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Properties

Label 32-1840e16-1.1-c1e16-0-3
Degree $32$
Conductor $1.726\times 10^{52}$
Sign $1$
Analytic cond. $4.71540\times 10^{18}$
Root an. cond. $3.83307$
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  + 8·9-s + 16·13-s − 8·25-s + 72·29-s − 4·41-s − 72·49-s − 20·73-s + 8·81-s + 28·101-s + 128·117-s − 46·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 16·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯
L(s)  = 1  + 8/3·9-s + 4.43·13-s − 8/5·25-s + 13.3·29-s − 0.624·41-s − 10.2·49-s − 2.34·73-s + 8/9·81-s + 2.78·101-s + 11.8·117-s − 4.18·121-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 + 1.23·169-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 5^{16} \cdot 23^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 5^{16} \cdot 23^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(32\)
Conductor: \(2^{64} \cdot 5^{16} \cdot 23^{16}\)
Sign: $1$
Analytic conductor: \(4.71540\times 10^{18}\)
Root analytic conductor: \(3.83307\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((32,\ 2^{64} \cdot 5^{16} \cdot 23^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(52.92358987\)
\(L(\frac12)\) \(\approx\) \(52.92358987\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( ( 1 + T^{2} )^{8} \)
23 \( 1 + 36 T^{2} + 900 T^{4} + 26268 T^{6} + 598070 T^{8} + 26268 p^{2} T^{10} + 900 p^{4} T^{12} + 36 p^{6} T^{14} + p^{8} T^{16} \)
good3 \( ( 1 - 4 T^{2} + 20 T^{4} - 65 T^{6} + 184 T^{8} - 65 p^{2} T^{10} + 20 p^{4} T^{12} - 4 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
7 \( ( 1 + 36 T^{2} + 12 p^{2} T^{4} + 6033 T^{6} + 46736 T^{8} + 6033 p^{2} T^{10} + 12 p^{6} T^{12} + 36 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
11 \( ( 1 + 23 T^{2} + 219 T^{4} + 287 p T^{6} + 49184 T^{8} + 287 p^{3} T^{10} + 219 p^{4} T^{12} + 23 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
13 \( ( 1 - 4 T + 36 T^{2} - 119 T^{3} + 590 T^{4} - 119 p T^{5} + 36 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{4} \)
17 \( ( 1 - 92 T^{2} + 4108 T^{4} - 118211 T^{6} + 2381800 T^{8} - 118211 p^{2} T^{10} + 4108 p^{4} T^{12} - 92 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
19 \( ( 1 + p T^{2} + 495 T^{4} - 2875 T^{6} - 4240 T^{8} - 2875 p^{2} T^{10} + 495 p^{4} T^{12} + p^{7} T^{14} + p^{8} T^{16} )^{2} \)
29 \( ( 1 - 18 T + 175 T^{2} - 1236 T^{3} + 7296 T^{4} - 1236 p T^{5} + 175 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} )^{4} \)
31 \( ( 1 + 7 T^{2} + 1294 T^{4} + 3832 T^{6} + 1498441 T^{8} + 3832 p^{2} T^{10} + 1294 p^{4} T^{12} + 7 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
37 \( ( 1 - 159 T^{2} + 358 p T^{4} - 749097 T^{6} + 31678626 T^{8} - 749097 p^{2} T^{10} + 358 p^{5} T^{12} - 159 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
41 \( ( 1 + T + 106 T^{2} + 296 T^{3} + 5241 T^{4} + 296 p T^{5} + 106 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} )^{4} \)
43 \( ( 1 + 264 T^{2} + 32028 T^{4} + 55464 p T^{6} + 121809446 T^{8} + 55464 p^{3} T^{10} + 32028 p^{4} T^{12} + 264 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
47 \( ( 1 - 333 T^{2} + 49950 T^{4} - 4431075 T^{6} + 255228770 T^{8} - 4431075 p^{2} T^{10} + 49950 p^{4} T^{12} - 333 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
53 \( ( 1 - 295 T^{2} + 41142 T^{4} - 3634961 T^{6} + 226230674 T^{8} - 3634961 p^{2} T^{10} + 41142 p^{4} T^{12} - 295 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
59 \( ( 1 - 57 T^{2} - 3558 T^{4} - 77631 T^{6} + 33572138 T^{8} - 77631 p^{2} T^{10} - 3558 p^{4} T^{12} - 57 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
61 \( ( 1 - 241 T^{2} + 31543 T^{4} - 2777719 T^{6} + 189980896 T^{8} - 2777719 p^{2} T^{10} + 31543 p^{4} T^{12} - 241 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
67 \( ( 1 + 153 T^{2} + 294 p T^{4} + 1740831 T^{6} + 128224346 T^{8} + 1740831 p^{2} T^{10} + 294 p^{5} T^{12} + 153 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
71 \( ( 1 - 377 T^{2} + 1002 p T^{4} - 8598256 T^{6} + 724206761 T^{8} - 8598256 p^{2} T^{10} + 1002 p^{5} T^{12} - 377 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
73 \( ( 1 + 5 T + 156 T^{2} + 127 T^{3} + 10478 T^{4} + 127 p T^{5} + 156 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} )^{4} \)
79 \( ( 1 - 8 T^{2} + 10716 T^{4} - 65272 T^{6} + 104007110 T^{8} - 65272 p^{2} T^{10} + 10716 p^{4} T^{12} - 8 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
83 \( ( 1 + 281 T^{2} + 50130 T^{4} + 6200383 T^{6} + 585070490 T^{8} + 6200383 p^{2} T^{10} + 50130 p^{4} T^{12} + 281 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
89 \( ( 1 - 472 T^{2} + 105468 T^{4} - 14983592 T^{6} + 1537064390 T^{8} - 14983592 p^{2} T^{10} + 105468 p^{4} T^{12} - 472 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
97 \( ( 1 - 345 T^{2} + 61363 T^{4} - 8335971 T^{6} + 917128992 T^{8} - 8335971 p^{2} T^{10} + 61363 p^{4} T^{12} - 345 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
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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

−2.37631553403519783934550688767, −2.25570047114611542828772400996, −2.19208224118713960449067571079, −2.12387012026574227660536033447, −2.09840006875943719599028934688, −1.79769347529210448725323096573, −1.66466431979361775660386262455, −1.66049255822749892677399638866, −1.63991727657831619604645567217, −1.51594721080367911232351218737, −1.49642163045498267257201289864, −1.43367758831613168919830568559, −1.34583928775719668556693600370, −1.24436456657681677008963060912, −1.14703374595006478265068685893, −1.14195735334949791121489755555, −1.09664010888153218031563642288, −1.06877630525124474202250993388, −1.05853302137433780214726886843, −0.927847206142963891319202277853, −0.71469444671341381058085645388, −0.37471596111146271455900985568, −0.36958709871258417352989823337, −0.27616763972032133614618164779, −0.22197047681885647524057009489, 0.22197047681885647524057009489, 0.27616763972032133614618164779, 0.36958709871258417352989823337, 0.37471596111146271455900985568, 0.71469444671341381058085645388, 0.927847206142963891319202277853, 1.05853302137433780214726886843, 1.06877630525124474202250993388, 1.09664010888153218031563642288, 1.14195735334949791121489755555, 1.14703374595006478265068685893, 1.24436456657681677008963060912, 1.34583928775719668556693600370, 1.43367758831613168919830568559, 1.49642163045498267257201289864, 1.51594721080367911232351218737, 1.63991727657831619604645567217, 1.66049255822749892677399638866, 1.66466431979361775660386262455, 1.79769347529210448725323096573, 2.09840006875943719599028934688, 2.12387012026574227660536033447, 2.19208224118713960449067571079, 2.25570047114611542828772400996, 2.37631553403519783934550688767

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

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