Properties

Label 32-42e32-1.1-c1e16-0-1
Degree $32$
Conductor $8.790\times 10^{51}$
Sign $1$
Analytic cond. $2.40110\times 10^{18}$
Root an. cond. $3.75308$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 8·13-s − 16-s + 28·25-s − 8·37-s + 16·61-s + 10·64-s + 8·73-s − 88·97-s + 24·109-s − 60·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 68·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯
L(s)  = 1  + 2.21·13-s − 1/4·16-s + 28/5·25-s − 1.31·37-s + 2.04·61-s + 5/4·64-s + 0.936·73-s − 8.93·97-s + 2.29·109-s − 5.45·121-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 − 5.23·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + ⋯

Functional equation

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

Invariants

Degree: \(32\)
Conductor: \(2^{32} \cdot 3^{32} \cdot 7^{32}\)
Sign: $1$
Analytic conductor: \(2.40110\times 10^{18}\)
Root analytic conductor: \(3.75308\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1764} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((32,\ 2^{32} \cdot 3^{32} \cdot 7^{32} ,\ ( \ : [1/2]^{16} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.2142098437\)
\(L(\frac12)\) \(\approx\) \(0.2142098437\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T^{4} - 5 p T^{6} + p^{2} T^{8} - 5 p^{3} T^{10} + p^{4} T^{12} + p^{8} T^{16} \)
3 \( 1 \)
7 \( 1 \)
good5 \( ( 1 - 14 T^{2} + 117 T^{4} - 698 T^{6} + 3576 T^{8} - 698 p^{2} T^{10} + 117 p^{4} T^{12} - 14 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
11 \( ( 1 + 30 T^{2} + 757 T^{4} + 11666 T^{6} + 154360 T^{8} + 11666 p^{2} T^{10} + 757 p^{4} T^{12} + 30 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
13 \( ( 1 - 2 T + 27 T^{2} - 38 T^{3} + 378 T^{4} - 38 p T^{5} + 27 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{4} \)
17 \( ( 1 - 76 T^{2} + 3096 T^{4} - 83940 T^{6} + 1656750 T^{8} - 83940 p^{2} T^{10} + 3096 p^{4} T^{12} - 76 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
19 \( ( 1 - 34 T^{2} + 1041 T^{4} - 25110 T^{6} + 610032 T^{8} - 25110 p^{2} T^{10} + 1041 p^{4} T^{12} - 34 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
23 \( ( 1 + 116 T^{2} + 6936 T^{4} + 269340 T^{6} + 7319790 T^{8} + 269340 p^{2} T^{10} + 6936 p^{4} T^{12} + 116 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
29 \( ( 1 - 146 T^{2} + 10849 T^{4} - 18010 p T^{6} + 17802004 T^{8} - 18010 p^{3} T^{10} + 10849 p^{4} T^{12} - 146 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
31 \( ( 1 - 132 T^{2} + 6430 T^{4} - 130264 T^{6} + 1736719 T^{8} - 130264 p^{2} T^{10} + 6430 p^{4} T^{12} - 132 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
37 \( ( 1 + 2 T + 115 T^{2} + 282 T^{3} + 5758 T^{4} + 282 p T^{5} + 115 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{4} \)
41 \( ( 1 - 188 T^{2} + 15912 T^{4} - 846644 T^{6} + 36194574 T^{8} - 846644 p^{2} T^{10} + 15912 p^{4} T^{12} - 188 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
43 \( ( 1 - 202 T^{2} + 17289 T^{4} - 881742 T^{6} + 37179072 T^{8} - 881742 p^{2} T^{10} + 17289 p^{4} T^{12} - 202 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
47 \( ( 1 + 200 T^{2} + 22212 T^{4} + 1666392 T^{6} + 91084806 T^{8} + 1666392 p^{2} T^{10} + 22212 p^{4} T^{12} + 200 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
53 \( ( 1 - 282 T^{2} + 34177 T^{4} - 2497906 T^{6} + 140957380 T^{8} - 2497906 p^{2} T^{10} + 34177 p^{4} T^{12} - 282 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
59 \( ( 1 + 186 T^{2} + 18161 T^{4} + 1259962 T^{6} + 77631844 T^{8} + 1259962 p^{2} T^{10} + 18161 p^{4} T^{12} + 186 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
61 \( ( 1 - 4 T + 162 T^{2} - 588 T^{3} + 12546 T^{4} - 588 p T^{5} + 162 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{4} \)
67 \( ( 1 - 254 T^{2} + 32925 T^{4} - 3201102 T^{6} + 246095804 T^{8} - 3201102 p^{2} T^{10} + 32925 p^{4} T^{12} - 254 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
71 \( ( 1 + 360 T^{2} + 62596 T^{4} + 7074488 T^{6} + 580384198 T^{8} + 7074488 p^{2} T^{10} + 62596 p^{4} T^{12} + 360 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
73 \( ( 1 - 2 T + 141 T^{2} + 58 T^{3} + 12336 T^{4} + 58 p T^{5} + 141 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{4} \)
79 \( ( 1 - 332 T^{2} + 56998 T^{4} - 6691240 T^{6} + 595616087 T^{8} - 6691240 p^{2} T^{10} + 56998 p^{4} T^{12} - 332 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
83 \( ( 1 + 110 T^{2} + 7989 T^{4} + 466194 T^{6} + 66686616 T^{8} + 466194 p^{2} T^{10} + 7989 p^{4} T^{12} + 110 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
89 \( ( 1 - 352 T^{2} + 72316 T^{4} - 10162848 T^{6} + 1047015622 T^{8} - 10162848 p^{2} T^{10} + 72316 p^{4} T^{12} - 352 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
97 \( ( 1 + 22 T + 373 T^{2} + 3806 T^{3} + 41336 T^{4} + 3806 p T^{5} + 373 p^{2} T^{6} + 22 p^{3} T^{7} + p^{4} T^{8} )^{4} \)
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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

−2.30928439750821853264113592489, −2.29310300794628704004090109905, −2.22962146986591607316730576467, −2.19776007444181731637267097585, −2.11747418113496779458162954828, −1.96296669641487703332882896907, −1.75667253381758288250559976270, −1.65230077352672996673022997976, −1.62233611148280031140669214392, −1.57463346306443153012111233324, −1.55345536084745322797682502665, −1.44687384864285138411337412489, −1.42852357792577046662499520543, −1.41868949283161618256399151777, −1.20098408421241316009927911398, −1.11586910554042702533833135339, −0.915516864024767389906771201490, −0.913777484020455147920030241309, −0.902743975397020308276624471876, −0.868071344142933998047553800645, −0.73651442347996414876479456447, −0.54484806619979619644088151832, −0.37979472306125781040975661393, −0.12335289185746413892489917120, −0.02974917700047077227018543910, 0.02974917700047077227018543910, 0.12335289185746413892489917120, 0.37979472306125781040975661393, 0.54484806619979619644088151832, 0.73651442347996414876479456447, 0.868071344142933998047553800645, 0.902743975397020308276624471876, 0.913777484020455147920030241309, 0.915516864024767389906771201490, 1.11586910554042702533833135339, 1.20098408421241316009927911398, 1.41868949283161618256399151777, 1.42852357792577046662499520543, 1.44687384864285138411337412489, 1.55345536084745322797682502665, 1.57463346306443153012111233324, 1.62233611148280031140669214392, 1.65230077352672996673022997976, 1.75667253381758288250559976270, 1.96296669641487703332882896907, 2.11747418113496779458162954828, 2.19776007444181731637267097585, 2.22962146986591607316730576467, 2.29310300794628704004090109905, 2.30928439750821853264113592489

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

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