Properties

Label 32-8820e16-1.1-c1e16-0-1
Degree $32$
Conductor $1.341\times 10^{63}$
Sign $1$
Analytic cond. $3.66379\times 10^{29}$
Root an. cond. $8.39214$
Motivic weight $1$
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  − 16·5-s + 136·25-s − 32·41-s − 32·43-s + 32·47-s + 32·59-s − 32·67-s + 64·89-s + 64·101-s + 32·109-s + 48·121-s − 816·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 96·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
L(s)  = 1  − 7.15·5-s + 27.1·25-s − 4.99·41-s − 4.87·43-s + 4.66·47-s + 4.16·59-s − 3.90·67-s + 6.78·89-s + 6.36·101-s + 3.06·109-s + 4.36·121-s − 72.9·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 7.38·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯

Functional equation

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

Invariants

Degree: \(32\)
Conductor: \(2^{32} \cdot 3^{32} \cdot 5^{16} \cdot 7^{32}\)
Sign: $1$
Analytic conductor: \(3.66379\times 10^{29}\)
Root analytic conductor: \(8.39214\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((32,\ 2^{32} \cdot 3^{32} \cdot 5^{16} \cdot 7^{32} ,\ ( \ : [1/2]^{16} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(4.597712165\)
\(L(\frac12)\) \(\approx\) \(4.597712165\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( ( 1 + T )^{16} \)
7 \( 1 \)
good11 \( 1 - 48 T^{2} + 8 p^{2} T^{4} - 1008 p T^{6} + 90588 T^{8} - 784240 T^{10} + 66744 p^{2} T^{12} - 63417872 T^{14} + 459693958 T^{16} - 63417872 p^{2} T^{18} + 66744 p^{6} T^{20} - 784240 p^{6} T^{22} + 90588 p^{8} T^{24} - 1008 p^{11} T^{26} + 8 p^{14} T^{28} - 48 p^{14} T^{30} + p^{16} T^{32} \)
13 \( 1 - 96 T^{2} + 4632 T^{4} - 149792 T^{6} + 3658524 T^{8} - 72318816 T^{10} + 1215320616 T^{12} - 18065594656 T^{14} + 244647903878 T^{16} - 18065594656 p^{2} T^{18} + 1215320616 p^{4} T^{20} - 72318816 p^{6} T^{22} + 3658524 p^{8} T^{24} - 149792 p^{10} T^{26} + 4632 p^{12} T^{28} - 96 p^{14} T^{30} + p^{16} T^{32} \)
17 \( ( 1 + 64 T^{2} + 32 T^{3} + 2036 T^{4} + 2592 T^{5} + 45504 T^{6} + 4928 p T^{7} + 833702 T^{8} + 4928 p^{2} T^{9} + 45504 p^{2} T^{10} + 2592 p^{3} T^{11} + 2036 p^{4} T^{12} + 32 p^{5} T^{13} + 64 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
19 \( 1 - 160 T^{2} + 12544 T^{4} - 639648 T^{6} + 23808580 T^{8} - 692496608 T^{10} + 16645276416 T^{12} - 350492300512 T^{14} + 6834420715206 T^{16} - 350492300512 p^{2} T^{18} + 16645276416 p^{4} T^{20} - 692496608 p^{6} T^{22} + 23808580 p^{8} T^{24} - 639648 p^{10} T^{26} + 12544 p^{12} T^{28} - 160 p^{14} T^{30} + p^{16} T^{32} \)
23 \( 1 - 192 T^{2} + 18448 T^{4} - 1184192 T^{6} + 57427748 T^{8} - 2253407040 T^{10} + 74467088688 T^{12} - 2116543941696 T^{14} + 52201172224966 T^{16} - 2116543941696 p^{2} T^{18} + 74467088688 p^{4} T^{20} - 2253407040 p^{6} T^{22} + 57427748 p^{8} T^{24} - 1184192 p^{10} T^{26} + 18448 p^{12} T^{28} - 192 p^{14} T^{30} + p^{16} T^{32} \)
29 \( 1 - 256 T^{2} + 32456 T^{4} - 2717440 T^{6} + 169228764 T^{8} - 8381414656 T^{10} + 345090227448 T^{12} - 12186396826368 T^{14} + 376509256399366 T^{16} - 12186396826368 p^{2} T^{18} + 345090227448 p^{4} T^{20} - 8381414656 p^{6} T^{22} + 169228764 p^{8} T^{24} - 2717440 p^{10} T^{26} + 32456 p^{12} T^{28} - 256 p^{14} T^{30} + p^{16} T^{32} \)
31 \( 1 - 192 T^{2} + 19392 T^{4} - 1347008 T^{6} + 72642564 T^{8} - 3278733120 T^{10} + 130150499136 T^{12} - 4659571780672 T^{14} + 151538316525062 T^{16} - 4659571780672 p^{2} T^{18} + 130150499136 p^{4} T^{20} - 3278733120 p^{6} T^{22} + 72642564 p^{8} T^{24} - 1347008 p^{10} T^{26} + 19392 p^{12} T^{28} - 192 p^{14} T^{30} + p^{16} T^{32} \)
37 \( ( 1 + 168 T^{2} - 288 T^{3} + 14604 T^{4} - 34080 T^{5} + 873752 T^{6} - 2045248 T^{7} + 37651206 T^{8} - 2045248 p T^{9} + 873752 p^{2} T^{10} - 34080 p^{3} T^{11} + 14604 p^{4} T^{12} - 288 p^{5} T^{13} + 168 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
41 \( ( 1 + 16 T + 296 T^{2} + 2672 T^{3} + 27196 T^{4} + 156880 T^{5} + 1186328 T^{6} + 4920752 T^{7} + 41340230 T^{8} + 4920752 p T^{9} + 1186328 p^{2} T^{10} + 156880 p^{3} T^{11} + 27196 p^{4} T^{12} + 2672 p^{5} T^{13} + 296 p^{6} T^{14} + 16 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
43 \( ( 1 + 16 T + 312 T^{2} + 3344 T^{3} + 40188 T^{4} + 7856 p T^{5} + 3104584 T^{6} + 500528 p T^{7} + 161121254 T^{8} + 500528 p^{2} T^{9} + 3104584 p^{2} T^{10} + 7856 p^{4} T^{11} + 40188 p^{4} T^{12} + 3344 p^{5} T^{13} + 312 p^{6} T^{14} + 16 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
47 \( ( 1 - 16 T + 368 T^{2} - 4368 T^{3} + 58132 T^{4} - 543632 T^{5} + 5291728 T^{6} - 40033936 T^{7} + 307242726 T^{8} - 40033936 p T^{9} + 5291728 p^{2} T^{10} - 543632 p^{3} T^{11} + 58132 p^{4} T^{12} - 4368 p^{5} T^{13} + 368 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
53 \( 1 - 512 T^{2} + 133456 T^{4} - 23285504 T^{6} + 3029216868 T^{8} - 310566767360 T^{10} + 25903385843696 T^{12} - 1790479941488640 T^{14} + 103573043157885062 T^{16} - 1790479941488640 p^{2} T^{18} + 25903385843696 p^{4} T^{20} - 310566767360 p^{6} T^{22} + 3029216868 p^{8} T^{24} - 23285504 p^{10} T^{26} + 133456 p^{12} T^{28} - 512 p^{14} T^{30} + p^{16} T^{32} \)
59 \( ( 1 - 16 T + 392 T^{2} - 4816 T^{3} + 70508 T^{4} - 698448 T^{5} + 7628984 T^{6} - 62374160 T^{7} + 546603078 T^{8} - 62374160 p T^{9} + 7628984 p^{2} T^{10} - 698448 p^{3} T^{11} + 70508 p^{4} T^{12} - 4816 p^{5} T^{13} + 392 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
61 \( 1 - 352 T^{2} + 64320 T^{4} - 8296544 T^{6} + 836028740 T^{8} - 69454203424 T^{10} + 5009254743488 T^{12} - 328522585966368 T^{14} + 20395558149127622 T^{16} - 328522585966368 p^{2} T^{18} + 5009254743488 p^{4} T^{20} - 69454203424 p^{6} T^{22} + 836028740 p^{8} T^{24} - 8296544 p^{10} T^{26} + 64320 p^{12} T^{28} - 352 p^{14} T^{30} + p^{16} T^{32} \)
67 \( ( 1 + 16 T + 392 T^{2} + 80 p T^{3} + 74188 T^{4} + 821680 T^{5} + 8772536 T^{6} + 78539728 T^{7} + 705009222 T^{8} + 78539728 p T^{9} + 8772536 p^{2} T^{10} + 821680 p^{3} T^{11} + 74188 p^{4} T^{12} + 80 p^{6} T^{13} + 392 p^{6} T^{14} + 16 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
71 \( 1 - 624 T^{2} + 202632 T^{4} - 44752016 T^{6} + 7464639260 T^{8} - 992199207856 T^{10} + 108367501192120 T^{12} - 9900593807807312 T^{14} + 763821242557693894 T^{16} - 9900593807807312 p^{2} T^{18} + 108367501192120 p^{4} T^{20} - 992199207856 p^{6} T^{22} + 7464639260 p^{8} T^{24} - 44752016 p^{10} T^{26} + 202632 p^{12} T^{28} - 624 p^{14} T^{30} + p^{16} T^{32} \)
73 \( 1 - 384 T^{2} + 77592 T^{4} - 10701696 T^{6} + 1074979676 T^{8} - 79144926336 T^{10} + 4125267949992 T^{12} - 139972779128448 T^{14} + 4755000718224454 T^{16} - 139972779128448 p^{2} T^{18} + 4125267949992 p^{4} T^{20} - 79144926336 p^{6} T^{22} + 1074979676 p^{8} T^{24} - 10701696 p^{10} T^{26} + 77592 p^{12} T^{28} - 384 p^{14} T^{30} + p^{16} T^{32} \)
79 \( ( 1 + 416 T^{2} + 192 T^{3} + 83076 T^{4} + 52032 T^{5} + 10694880 T^{6} + 6451584 T^{7} + 985301638 T^{8} + 6451584 p T^{9} + 10694880 p^{2} T^{10} + 52032 p^{3} T^{11} + 83076 p^{4} T^{12} + 192 p^{5} T^{13} + 416 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
83 \( ( 1 + 496 T^{2} - 32 T^{3} + 115444 T^{4} - 16224 T^{5} + 16658640 T^{6} - 2884160 T^{7} + 1647118406 T^{8} - 2884160 p T^{9} + 16658640 p^{2} T^{10} - 16224 p^{3} T^{11} + 115444 p^{4} T^{12} - 32 p^{5} T^{13} + 496 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
89 \( ( 1 - 32 T + 968 T^{2} - 19104 T^{3} + 344908 T^{4} - 4946272 T^{5} + 65211064 T^{6} - 722195936 T^{7} + 7382860134 T^{8} - 722195936 p T^{9} + 65211064 p^{2} T^{10} - 4946272 p^{3} T^{11} + 344908 p^{4} T^{12} - 19104 p^{5} T^{13} + 968 p^{6} T^{14} - 32 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
97 \( 1 - 608 T^{2} + 193368 T^{4} - 42922528 T^{6} + 7531732828 T^{8} - 1117268953952 T^{10} + 144903090270312 T^{12} - 16670739116503072 T^{14} + 1710880177471399878 T^{16} - 16670739116503072 p^{2} T^{18} + 144903090270312 p^{4} T^{20} - 1117268953952 p^{6} T^{22} + 7531732828 p^{8} T^{24} - 42922528 p^{10} T^{26} + 193368 p^{12} T^{28} - 608 p^{14} T^{30} + p^{16} T^{32} \)
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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

−1.76899619574109735621365308559, −1.72908400749233652273720206012, −1.72879957059964908916985539985, −1.58941275380998000554506380211, −1.55952692746533215731567345564, −1.47484320468127888511685169981, −1.46047330741662008492655248583, −1.45585146461531174200398765782, −1.45283928296874941009441738819, −0.978150336955341883212302911253, −0.929658355575968870177124681229, −0.907061268751395042823396018084, −0.858802708416759994777815169695, −0.77628782660267031131196941781, −0.76763232343447019691591995987, −0.70921426319946553571939709467, −0.65034781877424118907420366383, −0.57974939924754021019945768657, −0.51718382220880686396917511109, −0.44208018594293744005032887552, −0.42739144337368547145696264360, −0.41810585730735435980416051867, −0.38666890158088665356456198093, −0.29205130640146791847317550704, −0.05198036614835496817113155921, 0.05198036614835496817113155921, 0.29205130640146791847317550704, 0.38666890158088665356456198093, 0.41810585730735435980416051867, 0.42739144337368547145696264360, 0.44208018594293744005032887552, 0.51718382220880686396917511109, 0.57974939924754021019945768657, 0.65034781877424118907420366383, 0.70921426319946553571939709467, 0.76763232343447019691591995987, 0.77628782660267031131196941781, 0.858802708416759994777815169695, 0.907061268751395042823396018084, 0.929658355575968870177124681229, 0.978150336955341883212302911253, 1.45283928296874941009441738819, 1.45585146461531174200398765782, 1.46047330741662008492655248583, 1.47484320468127888511685169981, 1.55952692746533215731567345564, 1.58941275380998000554506380211, 1.72879957059964908916985539985, 1.72908400749233652273720206012, 1.76899619574109735621365308559

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

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