Properties

Label 32-42e32-1.1-c2e16-0-1
Degree $32$
Conductor $8.790\times 10^{51}$
Sign $1$
Analytic cond. $8.11592\times 10^{26}$
Root an. cond. $6.93293$
Motivic weight $2$
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  + 8·25-s − 128·37-s − 160·43-s + 384·67-s + 560·79-s + 400·109-s − 624·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2.56e3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯
L(s)  = 1  + 8/25·25-s − 3.45·37-s − 3.72·43-s + 5.73·67-s + 7.08·79-s + 3.66·109-s − 5.15·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 15.1·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + 0.00473·211-s + 0.00448·223-s + 0.00440·227-s + ⋯

Functional equation

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

Invariants

Degree: \(32\)
Conductor: \(2^{32} \cdot 3^{32} \cdot 7^{32}\)
Sign: $1$
Analytic conductor: \(8.11592\times 10^{26}\)
Root analytic conductor: \(6.93293\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1764} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((32,\ 2^{32} \cdot 3^{32} \cdot 7^{32} ,\ ( \ : [1]^{16} ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(11.47078336\)
\(L(\frac12)\) \(\approx\) \(11.47078336\)
\(L(2)\) 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 \( 1 \)
7 \( 1 \)
good5 \( ( 1 - 4 T^{2} + 62 p T^{4} + 6176 T^{6} - 314189 T^{8} + 6176 p^{4} T^{10} + 62 p^{9} T^{12} - 4 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
11 \( ( 1 + 156 T^{2} + 9695 T^{4} + 156 p^{4} T^{6} + p^{8} T^{8} )^{4} \)
13 \( ( 1 + 320 T^{2} + p^{4} T^{4} )^{8} \)
17 \( ( 1 + 908 T^{2} + 465238 T^{4} + 174503072 T^{6} + 53207503891 T^{8} + 174503072 p^{4} T^{10} + 465238 p^{8} T^{12} + 908 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
19 \( ( 1 + 204 T^{2} - 74630 T^{4} - 29456784 T^{6} - 8893869261 T^{8} - 29456784 p^{4} T^{10} - 74630 p^{8} T^{12} + 204 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
23 \( ( 1 + 1800 T^{2} + 1895086 T^{4} + 1413417600 T^{6} + 823797842115 T^{8} + 1413417600 p^{4} T^{10} + 1895086 p^{8} T^{12} + 1800 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
29 \( ( 1 - 2616 T^{2} + 3026354 T^{4} - 2616 p^{4} T^{6} + p^{8} T^{8} )^{4} \)
31 \( ( 1 - 2260 T^{2} + 2039386 T^{4} - 2759848720 T^{6} + 3844070252755 T^{8} - 2759848720 p^{4} T^{10} + 2039386 p^{8} T^{12} - 2260 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
37 \( ( 1 + 32 T - 422 T^{2} - 41344 T^{3} - 1185101 T^{4} - 41344 p^{2} T^{5} - 422 p^{4} T^{6} + 32 p^{6} T^{7} + p^{8} T^{8} )^{4} \)
41 \( ( 1 - 2588 T^{2} + 6977658 T^{4} - 2588 p^{4} T^{6} + p^{8} T^{8} )^{4} \)
43 \( ( 1 + 20 T + 2250 T^{2} + 20 p^{2} T^{3} + p^{4} T^{4} )^{8} \)
47 \( ( 1 + 3932 T^{2} + 6690634 T^{4} - 3890210704 T^{6} - 22341475242605 T^{8} - 3890210704 p^{4} T^{10} + 6690634 p^{8} T^{12} + 3932 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
53 \( ( 1 + 5568 T^{2} + 15495838 T^{4} - 1526611968 T^{6} - 66424893983709 T^{8} - 1526611968 p^{4} T^{10} + 15495838 p^{8} T^{12} + 5568 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
59 \( ( 1 + 9020 T^{2} + 41640106 T^{4} + 139679859440 T^{6} + 444997979937715 T^{8} + 139679859440 p^{4} T^{10} + 41640106 p^{8} T^{12} + 9020 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
61 \( ( 1 - 12552 T^{2} + 91686478 T^{4} - 479166876288 T^{6} + 1965202725781731 T^{8} - 479166876288 p^{4} T^{10} + 91686478 p^{8} T^{12} - 12552 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
67 \( ( 1 - 96 T - 518 T^{2} - 72576 T^{3} + 33229011 T^{4} - 72576 p^{2} T^{5} - 518 p^{4} T^{6} - 96 p^{6} T^{7} + p^{8} T^{8} )^{4} \)
71 \( ( 1 - 9400 T^{2} + 11634 p^{2} T^{4} - 9400 p^{4} T^{6} + p^{8} T^{8} )^{4} \)
73 \( ( 1 - 12024 T^{2} + 63623662 T^{4} - 290456938368 T^{6} + 1641193744994499 T^{8} - 290456938368 p^{4} T^{10} + 63623662 p^{8} T^{12} - 12024 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
79 \( ( 1 - 140 T + 8410 T^{2} + 180880 T^{3} - 36057581 T^{4} + 180880 p^{2} T^{5} + 8410 p^{4} T^{6} - 140 p^{6} T^{7} + p^{8} T^{8} )^{4} \)
83 \( ( 1 - 6404 T^{2} + 98011494 T^{4} - 6404 p^{4} T^{6} + p^{8} T^{8} )^{4} \)
89 \( ( 1 + 16988 T^{2} + 93314134 T^{4} + 1185652453664 T^{6} + 16089994534303315 T^{8} + 1185652453664 p^{4} T^{10} + 93314134 p^{8} T^{12} + 16988 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
97 \( ( 1 + 13704 T^{2} + 84688466 T^{4} + 13704 p^{4} T^{6} + p^{8} T^{8} )^{4} \)
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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

−1.98298108698039730137052035918, −1.96101497545133239488078605254, −1.82391658542711681985843711013, −1.81804845513365472230401879560, −1.80569821669473816115952609278, −1.80274517299070947311931120747, −1.80182688773801523221541952365, −1.73400185805229750360507385463, −1.43919263872232689591407476828, −1.42579879241939519933458607247, −1.36059618670705872234193765402, −1.18644191708580501999894605701, −1.17639515570394416088143326419, −1.09578999923154994246469257291, −1.09530232779165591545089677520, −0.961391339038250078067943616689, −0.74562766224195698735702903457, −0.71003948538679794079584614848, −0.53209870470383764361640049525, −0.52897277688496624073042839894, −0.47846291159754770010509120627, −0.36382607788023065819112502747, −0.31352120203443641693843840971, −0.21693063475097063078294040479, −0.079165499004023745110933366916, 0.079165499004023745110933366916, 0.21693063475097063078294040479, 0.31352120203443641693843840971, 0.36382607788023065819112502747, 0.47846291159754770010509120627, 0.52897277688496624073042839894, 0.53209870470383764361640049525, 0.71003948538679794079584614848, 0.74562766224195698735702903457, 0.961391339038250078067943616689, 1.09530232779165591545089677520, 1.09578999923154994246469257291, 1.17639515570394416088143326419, 1.18644191708580501999894605701, 1.36059618670705872234193765402, 1.42579879241939519933458607247, 1.43919263872232689591407476828, 1.73400185805229750360507385463, 1.80182688773801523221541952365, 1.80274517299070947311931120747, 1.80569821669473816115952609278, 1.81804845513365472230401879560, 1.82391658542711681985843711013, 1.96101497545133239488078605254, 1.98298108698039730137052035918

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

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