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Properties

Label 32-1080e16-1.1-c1e16-0-0
Degree $32$
Conductor $3.426\times 10^{48}$
Sign $1$
Analytic cond. $9.35866\times 10^{14}$
Root an. cond. $2.93663$
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  − 2·2-s − 4-s − 6·5-s + 4·8-s + 12·10-s − 7·16-s + 6·20-s + 7·25-s + 14·32-s − 24·40-s + 34·49-s − 14·50-s + 96·53-s − 64-s + 4·79-s + 42·80-s + 64·83-s − 68·98-s − 7·100-s − 192·106-s − 84·107-s + 90·121-s + 22·125-s + 127-s − 32·128-s + 131-s + 137-s + ⋯
L(s)  = 1  − 1.41·2-s − 1/2·4-s − 2.68·5-s + 1.41·8-s + 3.79·10-s − 7/4·16-s + 1.34·20-s + 7/5·25-s + 2.47·32-s − 3.79·40-s + 34/7·49-s − 1.97·50-s + 13.1·53-s − 1/8·64-s + 0.450·79-s + 4.69·80-s + 7.02·83-s − 6.86·98-s − 0.699·100-s − 18.6·106-s − 8.12·107-s + 8.18·121-s + 1.96·125-s + 0.0887·127-s − 2.82·128-s + 0.0873·131-s + 0.0854·137-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{48} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{48} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(32\)
Conductor: \(2^{48} \cdot 3^{48} \cdot 5^{16}\)
Sign: $1$
Analytic conductor: \(9.35866\times 10^{14}\)
Root analytic conductor: \(2.93663\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((32,\ 2^{48} \cdot 3^{48} \cdot 5^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.008073647059\)
\(L(\frac12)\) \(\approx\) \(0.008073647059\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 + T + p T^{2} + p T^{3} + 3 p T^{4} + p^{2} T^{5} + p^{3} T^{6} + p^{3} T^{7} + p^{4} T^{8} )^{2} \)
3 \( 1 \)
5 \( ( 1 + 3 T + 2 p T^{2} + p^{2} T^{3} + 74 T^{4} + p^{3} T^{5} + 2 p^{3} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
good7 \( ( 1 - 17 T^{2} + 234 T^{4} - 2187 T^{6} + 17102 T^{8} - 2187 p^{2} T^{10} + 234 p^{4} T^{12} - 17 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
11 \( ( 1 - 45 T^{2} + 1106 T^{4} - 18923 T^{6} + 239446 T^{8} - 18923 p^{2} T^{10} + 1106 p^{4} T^{12} - 45 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
13 \( ( 1 + 16 T^{2} + 360 T^{4} + 2308 T^{6} + 52614 T^{8} + 2308 p^{2} T^{10} + 360 p^{4} T^{12} + 16 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
17 \( ( 1 - 59 T^{2} + 1725 T^{4} - 34014 T^{6} + 572474 T^{8} - 34014 p^{2} T^{10} + 1725 p^{4} T^{12} - 59 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
19 \( ( 1 - 71 T^{2} + 2517 T^{4} - 58362 T^{6} + 1142654 T^{8} - 58362 p^{2} T^{10} + 2517 p^{4} T^{12} - 71 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
23 \( ( 1 - 111 T^{2} + 223 p T^{4} - 5974 p T^{6} + 3035758 T^{8} - 5974 p^{3} T^{10} + 223 p^{5} T^{12} - 111 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
29 \( ( 1 - 152 T^{2} + 11668 T^{4} - 572532 T^{6} + 19653678 T^{8} - 572532 p^{2} T^{10} + 11668 p^{4} T^{12} - 152 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
31 \( ( 1 + 80 T^{2} + 22 T^{3} + 3295 T^{4} + 22 p T^{5} + 80 p^{2} T^{6} + p^{4} T^{8} )^{4} \)
37 \( ( 1 + 1228 T^{4} + 81728 T^{6} + 554950 T^{8} + 81728 p^{2} T^{10} + 1228 p^{4} T^{12} + p^{8} T^{16} )^{2} \)
41 \( ( 1 + 64 T^{2} + 2652 T^{4} + 170548 T^{6} + 9149214 T^{8} + 170548 p^{2} T^{10} + 2652 p^{4} T^{12} + 64 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
43 \( ( 1 + 128 T^{2} + 10900 T^{4} + 681108 T^{6} + 33400942 T^{8} + 681108 p^{2} T^{10} + 10900 p^{4} T^{12} + 128 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
47 \( ( 1 - 236 T^{2} + 27732 T^{4} - 2122980 T^{6} + 116308022 T^{8} - 2122980 p^{2} T^{10} + 27732 p^{4} T^{12} - 236 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
53 \( ( 1 - 24 T + 418 T^{2} - 4570 T^{3} + 39689 T^{4} - 4570 p T^{5} + 418 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} )^{4} \)
59 \( ( 1 - 264 T^{2} + 38216 T^{4} - 3631076 T^{6} + 4246898 p T^{8} - 3631076 p^{2} T^{10} + 38216 p^{4} T^{12} - 264 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
61 \( ( 1 - 99 T^{2} + 12953 T^{4} - 874538 T^{6} + 72160462 T^{8} - 874538 p^{2} T^{10} + 12953 p^{4} T^{12} - 99 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
67 \( ( 1 + 68 T^{2} + 9028 T^{4} + 893340 T^{6} + 45726358 T^{8} + 893340 p^{2} T^{10} + 9028 p^{4} T^{12} + 68 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
71 \( ( 1 + 116 T^{2} + 12664 T^{4} + 397056 T^{6} + 487610 p T^{8} + 397056 p^{2} T^{10} + 12664 p^{4} T^{12} + 116 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
73 \( ( 1 - 449 T^{2} + 95506 T^{4} - 12503287 T^{6} + 1100324774 T^{8} - 12503287 p^{2} T^{10} + 95506 p^{4} T^{12} - 449 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
79 \( ( 1 - T + 151 T^{2} + 30 T^{3} + 14740 T^{4} + 30 p T^{5} + 151 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} )^{4} \)
83 \( ( 1 - 16 T + 298 T^{2} - 3820 T^{3} + 35903 T^{4} - 3820 p T^{5} + 298 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} )^{4} \)
89 \( ( 1 + 248 T^{2} + 37972 T^{4} + 4284828 T^{6} + 4697582 p T^{8} + 4284828 p^{2} T^{10} + 37972 p^{4} T^{12} + 248 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
97 \( ( 1 - 377 T^{2} + 838 p T^{4} - 11780311 T^{6} + 1306138322 T^{8} - 11780311 p^{2} T^{10} + 838 p^{5} T^{12} - 377 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
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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

−2.46583653645929372272848932673, −2.42089378725259448193770027747, −2.34790087653004125181365708221, −2.29421114174641945416286433242, −2.27443036991164184712954681481, −2.27020839627533127264621301545, −2.26038627807539037808346094844, −2.14522237272135846825053611742, −1.95312052686502374880234564358, −1.78275096771216924576576463477, −1.65688991798993456478147761429, −1.63109577663292307686131582235, −1.55166619545618672690997107173, −1.32978592468127240235832491560, −1.24233743507634755884120030953, −1.09757803766409354003279897486, −0.983759902139575316908129899741, −0.957576991480330944234689347976, −0.858694487340985425877040955551, −0.69287796486885197479319011522, −0.63344164019357881693455238361, −0.52809814314771670209715015247, −0.47023633300147153080128057410, −0.36108278711076275167114763181, −0.01172639169490916213985049993, 0.01172639169490916213985049993, 0.36108278711076275167114763181, 0.47023633300147153080128057410, 0.52809814314771670209715015247, 0.63344164019357881693455238361, 0.69287796486885197479319011522, 0.858694487340985425877040955551, 0.957576991480330944234689347976, 0.983759902139575316908129899741, 1.09757803766409354003279897486, 1.24233743507634755884120030953, 1.32978592468127240235832491560, 1.55166619545618672690997107173, 1.63109577663292307686131582235, 1.65688991798993456478147761429, 1.78275096771216924576576463477, 1.95312052686502374880234564358, 2.14522237272135846825053611742, 2.26038627807539037808346094844, 2.27020839627533127264621301545, 2.27443036991164184712954681481, 2.29421114174641945416286433242, 2.34790087653004125181365708221, 2.42089378725259448193770027747, 2.46583653645929372272848932673

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

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