Properties

Degree $40$
Conductor $1.153\times 10^{38}$
Sign $1$
Motivic weight $5$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 44·5-s + 804·13-s − 2.23e3·17-s + 1.44e3·25-s + 4.42e4·37-s − 6.76e3·41-s + 1.82e5·53-s − 4.10e4·61-s − 3.53e4·65-s + 2.64e5·73-s − 4.91e3·81-s + 9.83e4·85-s + 3.74e5·97-s − 5.14e5·101-s + 1.26e6·113-s + 1.26e6·121-s − 3.17e5·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 3.23e5·169-s + ⋯
L(s)  = 1  − 0.787·5-s + 1.31·13-s − 1.87·17-s + 0.461·25-s + 5.31·37-s − 0.628·41-s + 8.92·53-s − 1.41·61-s − 1.03·65-s + 5.80·73-s − 0.0831·81-s + 1.47·85-s + 4.04·97-s − 5.01·101-s + 9.34·113-s + 7.82·121-s − 1.81·125-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-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 + 0.870·169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{80} \cdot 5^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{80} \cdot 5^{20}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(40\)
Conductor: \(2^{80} \cdot 5^{20}\)
Sign: $1$
Motivic weight: \(5\)
Character: induced by $\chi_{80} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((40,\ 2^{80} \cdot 5^{20} ,\ ( \ : [5/2]^{20} ),\ 1 )\)

Particular Values

\(L(3)\) \(\approx\) \(36.2237\)
\(L(\frac12)\) \(\approx\) \(36.2237\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( ( 1 + 22 T + p T^{2} + 27552 p T^{3} + 18782 p^{4} T^{4} - 7372 p^{4} T^{5} + 18782 p^{9} T^{6} + 27552 p^{11} T^{7} + p^{16} T^{8} + 22 p^{20} T^{9} + p^{25} T^{10} )^{2} \)
good3 \( 1 + 4910 T^{4} + 4404762445 T^{8} + 15335701222280 p^{2} T^{12} - 84567556689796190 p^{4} T^{16} + 14403811239667020148 p^{10} T^{20} - 84567556689796190 p^{24} T^{24} + 15335701222280 p^{42} T^{28} + 4404762445 p^{60} T^{32} + 4910 p^{80} T^{36} + p^{100} T^{40} \)
7 \( 1 - 50878370 T^{4} + 88119764714863965 T^{8} + \)\(17\!\cdots\!80\)\( T^{12} + \)\(80\!\cdots\!70\)\( T^{16} + \)\(16\!\cdots\!76\)\( T^{20} + \)\(80\!\cdots\!70\)\( p^{20} T^{24} + \)\(17\!\cdots\!80\)\( p^{40} T^{28} + 88119764714863965 p^{60} T^{32} - 50878370 p^{80} T^{36} + p^{100} T^{40} \)
11 \( ( 1 - 630250 T^{2} + 230184799605 T^{4} - 63113737617167000 T^{6} + \)\(13\!\cdots\!10\)\( T^{8} - \)\(24\!\cdots\!00\)\( T^{10} + \)\(13\!\cdots\!10\)\( p^{10} T^{12} - 63113737617167000 p^{20} T^{14} + 230184799605 p^{30} T^{16} - 630250 p^{40} T^{18} + p^{50} T^{20} )^{2} \)
13 \( ( 1 - 402 T + 80802 T^{2} - 412195466 T^{3} + 274076716325 T^{4} + 98802411059208 T^{5} + 23088035017184312 T^{6} - 33522742681396790296 T^{7} - \)\(32\!\cdots\!50\)\( p^{2} T^{8} + \)\(26\!\cdots\!36\)\( p T^{9} + \)\(30\!\cdots\!32\)\( T^{10} + \)\(26\!\cdots\!36\)\( p^{6} T^{11} - \)\(32\!\cdots\!50\)\( p^{12} T^{12} - 33522742681396790296 p^{15} T^{13} + 23088035017184312 p^{20} T^{14} + 98802411059208 p^{25} T^{15} + 274076716325 p^{30} T^{16} - 412195466 p^{35} T^{17} + 80802 p^{40} T^{18} - 402 p^{45} T^{19} + p^{50} T^{20} )^{2} \)
17 \( ( 1 + 1118 T + 624962 T^{2} + 69923678 p T^{3} - 1739300659939 T^{4} - 3459207301051384 T^{5} - 2073890095879259656 T^{6} - \)\(50\!\cdots\!88\)\( T^{7} + \)\(54\!\cdots\!26\)\( T^{8} + \)\(82\!\cdots\!84\)\( T^{9} + \)\(52\!\cdots\!56\)\( T^{10} + \)\(82\!\cdots\!84\)\( p^{5} T^{11} + \)\(54\!\cdots\!26\)\( p^{10} T^{12} - \)\(50\!\cdots\!88\)\( p^{15} T^{13} - 2073890095879259656 p^{20} T^{14} - 3459207301051384 p^{25} T^{15} - 1739300659939 p^{30} T^{16} + 69923678 p^{36} T^{17} + 624962 p^{40} T^{18} + 1118 p^{45} T^{19} + p^{50} T^{20} )^{2} \)
19 \( ( 1 + 8249550 T^{2} + 36376777568405 T^{4} + \)\(13\!\cdots\!00\)\( T^{6} + \)\(42\!\cdots\!10\)\( T^{8} + \)\(11\!\cdots\!00\)\( T^{10} + \)\(42\!\cdots\!10\)\( p^{10} T^{12} + \)\(13\!\cdots\!00\)\( p^{20} T^{14} + 36376777568405 p^{30} T^{16} + 8249550 p^{40} T^{18} + p^{50} T^{20} )^{2} \)
23 \( 1 - 207859623483490 T^{4} + \)\(17\!\cdots\!45\)\( T^{8} - \)\(53\!\cdots\!80\)\( T^{12} - \)\(11\!\cdots\!90\)\( T^{16} + \)\(13\!\cdots\!52\)\( T^{20} - \)\(11\!\cdots\!90\)\( p^{20} T^{24} - \)\(53\!\cdots\!80\)\( p^{40} T^{28} + \)\(17\!\cdots\!45\)\( p^{60} T^{32} - 207859623483490 p^{80} T^{36} + p^{100} T^{40} \)
29 \( ( 1 - 99799410 T^{2} + 5177263888867445 T^{4} - \)\(18\!\cdots\!20\)\( T^{6} + \)\(49\!\cdots\!10\)\( T^{8} - \)\(10\!\cdots\!52\)\( T^{10} + \)\(49\!\cdots\!10\)\( p^{10} T^{12} - \)\(18\!\cdots\!20\)\( p^{20} T^{14} + 5177263888867445 p^{30} T^{16} - 99799410 p^{40} T^{18} + p^{50} T^{20} )^{2} \)
31 \( ( 1 - 101180050 T^{2} + 5842226806223805 T^{4} - \)\(25\!\cdots\!00\)\( T^{6} + \)\(95\!\cdots\!10\)\( T^{8} - \)\(29\!\cdots\!00\)\( T^{10} + \)\(95\!\cdots\!10\)\( p^{10} T^{12} - \)\(25\!\cdots\!00\)\( p^{20} T^{14} + 5842226806223805 p^{30} T^{16} - 101180050 p^{40} T^{18} + p^{50} T^{20} )^{2} \)
37 \( ( 1 - 22130 T + 244868450 T^{2} - 1308308732410 T^{3} + 13809729158765845 T^{4} - \)\(24\!\cdots\!80\)\( T^{5} + \)\(28\!\cdots\!00\)\( T^{6} - \)\(15\!\cdots\!60\)\( T^{7} + \)\(10\!\cdots\!10\)\( T^{8} - \)\(15\!\cdots\!80\)\( T^{9} + \)\(18\!\cdots\!00\)\( T^{10} - \)\(15\!\cdots\!80\)\( p^{5} T^{11} + \)\(10\!\cdots\!10\)\( p^{10} T^{12} - \)\(15\!\cdots\!60\)\( p^{15} T^{13} + \)\(28\!\cdots\!00\)\( p^{20} T^{14} - \)\(24\!\cdots\!80\)\( p^{25} T^{15} + 13809729158765845 p^{30} T^{16} - 1308308732410 p^{35} T^{17} + 244868450 p^{40} T^{18} - 22130 p^{45} T^{19} + p^{50} T^{20} )^{2} \)
41 \( ( 1 + 1690 T + 164119545 T^{2} - 1028151329120 T^{3} + 25291870422416110 T^{4} - 93379183076463398452 T^{5} + 25291870422416110 p^{5} T^{6} - 1028151329120 p^{10} T^{7} + 164119545 p^{15} T^{8} + 1690 p^{20} T^{9} + p^{25} T^{10} )^{4} \)
43 \( 1 + 22292551011039310 T^{4} + \)\(40\!\cdots\!45\)\( T^{8} + \)\(23\!\cdots\!20\)\( T^{12} + \)\(67\!\cdots\!10\)\( T^{16} + \)\(21\!\cdots\!52\)\( T^{20} + \)\(67\!\cdots\!10\)\( p^{20} T^{24} + \)\(23\!\cdots\!20\)\( p^{40} T^{28} + \)\(40\!\cdots\!45\)\( p^{60} T^{32} + 22292551011039310 p^{80} T^{36} + p^{100} T^{40} \)
47 \( 1 + 74279974827999230 T^{4} + \)\(48\!\cdots\!65\)\( T^{8} + \)\(19\!\cdots\!80\)\( T^{12} + \)\(14\!\cdots\!70\)\( T^{16} - \)\(25\!\cdots\!24\)\( T^{20} + \)\(14\!\cdots\!70\)\( p^{20} T^{24} + \)\(19\!\cdots\!80\)\( p^{40} T^{28} + \)\(48\!\cdots\!65\)\( p^{60} T^{32} + 74279974827999230 p^{80} T^{36} + p^{100} T^{40} \)
53 \( ( 1 - 91226 T + 4161091538 T^{2} - 141622957033378 T^{3} + 3893612180777232821 T^{4} - \)\(81\!\cdots\!12\)\( T^{5} + \)\(12\!\cdots\!56\)\( T^{6} - \)\(10\!\cdots\!76\)\( T^{7} - \)\(19\!\cdots\!54\)\( T^{8} + \)\(10\!\cdots\!72\)\( T^{9} - \)\(26\!\cdots\!96\)\( T^{10} + \)\(10\!\cdots\!72\)\( p^{5} T^{11} - \)\(19\!\cdots\!54\)\( p^{10} T^{12} - \)\(10\!\cdots\!76\)\( p^{15} T^{13} + \)\(12\!\cdots\!56\)\( p^{20} T^{14} - \)\(81\!\cdots\!12\)\( p^{25} T^{15} + 3893612180777232821 p^{30} T^{16} - 141622957033378 p^{35} T^{17} + 4161091538 p^{40} T^{18} - 91226 p^{45} T^{19} + p^{50} T^{20} )^{2} \)
59 \( ( 1 + 4265422750 T^{2} + 8278744468566020805 T^{4} + \)\(98\!\cdots\!00\)\( T^{6} + \)\(85\!\cdots\!10\)\( T^{8} + \)\(63\!\cdots\!00\)\( T^{10} + \)\(85\!\cdots\!10\)\( p^{10} T^{12} + \)\(98\!\cdots\!00\)\( p^{20} T^{14} + 8278744468566020805 p^{30} T^{16} + 4265422750 p^{40} T^{18} + p^{50} T^{20} )^{2} \)
61 \( ( 1 + 10270 T + 3306972765 T^{2} + 29698468018720 T^{3} + 4981733552475787870 T^{4} + \)\(35\!\cdots\!24\)\( T^{5} + 4981733552475787870 p^{5} T^{6} + 29698468018720 p^{10} T^{7} + 3306972765 p^{15} T^{8} + 10270 p^{20} T^{9} + p^{25} T^{10} )^{4} \)
67 \( 1 - 2038648142805486290 T^{4} - \)\(25\!\cdots\!55\)\( T^{8} + \)\(14\!\cdots\!20\)\( T^{12} + \)\(73\!\cdots\!10\)\( T^{16} - \)\(57\!\cdots\!48\)\( T^{20} + \)\(73\!\cdots\!10\)\( p^{20} T^{24} + \)\(14\!\cdots\!20\)\( p^{40} T^{28} - \)\(25\!\cdots\!55\)\( p^{60} T^{32} - 2038648142805486290 p^{80} T^{36} + p^{100} T^{40} \)
71 \( ( 1 - 14421599170 T^{2} + 96751044929093627565 T^{4} - \)\(40\!\cdots\!20\)\( T^{6} + \)\(11\!\cdots\!70\)\( T^{8} - \)\(24\!\cdots\!24\)\( T^{10} + \)\(11\!\cdots\!70\)\( p^{10} T^{12} - \)\(40\!\cdots\!20\)\( p^{20} T^{14} + 96751044929093627565 p^{30} T^{16} - 14421599170 p^{40} T^{18} + p^{50} T^{20} )^{2} \)
73 \( ( 1 - 132186 T + 8736569298 T^{2} - 431249337673738 T^{3} + 10750357541220477581 T^{4} + \)\(12\!\cdots\!88\)\( T^{5} - \)\(17\!\cdots\!84\)\( T^{6} + \)\(58\!\cdots\!44\)\( T^{7} + \)\(47\!\cdots\!06\)\( T^{8} - \)\(60\!\cdots\!08\)\( T^{9} + \)\(32\!\cdots\!04\)\( T^{10} - \)\(60\!\cdots\!08\)\( p^{5} T^{11} + \)\(47\!\cdots\!06\)\( p^{10} T^{12} + \)\(58\!\cdots\!44\)\( p^{15} T^{13} - \)\(17\!\cdots\!84\)\( p^{20} T^{14} + \)\(12\!\cdots\!88\)\( p^{25} T^{15} + 10750357541220477581 p^{30} T^{16} - 431249337673738 p^{35} T^{17} + 8736569298 p^{40} T^{18} - 132186 p^{45} T^{19} + p^{50} T^{20} )^{2} \)
79 \( ( 1 + 18614864790 T^{2} + \)\(16\!\cdots\!45\)\( T^{4} + \)\(10\!\cdots\!80\)\( T^{6} + \)\(44\!\cdots\!10\)\( T^{8} + \)\(15\!\cdots\!48\)\( T^{10} + \)\(44\!\cdots\!10\)\( p^{10} T^{12} + \)\(10\!\cdots\!80\)\( p^{20} T^{14} + \)\(16\!\cdots\!45\)\( p^{30} T^{16} + 18614864790 p^{40} T^{18} + p^{50} T^{20} )^{2} \)
83 \( 1 - 18609684090616711570 T^{4} - \)\(46\!\cdots\!35\)\( T^{8} + \)\(34\!\cdots\!80\)\( T^{12} + \)\(25\!\cdots\!70\)\( T^{16} - \)\(17\!\cdots\!24\)\( T^{20} + \)\(25\!\cdots\!70\)\( p^{20} T^{24} + \)\(34\!\cdots\!80\)\( p^{40} T^{28} - \)\(46\!\cdots\!35\)\( p^{60} T^{32} - 18609684090616711570 p^{80} T^{36} + p^{100} T^{40} \)
89 \( ( 1 - 33120686970 T^{2} + \)\(41\!\cdots\!65\)\( T^{4} - \)\(20\!\cdots\!20\)\( T^{6} - \)\(32\!\cdots\!30\)\( T^{8} + \)\(73\!\cdots\!76\)\( T^{10} - \)\(32\!\cdots\!30\)\( p^{10} T^{12} - \)\(20\!\cdots\!20\)\( p^{20} T^{14} + \)\(41\!\cdots\!65\)\( p^{30} T^{16} - 33120686970 p^{40} T^{18} + p^{50} T^{20} )^{2} \)
97 \( ( 1 - 187386 T + 17556756498 T^{2} - 497237634071002 T^{3} - \)\(23\!\cdots\!35\)\( T^{4} + \)\(42\!\cdots\!24\)\( T^{5} - \)\(37\!\cdots\!32\)\( T^{6} + \)\(18\!\cdots\!68\)\( T^{7} + \)\(11\!\cdots\!50\)\( T^{8} - \)\(33\!\cdots\!76\)\( T^{9} + \)\(36\!\cdots\!68\)\( T^{10} - \)\(33\!\cdots\!76\)\( p^{5} T^{11} + \)\(11\!\cdots\!50\)\( p^{10} T^{12} + \)\(18\!\cdots\!68\)\( p^{15} T^{13} - \)\(37\!\cdots\!32\)\( p^{20} T^{14} + \)\(42\!\cdots\!24\)\( p^{25} T^{15} - \)\(23\!\cdots\!35\)\( p^{30} T^{16} - 497237634071002 p^{35} T^{17} + 17556756498 p^{40} T^{18} - 187386 p^{45} T^{19} + p^{50} T^{20} )^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{40} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−2.60958425965785299297251851363, −2.53306642534811098433529980698, −2.45270932408861420428421830277, −2.41472905427548108092111930753, −2.37607864811647305912262463685, −2.18052429341612950612465615346, −2.10215449842015770000186826855, −2.06616571229224580739448636445, −2.02105950092773251258036991155, −1.74441135995439221018530639549, −1.74226003942223379744404240138, −1.72768310764690907379276228452, −1.62392508785158906151156088388, −1.25480048903888908127183800244, −1.04881376529821229929926917762, −0.956463349721247668034993931006, −0.868767072660102397329778793329, −0.842037222939004802593891563673, −0.814393730904499625618036560738, −0.76124038033039586460059235960, −0.67867472357543196246462170477, −0.47550912474881410405575988855, −0.39061172290722094274041907018, −0.37522031494584147842910580944, −0.10223058955846150327380687089, 0.10223058955846150327380687089, 0.37522031494584147842910580944, 0.39061172290722094274041907018, 0.47550912474881410405575988855, 0.67867472357543196246462170477, 0.76124038033039586460059235960, 0.814393730904499625618036560738, 0.842037222939004802593891563673, 0.868767072660102397329778793329, 0.956463349721247668034993931006, 1.04881376529821229929926917762, 1.25480048903888908127183800244, 1.62392508785158906151156088388, 1.72768310764690907379276228452, 1.74226003942223379744404240138, 1.74441135995439221018530639549, 2.02105950092773251258036991155, 2.06616571229224580739448636445, 2.10215449842015770000186826855, 2.18052429341612950612465615346, 2.37607864811647305912262463685, 2.41472905427548108092111930753, 2.45270932408861420428421830277, 2.53306642534811098433529980698, 2.60958425965785299297251851363

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

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