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Properties

Label 32-1520e16-1.1-c1e16-0-2
Degree $32$
Conductor $8.119\times 10^{50}$
Sign $1$
Analytic cond. $2.21784\times 10^{17}$
Root an. cond. $3.48385$
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 − 8·19-s − 4·25-s + 96·31-s − 60·49-s − 72·59-s − 24·61-s − 96·71-s + 32·79-s + 6·81-s − 72·101-s + 96·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 96·169-s − 64·171-s + 173-s + 179-s + 181-s + 191-s + ⋯
L(s)  = 1  + 8/3·9-s − 1.83·19-s − 4/5·25-s + 17.2·31-s − 8.57·49-s − 9.37·59-s − 3.07·61-s − 11.3·71-s + 3.60·79-s + 2/3·81-s − 7.16·101-s + 8.72·121-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 − 7.38·169-s − 4.89·171-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 5^{16} \cdot 19^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 5^{16} \cdot 19^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(32\)
Conductor: \(2^{64} \cdot 5^{16} \cdot 19^{16}\)
Sign: $1$
Analytic conductor: \(2.21784\times 10^{17}\)
Root analytic conductor: \(3.48385\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((32,\ 2^{64} \cdot 5^{16} \cdot 19^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(9.234082775\)
\(L(\frac12)\) \(\approx\) \(9.234082775\)
\(L(\frac{3}{2})\) 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 \( ( 1 + 2 T^{2} - 6 T^{4} + 2 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
19 \( ( 1 + 2 T - 18 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{4} \)
good3 \( ( 1 - 4 T^{2} + 7 p T^{4} - 70 T^{6} + 292 T^{8} - 70 p^{2} T^{10} + 7 p^{5} T^{12} - 4 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
7 \( ( 1 + 30 T^{2} + 433 T^{4} + 4134 T^{6} + 31284 T^{8} + 4134 p^{2} T^{10} + 433 p^{4} T^{12} + 30 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
11 \( ( 1 - 48 T^{2} + 1264 T^{4} - 22320 T^{6} + 286782 T^{8} - 22320 p^{2} T^{10} + 1264 p^{4} T^{12} - 48 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
13 \( ( 1 + 48 T^{2} + 73 p T^{4} + 10566 T^{6} + 107940 T^{8} + 10566 p^{2} T^{10} + 73 p^{5} T^{12} + 48 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
17 \( ( 1 - 47 T^{2} + 1116 T^{4} - 47 p^{2} T^{6} + p^{4} T^{8} )^{4} \)
23 \( ( 1 + 94 T^{2} + 4753 T^{4} + 165190 T^{6} + 4326004 T^{8} + 165190 p^{2} T^{10} + 4753 p^{4} T^{12} + 94 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
29 \( ( 1 - 126 T^{2} + 8593 T^{4} - 399558 T^{6} + 13464948 T^{8} - 399558 p^{2} T^{10} + 8593 p^{4} T^{12} - 126 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
31 \( ( 1 - 24 T + 322 T^{2} - 2880 T^{3} + 18714 T^{4} - 2880 p T^{5} + 322 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} )^{4} \)
37 \( ( 1 + 186 T^{2} + 17116 T^{4} + 1021830 T^{6} + 43896582 T^{8} + 1021830 p^{2} T^{10} + 17116 p^{4} T^{12} + 186 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
41 \( ( 1 - 36 T^{2} + 4456 T^{4} - 69228 T^{6} + 8119950 T^{8} - 69228 p^{2} T^{10} + 4456 p^{4} T^{12} - 36 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
43 \( ( 1 + 228 T^{2} + 25432 T^{4} + 1822860 T^{6} + 92148558 T^{8} + 1822860 p^{2} T^{10} + 25432 p^{4} T^{12} + 228 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
47 \( ( 1 + 136 T^{2} + 12748 T^{4} + 852856 T^{6} + 46392550 T^{8} + 852856 p^{2} T^{10} + 12748 p^{4} T^{12} + 136 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
53 \( ( 1 + 136 T^{2} + 15205 T^{4} + 1092982 T^{6} + 69120628 T^{8} + 1092982 p^{2} T^{10} + 15205 p^{4} T^{12} + 136 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
59 \( ( 1 + 18 T + 245 T^{2} + 2310 T^{3} + 19596 T^{4} + 2310 p T^{5} + 245 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} )^{4} \)
61 \( ( 1 + 6 T + 202 T^{2} + 774 T^{3} + 16698 T^{4} + 774 p T^{5} + 202 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{4} \)
67 \( ( 1 - 236 T^{2} + 28693 T^{4} - 2607206 T^{6} + 193504180 T^{8} - 2607206 p^{2} T^{10} + 28693 p^{4} T^{12} - 236 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
71 \( ( 1 + 24 T + 482 T^{2} + 5760 T^{3} + 59034 T^{4} + 5760 p T^{5} + 482 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} )^{4} \)
73 \( ( 1 - 350 T^{2} + 60601 T^{4} - 7105478 T^{6} + 607038964 T^{8} - 7105478 p^{2} T^{10} + 60601 p^{4} T^{12} - 350 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
79 \( ( 1 - 8 T + 232 T^{2} - 1496 T^{3} + 24622 T^{4} - 1496 p T^{5} + 232 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{4} \)
83 \( ( 1 + 448 T^{2} + 98188 T^{4} + 13786240 T^{6} + 1353675910 T^{8} + 13786240 p^{2} T^{10} + 98188 p^{4} T^{12} + 448 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
89 \( ( 1 - 252 T^{2} + 21784 T^{4} + 196140 T^{6} - 149392338 T^{8} + 196140 p^{2} T^{10} + 21784 p^{4} T^{12} - 252 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
97 \( ( 1 + 546 T^{2} + 135580 T^{4} + 20962062 T^{6} + 2334879078 T^{8} + 20962062 p^{2} T^{10} + 135580 p^{4} T^{12} + 546 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
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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

−2.62239111081443246126207628773, −2.38214213264734303706798544015, −2.32335637581479852664288760695, −2.01581029113973870043303630680, −1.88596876808209692780398768956, −1.77796873241722336542768854521, −1.76918114113374262412491335229, −1.73942206217852778256860518330, −1.73800565588641834667367341936, −1.73677052887819133750636344361, −1.68028003786391137015713515163, −1.55206752781555973603390793824, −1.51926885433376429050225351360, −1.39390047777222381829631046961, −1.33856902228024375439641601795, −1.28925175777824011607726910756, −1.04469315014153154970751093589, −0.990284825456351434254407180204, −0.871188889111057589437183071083, −0.75816375718090786783272029418, −0.69396088469972765458066774308, −0.52277526845661624651651874960, −0.37896076344505265105169810288, −0.35348661585823347303987468156, −0.097227394299152329788986388307, 0.097227394299152329788986388307, 0.35348661585823347303987468156, 0.37896076344505265105169810288, 0.52277526845661624651651874960, 0.69396088469972765458066774308, 0.75816375718090786783272029418, 0.871188889111057589437183071083, 0.990284825456351434254407180204, 1.04469315014153154970751093589, 1.28925175777824011607726910756, 1.33856902228024375439641601795, 1.39390047777222381829631046961, 1.51926885433376429050225351360, 1.55206752781555973603390793824, 1.68028003786391137015713515163, 1.73677052887819133750636344361, 1.73800565588641834667367341936, 1.73942206217852778256860518330, 1.76918114113374262412491335229, 1.77796873241722336542768854521, 1.88596876808209692780398768956, 2.01581029113973870043303630680, 2.32335637581479852664288760695, 2.38214213264734303706798544015, 2.62239111081443246126207628773

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

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