!\cdots\!16\)\( p^{10} T^{10} + 104784626889300316 p^{20} T^{12} + 417110896 p^{30} T^{14} + p^{40} T^{16} )^{2} \)
43 \( ( 1 - 27763 T + 699003004 T^{2} - 11223042578867 T^{3} + 157088420435352182 T^{4} - 11223042578867 p^{5} T^{5} + 699003004 p^{10} T^{6} - 27763 p^{15} T^{7} + p^{20} T^{8} )^{4} \)
47 \( ( 1 + 1659035398 T^{2} + 1241539376543157337 T^{4} + \)\(54\!\cdots\!58\)\( T^{6} + \)\(15\!\cdots\!08\)\( T^{8} + \)\(54\!\cdots\!58\)\( p^{10} T^{10} + 1241539376543157337 p^{20} T^{12} + 1659035398 p^{30} T^{14} + p^{40} T^{16} )^{2} \)
53 \( ( 1 - 1522005769 T^{2} + 20033470999258949 p T^{4} - \)\(45\!\cdots\!66\)\( T^{6} + \)\(17\!\cdots\!18\)\( T^{8} - \)\(45\!\cdots\!66\)\( p^{10} T^{10} + 20033470999258949 p^{21} T^{12} - 1522005769 p^{30} T^{14} + p^{40} T^{16} )^{2} \)
59 \( ( 1 + 3710257099 T^{2} + 6384076097381414905 T^{4} + \)\(69\!\cdots\!98\)\( T^{6} + \)\(56\!\cdots\!90\)\( T^{8} + \)\(69\!\cdots\!98\)\( p^{10} T^{10} + 6384076097381414905 p^{20} T^{12} + 3710257099 p^{30} T^{14} + p^{40} T^{16} )^{2} \)
61 \( ( 1 - 4331296586 T^{2} + 9541896422707738105 T^{4} - \)\(13\!\cdots\!82\)\( T^{6} + \)\(13\!\cdots\!00\)\( T^{8} - \)\(13\!\cdots\!82\)\( p^{10} T^{10} + 9541896422707738105 p^{20} T^{12} - 4331296586 p^{30} T^{14} + p^{40} T^{16} )^{2} \)
67 \( ( 1 - 51101 T + 5544456325 T^{2} - 198345529855244 T^{3} + 11276127169607793166 T^{4} - 198345529855244 p^{5} T^{5} + 5544456325 p^{10} T^{6} - 51101 p^{15} T^{7} + p^{20} T^{8} )^{4} \)
71 \( ( 1 - 8041496464 T^{2} + 34349295256214414812 T^{4} - \)\(96\!\cdots\!88\)\( T^{6} + \)\(20\!\cdots\!74\)\( T^{8} - \)\(96\!\cdots\!88\)\( p^{10} T^{10} + 34349295256214414812 p^{20} T^{12} - 8041496464 p^{30} T^{14} + p^{40} T^{16} )^{2} \)
73 \( ( 1 - 11067072335 T^{2} + 59992951581405223777 T^{4} - \)\(21\!\cdots\!18\)\( T^{6} + \)\(51\!\cdots\!26\)\( T^{8} - \)\(21\!\cdots\!18\)\( p^{10} T^{10} + 59992951581405223777 p^{20} T^{12} - 11067072335 p^{30} T^{14} + p^{40} T^{16} )^{2} \)
79 \( ( 1 - 125654 T + 11660654632 T^{2} - 808762458316304 T^{3} + 49927374895774090681 T^{4} - 808762458316304 p^{5} T^{5} + 11660654632 p^{10} T^{6} - 125654 p^{15} T^{7} + p^{20} T^{8} )^{4} \)
83 \( ( 1 + 12475595005 T^{2} + 77581754742007923190 T^{4} + \)\(37\!\cdots\!39\)\( T^{6} + \)\(16\!\cdots\!06\)\( T^{8} + \)\(37\!\cdots\!39\)\( p^{10} T^{10} + 77581754742007923190 p^{20} T^{12} + 12475595005 p^{30} T^{14} + p^{40} T^{16} )^{2} \)
89 \( ( 1 + 24614586730 T^{2} + \)\(22\!\cdots\!57\)\( T^{4} + \)\(93\!\cdots\!70\)\( T^{6} + \)\(29\!\cdots\!80\)\( T^{8} + \)\(93\!\cdots\!70\)\( p^{10} T^{10} + \)\(22\!\cdots\!57\)\( p^{20} T^{12} + 24614586730 p^{30} T^{14} + p^{40} T^{16} )^{2} \)
97 \( ( 1 - 33885078893 T^{2} + \)\(57\!\cdots\!86\)\( T^{4} - \)\(72\!\cdots\!79\)\( T^{6} + \)\(70\!\cdots\!98\)\( T^{8} - \)\(72\!\cdots\!79\)\( p^{10} T^{10} + \)\(57\!\cdots\!86\)\( p^{20} T^{12} - 33885078893 p^{30} T^{14} + p^{40} T^{16} )^{2} \)
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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

−2.64495143247003017007130337632, −2.54002968310498542098245279276, −2.39687537824437042594724644325, −2.29843476779164296233528912695, −2.16117087481376137877233415593, −2.14406737068596101996830743370, −2.12013705425074645662111200101, −2.10562879603227127425838345521, −2.08166107527363674109163709690, −2.06772490223051368575219469616, −1.93367724374874536721865426325, −1.87429403745950027291544769882, −1.67259314751575459525657570244, −1.33805201463579848369915760264, −1.33631604950505413491778201367, −1.18124574455535341243572340895, −1.09493869236322679610720811882, −0.832674219071741408468897922899, −0.807907840663200543866080197924, −0.76732301667370698493505831792, −0.71625500129336808105226755652, −0.65387480604710698956255772656, −0.62115573056224084981996490456, −0.29930013478146209963840675260, −0.10725954061039257545504679423, 0.10725954061039257545504679423, 0.29930013478146209963840675260, 0.62115573056224084981996490456, 0.65387480604710698956255772656, 0.71625500129336808105226755652, 0.76732301667370698493505831792, 0.807907840663200543866080197924, 0.832674219071741408468897922899, 1.09493869236322679610720811882, 1.18124574455535341243572340895, 1.33631604950505413491778201367, 1.33805201463579848369915760264, 1.67259314751575459525657570244, 1.87429403745950027291544769882, 1.93367724374874536721865426325, 2.06772490223051368575219469616, 2.08166107527363674109163709690, 2.10562879603227127425838345521, 2.12013705425074645662111200101, 2.14406737068596101996830743370, 2.16117087481376137877233415593, 2.29843476779164296233528912695, 2.39687537824437042594724644325, 2.54002968310498542098245279276, 2.64495143247003017007130337632

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

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