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Properties

Label 32-1014e16-1.1-c1e16-0-0
Degree $32$
Conductor $1.249\times 10^{48}$
Sign $1$
Analytic cond. $3.41225\times 10^{14}$
Root an. cond. $2.84549$
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  + 16·7-s − 4·16-s − 32·19-s − 16·31-s + 8·37-s + 128·49-s + 48·61-s − 32·67-s − 56·73-s − 96·79-s − 12·81-s − 16·97-s + 16·109-s − 64·112-s + 127-s + 131-s − 512·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + ⋯
L(s)  = 1  + 6.04·7-s − 16-s − 7.34·19-s − 2.87·31-s + 1.31·37-s + 18.2·49-s + 6.14·61-s − 3.90·67-s − 6.55·73-s − 10.8·79-s − 4/3·81-s − 1.62·97-s + 1.53·109-s − 6.04·112-s + 0.0887·127-s + 0.0873·131-s − 44.3·133-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 + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 13^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 13^{32}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(32\)
Conductor: \(2^{16} \cdot 3^{16} \cdot 13^{32}\)
Sign: $1$
Analytic conductor: \(3.41225\times 10^{14}\)
Root analytic conductor: \(2.84549\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((32,\ 2^{16} \cdot 3^{16} \cdot 13^{32} ,\ ( \ : [1/2]^{16} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.075870461\)
\(L(\frac12)\) \(\approx\) \(1.075870461\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 + T^{4} )^{4} \)
3 \( ( 1 + 2 p T^{4} + p^{4} T^{8} )^{2} \)
13 \( 1 \)
good5 \( 1 + 4 p T^{4} - 614 T^{8} - 1232 p T^{12} + 499171 T^{16} - 1232 p^{5} T^{20} - 614 p^{8} T^{24} + 4 p^{13} T^{28} + p^{16} T^{32} \)
7 \( ( 1 - 8 T + 32 T^{2} - 16 p T^{3} + 404 T^{4} - 1312 T^{5} + 3840 T^{6} - 11496 T^{7} + 32890 T^{8} - 11496 p T^{9} + 3840 p^{2} T^{10} - 1312 p^{3} T^{11} + 404 p^{4} T^{12} - 16 p^{6} T^{13} + 32 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
11 \( ( 1 + 4 T^{4} - 21414 T^{8} + 4 p^{4} T^{12} + p^{8} T^{16} )^{2} \)
17 \( ( 1 + 64 T^{2} + 2398 T^{4} + 63040 T^{6} + 1231459 T^{8} + 63040 p^{2} T^{10} + 2398 p^{4} T^{12} + 64 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
19 \( ( 1 + 16 T + 128 T^{2} + 800 T^{3} + 4772 T^{4} + 26240 T^{5} + 129024 T^{6} + 588816 T^{7} + 2594074 T^{8} + 588816 p T^{9} + 129024 p^{2} T^{10} + 26240 p^{3} T^{11} + 4772 p^{4} T^{12} + 800 p^{5} T^{13} + 128 p^{6} T^{14} + 16 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
23 \( ( 1 + 124 T^{2} + 7300 T^{4} + 272668 T^{6} + 7281418 T^{8} + 272668 p^{2} T^{10} + 7300 p^{4} T^{12} + 124 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
29 \( ( 1 - 100 T^{2} + 5650 T^{4} - 224896 T^{6} + 7187947 T^{8} - 224896 p^{2} T^{10} + 5650 p^{4} T^{12} - 100 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
31 \( ( 1 + 8 T + 32 T^{2} - 8 T^{3} + 500 T^{4} + 11608 T^{5} + 76896 T^{6} + 198312 T^{7} + 81958 T^{8} + 198312 p T^{9} + 76896 p^{2} T^{10} + 11608 p^{3} T^{11} + 500 p^{4} T^{12} - 8 p^{5} T^{13} + 32 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
37 \( ( 1 - 4 T + 8 T^{2} - 56 T^{3} - 826 T^{4} + 1084 T^{5} + 3840 T^{6} - 120828 T^{7} + 2893915 T^{8} - 120828 p T^{9} + 3840 p^{2} T^{10} + 1084 p^{3} T^{11} - 826 p^{4} T^{12} - 56 p^{5} T^{13} + 8 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
41 \( 1 - 9604 T^{4} + 43285642 T^{8} - 122184032656 T^{12} + 241429707746323 T^{16} - 122184032656 p^{4} T^{20} + 43285642 p^{8} T^{24} - 9604 p^{12} T^{28} + p^{16} T^{32} \)
43 \( ( 1 - 154 T^{2} + 9600 T^{4} - 154 p^{2} T^{6} + p^{4} T^{8} )^{4} \)
47 \( 1 - 2872 T^{4} + 3290884 T^{8} + 7102487576 T^{12} - 16252261348538 T^{16} + 7102487576 p^{4} T^{20} + 3290884 p^{8} T^{24} - 2872 p^{12} T^{28} + p^{16} T^{32} \)
53 \( ( 1 - 268 T^{2} + 36418 T^{4} - 3215200 T^{6} + 200754715 T^{8} - 3215200 p^{2} T^{10} + 36418 p^{4} T^{12} - 268 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
59 \( 1 - 12376 T^{4} + 67732252 T^{8} - 215804963560 T^{12} + 618028375734790 T^{16} - 215804963560 p^{4} T^{20} + 67732252 p^{8} T^{24} - 12376 p^{12} T^{28} + p^{16} T^{32} \)
61 \( ( 1 - 12 T + 184 T^{2} - 1380 T^{3} + 14787 T^{4} - 1380 p T^{5} + 184 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} )^{4} \)
67 \( ( 1 + 16 T + 128 T^{2} + 1328 T^{3} + 16436 T^{4} + 140816 T^{5} + 1031040 T^{6} + 10481520 T^{7} + 105435226 T^{8} + 10481520 p T^{9} + 1031040 p^{2} T^{10} + 140816 p^{3} T^{11} + 16436 p^{4} T^{12} + 1328 p^{5} T^{13} + 128 p^{6} T^{14} + 16 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
71 \( 1 - 9880 T^{4} + 13979524 T^{8} + 212279401976 T^{12} - 1529872372511546 T^{16} + 212279401976 p^{4} T^{20} + 13979524 p^{8} T^{24} - 9880 p^{12} T^{28} + p^{16} T^{32} \)
73 \( ( 1 + 28 T + 392 T^{2} + 4232 T^{3} + 37106 T^{4} + 243188 T^{5} + 1218624 T^{6} + 2086596 T^{7} - 23078141 T^{8} + 2086596 p T^{9} + 1218624 p^{2} T^{10} + 243188 p^{3} T^{11} + 37106 p^{4} T^{12} + 4232 p^{5} T^{13} + 392 p^{6} T^{14} + 28 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
79 \( ( 1 + 24 T + 424 T^{2} + 5256 T^{3} + 54822 T^{4} + 5256 p T^{5} + 424 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} )^{4} \)
83 \( 1 + 8744 T^{4} + 13736260 T^{8} - 185286399304 T^{12} - 1538291441783546 T^{16} - 185286399304 p^{4} T^{20} + 13736260 p^{8} T^{24} + 8744 p^{12} T^{28} + p^{16} T^{32} \)
89 \( 1 - 31048 T^{4} + 529608604 T^{8} - 6228527319544 T^{12} + 55552390840330438 T^{16} - 6228527319544 p^{4} T^{20} + 529608604 p^{8} T^{24} - 31048 p^{12} T^{28} + p^{16} T^{32} \)
97 \( ( 1 + 8 T + 32 T^{2} - 728 T^{3} + 2084 T^{4} + 164152 T^{5} + 1511520 T^{6} + 2490456 T^{7} - 118106810 T^{8} + 2490456 p T^{9} + 1511520 p^{2} T^{10} + 164152 p^{3} T^{11} + 2084 p^{4} T^{12} - 728 p^{5} T^{13} + 32 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
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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

−2.43348622235331456247776463045, −2.31012103347525197637837438426, −2.30649655488144933651730123545, −2.30063708324845648652401159659, −2.23516137012976657834445795599, −2.16190075243257430093341138548, −2.03612844318724919991655564674, −2.02974335703256798022588039770, −1.88682903814204613497820954763, −1.86532960968696613001524439692, −1.84339397856940589898593485981, −1.60941799576055935726207400794, −1.55028457427433649377913096987, −1.50454690223312031860967011495, −1.50176938967315062121644026703, −1.43129228507840089909841717196, −1.36512432310787874508685886184, −1.31089078456558644923483431063, −0.978725589680313139758492770243, −0.910114651669904149486824249508, −0.846389763096417600080347137463, −0.49540309755127825585421892244, −0.37308024113706369849707344807, −0.18240040711233413754522185196, −0.092705263966179818209449985776, 0.092705263966179818209449985776, 0.18240040711233413754522185196, 0.37308024113706369849707344807, 0.49540309755127825585421892244, 0.846389763096417600080347137463, 0.910114651669904149486824249508, 0.978725589680313139758492770243, 1.31089078456558644923483431063, 1.36512432310787874508685886184, 1.43129228507840089909841717196, 1.50176938967315062121644026703, 1.50454690223312031860967011495, 1.55028457427433649377913096987, 1.60941799576055935726207400794, 1.84339397856940589898593485981, 1.86532960968696613001524439692, 1.88682903814204613497820954763, 2.02974335703256798022588039770, 2.03612844318724919991655564674, 2.16190075243257430093341138548, 2.23516137012976657834445795599, 2.30063708324845648652401159659, 2.30649655488144933651730123545, 2.31012103347525197637837438426, 2.43348622235331456247776463045

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

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