Properties

Label 32-1155e16-1.1-c1e16-0-0
Degree $32$
Conductor $1.003\times 10^{49}$
Sign $1$
Analytic cond. $2.73996\times 10^{15}$
Root an. cond. $3.03689$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·4-s − 8·9-s + 16·11-s + 2·16-s + 40·23-s − 8·25-s − 32·36-s − 40·37-s + 64·44-s + 8·53-s − 4·64-s + 80·67-s − 24·71-s + 36·81-s + 160·92-s − 128·99-s − 32·100-s − 88·113-s + 136·121-s + 127-s + 131-s + 137-s + 139-s − 16·144-s − 160·148-s + 149-s + 151-s + ⋯
L(s)  = 1  + 2·4-s − 8/3·9-s + 4.82·11-s + 1/2·16-s + 8.34·23-s − 8/5·25-s − 5.33·36-s − 6.57·37-s + 9.64·44-s + 1.09·53-s − 1/2·64-s + 9.77·67-s − 2.84·71-s + 4·81-s + 16.6·92-s − 12.8·99-s − 3.19·100-s − 8.27·113-s + 12.3·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 4/3·144-s − 13.1·148-s + 0.0819·149-s + 0.0813·151-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{16} \cdot 7^{16} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{16} \cdot 7^{16} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(32\)
Conductor: \(3^{16} \cdot 5^{16} \cdot 7^{16} \cdot 11^{16}\)
Sign: $1$
Analytic conductor: \(2.73996\times 10^{15}\)
Root analytic conductor: \(3.03689\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((32,\ 3^{16} \cdot 5^{16} \cdot 7^{16} \cdot 11^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(7.445261005\)
\(L(\frac12)\) \(\approx\) \(7.445261005\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( ( 1 + T^{2} )^{8} \)
5 \( ( 1 + T^{2} )^{8} \)
7 \( 1 - 4 T^{4} - 314 T^{8} - 4 p^{4} T^{12} + p^{8} T^{16} \)
11 \( ( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{4} \)
good2 \( ( 1 - p T^{2} + 5 T^{4} - 3 p^{2} T^{6} + 3 p^{3} T^{8} - 3 p^{4} T^{10} + 5 p^{4} T^{12} - p^{7} T^{14} + p^{8} T^{16} )^{2} \)
13 \( ( 1 + 20 T^{2} + 116 T^{4} + 3340 T^{6} + 82486 T^{8} + 3340 p^{2} T^{10} + 116 p^{4} T^{12} + 20 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
17 \( ( 1 + 62 T^{2} + 2165 T^{4} + 53542 T^{6} + 1011484 T^{8} + 53542 p^{2} T^{10} + 2165 p^{4} T^{12} + 62 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
19 \( ( 1 + 78 T^{2} + 3413 T^{4} + 101206 T^{6} + 2229420 T^{8} + 101206 p^{2} T^{10} + 3413 p^{4} T^{12} + 78 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
23 \( ( 1 - 10 T + 117 T^{2} - 30 p T^{3} + 4304 T^{4} - 30 p^{2} T^{5} + 117 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} )^{4} \)
29 \( ( 1 - 126 T^{2} + 8965 T^{4} - 422054 T^{6} + 494716 p T^{8} - 422054 p^{2} T^{10} + 8965 p^{4} T^{12} - 126 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
31 \( ( 1 - 148 T^{2} + 10868 T^{4} - 529836 T^{6} + 18939030 T^{8} - 529836 p^{2} T^{10} + 10868 p^{4} T^{12} - 148 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
37 \( ( 1 + 10 T + 4 p T^{2} + 990 T^{3} + 8294 T^{4} + 990 p T^{5} + 4 p^{3} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} )^{4} \)
41 \( ( 1 + 92 T^{2} + 7508 T^{4} + 417444 T^{6} + 19059990 T^{8} + 417444 p^{2} T^{10} + 7508 p^{4} T^{12} + 92 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
43 \( ( 1 + 2 T^{2} + 3245 T^{4} + 20322 T^{6} + 8864604 T^{8} + 20322 p^{2} T^{10} + 3245 p^{4} T^{12} + 2 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
47 \( ( 1 - 76 T^{2} + 2612 T^{4} - 154868 T^{6} + 10677910 T^{8} - 154868 p^{2} T^{10} + 2612 p^{4} T^{12} - 76 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
53 \( ( 1 - 2 T + 21 T^{2} + 214 T^{3} + 2384 T^{4} + 214 p T^{5} + 21 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{4} \)
59 \( ( 1 - 366 T^{2} + 63505 T^{4} - 6759614 T^{6} + 481830644 T^{8} - 6759614 p^{2} T^{10} + 63505 p^{4} T^{12} - 366 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
61 \( ( 1 + 262 T^{2} + 39293 T^{4} + 3890814 T^{6} + 278453340 T^{8} + 3890814 p^{2} T^{10} + 39293 p^{4} T^{12} + 262 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
67 \( ( 1 - 20 T + 328 T^{2} - 3940 T^{3} + 34654 T^{4} - 3940 p T^{5} + 328 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} )^{4} \)
71 \( ( 1 + 6 T + 200 T^{2} + 974 T^{3} + 18254 T^{4} + 974 p T^{5} + 200 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{4} \)
73 \( ( 1 + 408 T^{2} + 81340 T^{4} + 10280488 T^{6} + 894736454 T^{8} + 10280488 p^{2} T^{10} + 81340 p^{4} T^{12} + 408 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
79 \( ( 1 - 332 T^{2} + 59108 T^{4} - 7101044 T^{6} + 639039030 T^{8} - 7101044 p^{2} T^{10} + 59108 p^{4} T^{12} - 332 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
83 \( ( 1 + 198 T^{2} + 31805 T^{4} + 3108638 T^{6} + 297770844 T^{8} + 3108638 p^{2} T^{10} + 31805 p^{4} T^{12} + 198 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
89 \( ( 1 - 566 T^{2} + 144105 T^{4} - 22231174 T^{6} + 2346778004 T^{8} - 22231174 p^{2} T^{10} + 144105 p^{4} T^{12} - 566 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
97 \( ( 1 - 502 T^{2} + 125065 T^{4} - 20242182 T^{6} + 2314322804 T^{8} - 20242182 p^{2} T^{10} + 125065 p^{4} T^{12} - 502 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
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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

−2.53626196269337117016542394561, −2.39923891625424883576567120193, −2.30515013466704664716181023011, −2.27626766500729590777957418121, −2.22743377239064447345885648075, −2.15844752191180017376352628110, −2.06858519801141461562204298016, −2.04103866604268116015605706662, −1.90900441354173576814566048705, −1.80131268849815021044891662467, −1.65865002449760006242014662737, −1.62994805737110104805227979420, −1.60548767575954403437744871745, −1.44568908262504999772425635311, −1.22210892297016557366668286222, −1.20537364861600472949678927314, −1.12701429751422840745182061166, −1.00863072030114355832978182673, −0.976315719235533066704641242546, −0.916757727573715427590698231239, −0.912845436834453462189460528480, −0.885302855003492806782493718894, −0.25576990903571980420387422151, −0.24066639516278174519600788012, −0.15331985301659259996622565080, 0.15331985301659259996622565080, 0.24066639516278174519600788012, 0.25576990903571980420387422151, 0.885302855003492806782493718894, 0.912845436834453462189460528480, 0.916757727573715427590698231239, 0.976315719235533066704641242546, 1.00863072030114355832978182673, 1.12701429751422840745182061166, 1.20537364861600472949678927314, 1.22210892297016557366668286222, 1.44568908262504999772425635311, 1.60548767575954403437744871745, 1.62994805737110104805227979420, 1.65865002449760006242014662737, 1.80131268849815021044891662467, 1.90900441354173576814566048705, 2.04103866604268116015605706662, 2.06858519801141461562204298016, 2.15844752191180017376352628110, 2.22743377239064447345885648075, 2.27626766500729590777957418121, 2.30515013466704664716181023011, 2.39923891625424883576567120193, 2.53626196269337117016542394561

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

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