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Properties

Label 32-1096e16-1.1-c0e16-0-1
Degree $32$
Conductor $4.335\times 10^{48}$
Sign $1$
Analytic cond. $6.41922\times 10^{-5}$
Root an. cond. $0.739577$
Motivic weight $0$
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  + 2-s − 9-s + 2·11-s − 2·17-s − 18-s − 2·19-s + 2·22-s + 25-s − 2·34-s − 2·38-s − 49-s + 50-s − 2·59-s + 2·73-s + 17·97-s − 98-s − 2·99-s + 2·107-s − 2·118-s + 121-s + 127-s + 131-s + 137-s + 139-s + 2·146-s + 149-s + 151-s + ⋯
L(s)  = 1  + 2-s − 9-s + 2·11-s − 2·17-s − 18-s − 2·19-s + 2·22-s + 25-s − 2·34-s − 2·38-s − 49-s + 50-s − 2·59-s + 2·73-s + 17·97-s − 98-s − 2·99-s + 2·107-s − 2·118-s + 121-s + 127-s + 131-s + 137-s + 139-s + 2·146-s + 149-s + 151-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 137^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 137^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(32\)
Conductor: \(2^{48} \cdot 137^{16}\)
Sign: $1$
Analytic conductor: \(6.41922\times 10^{-5}\)
Root analytic conductor: \(0.739577\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((32,\ 2^{48} \cdot 137^{16} ,\ ( \ : [0]^{16} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7753795879\)
\(L(\frac12)\) \(\approx\) \(0.7753795879\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{15} + T^{16} \)
137 \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{15} + T^{16} \)
good3 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{15} + T^{16} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} ) \)
5 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} - T^{22} + T^{24} - T^{26} + T^{28} - T^{30} + T^{32} \)
7 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{15} + T^{16} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} ) \)
11 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{15} + T^{16} )^{2} \)
13 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} - T^{22} + T^{24} - T^{26} + T^{28} - T^{30} + T^{32} \)
17 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} )^{2} \)
19 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} )^{2} \)
23 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} - T^{22} + T^{24} - T^{26} + T^{28} - T^{30} + T^{32} \)
29 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} - T^{22} + T^{24} - T^{26} + T^{28} - T^{30} + T^{32} \)
31 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} - T^{22} + T^{24} - T^{26} + T^{28} - T^{30} + T^{32} \)
37 \( ( 1 - T )^{16}( 1 + T )^{16} \)
41 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{15} + T^{16} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} ) \)
43 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{15} + T^{16} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} ) \)
47 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} - T^{22} + T^{24} - T^{26} + T^{28} - T^{30} + T^{32} \)
53 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} - T^{22} + T^{24} - T^{26} + T^{28} - T^{30} + T^{32} \)
59 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} )^{2} \)
61 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{15} + T^{16} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} ) \)
67 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{15} + T^{16} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} ) \)
71 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} - T^{22} + T^{24} - T^{26} + T^{28} - T^{30} + T^{32} \)
73 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{15} + T^{16} )^{2} \)
79 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} - T^{22} + T^{24} - T^{26} + T^{28} - T^{30} + T^{32} \)
83 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{15} + T^{16} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} ) \)
89 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{15} + T^{16} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} ) \)
97 \( ( 1 - T )^{16}( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{15} + T^{16} ) \)
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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

−2.69030679784203010628568649902, −2.68098869046165183236369457980, −2.62886308779763231105896341788, −2.55951836659513763532795767306, −2.49098435373970367643632325493, −2.45641373273974074333005943518, −2.45600446548811054995214865167, −2.35012855607257515032882447433, −2.23119689894166442956091697173, −2.19049591183431777267456024837, −2.00503318586944893068336890234, −1.97823203230882573679646225142, −1.93025855130957768895837953360, −1.84938651237408403321539456709, −1.68052297616302506048396751733, −1.66436693593590726062270934204, −1.52404066549442540763923737229, −1.52001875603671119936875462236, −1.36254890265494419215471625493, −1.17079301280459413130868254061, −1.10670958962211824246488351342, −0.971710179453613684431399728071, −0.894525661276553151792556640464, −0.61725585943390422353596161546, −0.59117339510943719327275878779, 0.59117339510943719327275878779, 0.61725585943390422353596161546, 0.894525661276553151792556640464, 0.971710179453613684431399728071, 1.10670958962211824246488351342, 1.17079301280459413130868254061, 1.36254890265494419215471625493, 1.52001875603671119936875462236, 1.52404066549442540763923737229, 1.66436693593590726062270934204, 1.68052297616302506048396751733, 1.84938651237408403321539456709, 1.93025855130957768895837953360, 1.97823203230882573679646225142, 2.00503318586944893068336890234, 2.19049591183431777267456024837, 2.23119689894166442956091697173, 2.35012855607257515032882447433, 2.45600446548811054995214865167, 2.45641373273974074333005943518, 2.49098435373970367643632325493, 2.55951836659513763532795767306, 2.62886308779763231105896341788, 2.68098869046165183236369457980, 2.69030679784203010628568649902

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

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