Properties

Degree $32$
Conductor $8.790\times 10^{51}$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·4-s − 2·16-s + 48·25-s − 64·37-s + 36·64-s − 192·100-s − 96·109-s + 96·121-s + 127-s + 131-s + 137-s + 139-s + 256·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + 128·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯
L(s)  = 1  − 2·4-s − 1/2·16-s + 48/5·25-s − 10.5·37-s + 9/2·64-s − 19.1·100-s − 9.19·109-s + 8.72·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 21.0·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 9.84·169-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 + 0.0708·199-s + 0.0688·211-s + ⋯

Functional equation

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

Invariants

Degree: \(32\)
Conductor: \(2^{32} \cdot 3^{32} \cdot 7^{32}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{1764} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((32,\ 2^{32} \cdot 3^{32} \cdot 7^{32} ,\ ( \ : [1/2]^{16} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(2.759290982\)
\(L(\frac12)\) \(\approx\) \(2.759290982\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 + p T^{2} + 7 T^{4} + p^{3} T^{6} + p^{4} T^{8} )^{2} \)
3 \( 1 \)
7 \( 1 \)
good5 \( ( 1 - 12 T^{2} + 84 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} )^{4} \)
11 \( ( 1 - 24 T^{2} + 378 T^{4} - 24 p^{2} T^{6} + p^{4} T^{8} )^{4} \)
13 \( ( 1 - 32 T^{2} + 592 T^{4} - 32 p^{2} T^{6} + p^{4} T^{8} )^{4} \)
17 \( ( 1 - 60 T^{2} + 1476 T^{4} - 60 p^{2} T^{6} + p^{4} T^{8} )^{4} \)
19 \( ( 1 - 12 T^{2} + 110 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} )^{4} \)
23 \( ( 1 - 48 T^{2} + 54 p T^{4} - 48 p^{2} T^{6} + p^{4} T^{8} )^{4} \)
29 \( ( 1 + 32 T^{2} + 1546 T^{4} + 32 p^{2} T^{6} + p^{4} T^{8} )^{4} \)
31 \( ( 1 + 4 T^{2} - 386 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} )^{4} \)
37 \( ( 1 + 8 T + 72 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{8} \)
41 \( ( 1 - 20 T^{2} - 588 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} )^{4} \)
43 \( ( 1 - 68 T^{2} + 4726 T^{4} - 68 p^{2} T^{6} + p^{4} T^{8} )^{4} \)
47 \( ( 1 - 4 T^{2} + 4222 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} )^{4} \)
53 \( ( 1 + 132 T^{2} + 8822 T^{4} + 132 p^{2} T^{6} + p^{4} T^{8} )^{4} \)
59 \( ( 1 + 44 T^{2} + 7246 T^{4} + 44 p^{2} T^{6} + p^{4} T^{8} )^{4} \)
61 \( ( 1 - 224 T^{2} + 19984 T^{4} - 224 p^{2} T^{6} + p^{4} T^{8} )^{4} \)
67 \( ( 1 - 60 T^{2} + 9366 T^{4} - 60 p^{2} T^{6} + p^{4} T^{8} )^{4} \)
71 \( ( 1 - 104 T^{2} + 12138 T^{4} - 104 p^{2} T^{6} + p^{4} T^{8} )^{4} \)
73 \( ( 1 - 128 T^{2} + 12832 T^{4} - 128 p^{2} T^{6} + p^{4} T^{8} )^{4} \)
79 \( ( 1 - 68 T^{2} + 838 T^{4} - 68 p^{2} T^{6} + p^{4} T^{8} )^{4} \)
83 \( ( 1 - 52 T^{2} + 13654 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} )^{4} \)
89 \( ( 1 - 172 T^{2} + 14788 T^{4} - 172 p^{2} T^{6} + p^{4} T^{8} )^{4} \)
97 \( ( 1 - 192 T^{2} + 23232 T^{4} - 192 p^{2} T^{6} + p^{4} T^{8} )^{4} \)
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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

−2.40825013383874555099721681010, −2.25587598349830082165641903102, −2.09751363874314606187620955883, −1.95948837231462828078858027507, −1.95860187502535794551826074907, −1.94626271765675220942288039642, −1.85529435415175420698527321478, −1.80127367765379861982832855577, −1.71027101250864604033837361318, −1.69592881644758942337173436844, −1.53595776900806944372834322203, −1.45729274791901545824670615685, −1.41807147242628669580759384893, −1.30806444956721212805661182589, −1.24661757950307111564011727739, −1.10363390315352232626702084212, −1.00208866235109440875137000474, −0.895560000748961543923742238190, −0.72300504249796505748268645515, −0.59517265524206225672799480948, −0.52535952304816477550715255870, −0.51609221502480383096396745285, −0.48959446679038249125839966977, −0.19218112262037312034461936932, −0.12285932077456366592653348894, 0.12285932077456366592653348894, 0.19218112262037312034461936932, 0.48959446679038249125839966977, 0.51609221502480383096396745285, 0.52535952304816477550715255870, 0.59517265524206225672799480948, 0.72300504249796505748268645515, 0.895560000748961543923742238190, 1.00208866235109440875137000474, 1.10363390315352232626702084212, 1.24661757950307111564011727739, 1.30806444956721212805661182589, 1.41807147242628669580759384893, 1.45729274791901545824670615685, 1.53595776900806944372834322203, 1.69592881644758942337173436844, 1.71027101250864604033837361318, 1.80127367765379861982832855577, 1.85529435415175420698527321478, 1.94626271765675220942288039642, 1.95860187502535794551826074907, 1.95948837231462828078858027507, 2.09751363874314606187620955883, 2.25587598349830082165641903102, 2.40825013383874555099721681010

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

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