Properties

Label 32-42e32-1.1-c1e16-0-2
Degree $32$
Conductor $8.790\times 10^{51}$
Sign $1$
Analytic cond. $2.40110\times 10^{18}$
Root an. cond. $3.75308$
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  − 3·9-s − 6·11-s − 6·23-s + 16·25-s − 12·29-s − 2·37-s + 4·43-s + 36·53-s − 28·67-s − 40·79-s + 18·99-s − 6·107-s + 10·109-s + 90·113-s − 25·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 56·169-s + 173-s + 179-s + ⋯
L(s)  = 1  − 9-s − 1.80·11-s − 1.25·23-s + 16/5·25-s − 2.22·29-s − 0.328·37-s + 0.609·43-s + 4.94·53-s − 3.42·67-s − 4.50·79-s + 1.80·99-s − 0.580·107-s + 0.957·109-s + 8.46·113-s − 2.27·121-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 − 4.30·169-s + 0.0760·173-s + 0.0747·179-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: $1$
Analytic conductor: \(2.40110\times 10^{18}\)
Root analytic conductor: \(3.75308\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
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

\(L(1)\) \(\approx\) \(0.6673713958\)
\(L(\frac12)\) \(\approx\) \(0.6673713958\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + p T^{2} + p^{2} T^{4} - p^{3} T^{6} - 11 p^{2} T^{8} - p^{5} T^{10} + p^{6} T^{12} + p^{7} T^{14} + p^{8} T^{16} \)
7 \( 1 \)
good5 \( 1 - 16 T^{2} + 159 T^{4} - 998 T^{6} + 4298 T^{8} - 12066 T^{10} + 21316 T^{12} - 64633 T^{14} + 271566 T^{16} - 64633 p^{2} T^{18} + 21316 p^{4} T^{20} - 12066 p^{6} T^{22} + 4298 p^{8} T^{24} - 998 p^{10} T^{26} + 159 p^{12} T^{28} - 16 p^{14} T^{30} + p^{16} T^{32} \)
11 \( ( 1 + 3 T + 26 T^{2} + 69 T^{3} + 265 T^{4} + 756 T^{5} + 235 p T^{6} + 8259 T^{7} + 34846 T^{8} + 8259 p T^{9} + 235 p^{3} T^{10} + 756 p^{3} T^{11} + 265 p^{4} T^{12} + 69 p^{5} T^{13} + 26 p^{6} T^{14} + 3 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
13 \( 1 + 56 T^{2} + 1590 T^{4} + 26848 T^{6} + 274865 T^{8} + 1615656 T^{10} + 12863830 T^{12} + 363405056 T^{14} + 6601006116 T^{16} + 363405056 p^{2} T^{18} + 12863830 p^{4} T^{20} + 1615656 p^{6} T^{22} + 274865 p^{8} T^{24} + 26848 p^{10} T^{26} + 1590 p^{12} T^{28} + 56 p^{14} T^{30} + p^{16} T^{32} \)
17 \( 1 - 58 T^{2} + 1761 T^{4} - 32948 T^{6} + 331181 T^{8} + 945588 T^{10} - 132300833 T^{12} + 3706114724 T^{14} - 71840565063 T^{16} + 3706114724 p^{2} T^{18} - 132300833 p^{4} T^{20} + 945588 p^{6} T^{22} + 331181 p^{8} T^{24} - 32948 p^{10} T^{26} + 1761 p^{12} T^{28} - 58 p^{14} T^{30} + p^{16} T^{32} \)
19 \( 1 + 77 T^{2} + 2895 T^{4} + 64882 T^{6} + 896840 T^{8} + 7384440 T^{10} + 66022636 T^{12} + 2128839191 T^{14} + 54949527117 T^{16} + 2128839191 p^{2} T^{18} + 66022636 p^{4} T^{20} + 7384440 p^{6} T^{22} + 896840 p^{8} T^{24} + 64882 p^{10} T^{26} + 2895 p^{12} T^{28} + 77 p^{14} T^{30} + p^{16} T^{32} \)
23 \( ( 1 + 3 T + 56 T^{2} + 159 T^{3} + 1321 T^{4} + 3510 T^{5} + 30863 T^{6} + 67173 T^{7} + 842170 T^{8} + 67173 p T^{9} + 30863 p^{2} T^{10} + 3510 p^{3} T^{11} + 1321 p^{4} T^{12} + 159 p^{5} T^{13} + 56 p^{6} T^{14} + 3 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
29 \( ( 1 + 6 T + 53 T^{2} + 246 T^{3} + 20 p T^{4} + 540 T^{5} - 22066 T^{6} - 299625 T^{7} - 1268282 T^{8} - 299625 p T^{9} - 22066 p^{2} T^{10} + 540 p^{3} T^{11} + 20 p^{5} T^{12} + 246 p^{5} T^{13} + 53 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
31 \( ( 1 - 176 T^{2} + 14569 T^{4} - 759287 T^{6} + 27714295 T^{8} - 759287 p^{2} T^{10} + 14569 p^{4} T^{12} - 176 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
37 \( ( 1 + T - 81 T^{2} - 94 T^{3} + 2696 T^{4} + 2592 T^{5} - 94076 T^{6} - 24353 T^{7} + 4259061 T^{8} - 24353 p T^{9} - 94076 p^{2} T^{10} + 2592 p^{3} T^{11} + 2696 p^{4} T^{12} - 94 p^{5} T^{13} - 81 p^{6} T^{14} + p^{7} T^{15} + p^{8} T^{16} )^{2} \)
41 \( 1 - 151 T^{2} + 9564 T^{4} - 353897 T^{6} + 12352577 T^{8} - 588962616 T^{10} + 30503470855 T^{12} - 1573469011633 T^{14} + 72313421958936 T^{16} - 1573469011633 p^{2} T^{18} + 30503470855 p^{4} T^{20} - 588962616 p^{6} T^{22} + 12352577 p^{8} T^{24} - 353897 p^{10} T^{26} + 9564 p^{12} T^{28} - 151 p^{14} T^{30} + p^{16} T^{32} \)
43 \( ( 1 - 2 T - 93 T^{2} - 64 T^{3} + 92 p T^{4} + 12192 T^{5} - 158990 T^{6} - 321485 T^{7} + 7972668 T^{8} - 321485 p T^{9} - 158990 p^{2} T^{10} + 12192 p^{3} T^{11} + 92 p^{5} T^{12} - 64 p^{5} T^{13} - 93 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
47 \( ( 1 + 154 T^{2} + 16033 T^{4} + 23651 p T^{6} + 60842887 T^{8} + 23651 p^{3} T^{10} + 16033 p^{4} T^{12} + 154 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
53 \( ( 1 - 18 T + 167 T^{2} - 1062 T^{3} + 3655 T^{4} + 5670 T^{5} - 78073 T^{6} + 140364 T^{7} + 103069 T^{8} + 140364 p T^{9} - 78073 p^{2} T^{10} + 5670 p^{3} T^{11} + 3655 p^{4} T^{12} - 1062 p^{5} T^{13} + 167 p^{6} T^{14} - 18 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
59 \( ( 1 + 376 T^{2} + 65923 T^{4} + 7045963 T^{6} + 502678435 T^{8} + 7045963 p^{2} T^{10} + 65923 p^{4} T^{12} + 376 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
61 \( ( 1 - 137 T^{2} + 18463 T^{4} - 1448957 T^{6} + 109191469 T^{8} - 1448957 p^{2} T^{10} + 18463 p^{4} T^{12} - 137 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
67 \( ( 1 + 7 T + 157 T^{2} + 403 T^{3} + 10075 T^{4} + 403 p T^{5} + 157 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} )^{4} \)
71 \( ( 1 - 361 T^{2} + 64216 T^{4} - 7488268 T^{6} + 623512600 T^{8} - 7488268 p^{2} T^{10} + 64216 p^{4} T^{12} - 361 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
73 \( 1 + 341 T^{2} + 53883 T^{4} + 6153142 T^{6} + 667124744 T^{8} + 70476881112 T^{10} + 6578299196452 T^{12} + 524328155054171 T^{14} + 38507998242556701 T^{16} + 524328155054171 p^{2} T^{18} + 6578299196452 p^{4} T^{20} + 70476881112 p^{6} T^{22} + 667124744 p^{8} T^{24} + 6153142 p^{10} T^{26} + 53883 p^{12} T^{28} + 341 p^{14} T^{30} + p^{16} T^{32} \)
79 \( ( 1 + 10 T + 223 T^{2} + 1537 T^{3} + 23317 T^{4} + 1537 p T^{5} + 223 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} )^{4} \)
83 \( 1 - 397 T^{2} + 87990 T^{4} - 12501089 T^{6} + 1185510017 T^{8} - 57837981066 T^{10} - 2507933243939 T^{12} + 849901181197043 T^{14} - 92913215010718536 T^{16} + 849901181197043 p^{2} T^{18} - 2507933243939 p^{4} T^{20} - 57837981066 p^{6} T^{22} + 1185510017 p^{8} T^{24} - 12501089 p^{10} T^{26} + 87990 p^{12} T^{28} - 397 p^{14} T^{30} + p^{16} T^{32} \)
89 \( 1 - 64 T^{2} - 5391 T^{4} + 717502 T^{6} - 95745073 T^{8} + 6819998976 T^{10} - 66652775855 T^{12} - 65309085806572 T^{14} + 11564582627289681 T^{16} - 65309085806572 p^{2} T^{18} - 66652775855 p^{4} T^{20} + 6819998976 p^{6} T^{22} - 95745073 p^{8} T^{24} + 717502 p^{10} T^{26} - 5391 p^{12} T^{28} - 64 p^{14} T^{30} + p^{16} T^{32} \)
97 \( 1 + 404 T^{2} + 81609 T^{4} + 10963666 T^{6} + 1046175986 T^{8} + 591335472 p T^{10} - 1663618077194 T^{12} - 785812116013063 T^{14} - 99406691908304130 T^{16} - 785812116013063 p^{2} T^{18} - 1663618077194 p^{4} T^{20} + 591335472 p^{7} T^{22} + 1046175986 p^{8} T^{24} + 10963666 p^{10} T^{26} + 81609 p^{12} T^{28} + 404 p^{14} T^{30} + p^{16} T^{32} \)
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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

−2.34631145853591824246889842973, −2.28055663135964647972002781660, −2.22541130133437318826005066572, −2.15416088526027705291195562830, −1.98844458095156198311768873041, −1.97908341193160625936515877747, −1.93218964070625007688530356390, −1.86679387876951253230399295645, −1.63410573822088408946638619649, −1.60587256575830190171989087926, −1.48552753607615977139470773773, −1.48290345381601732344853640659, −1.42761042218002406715430008486, −1.41989455507221031046802955142, −1.32635802039883100932476658681, −1.06210670258376150423727226397, −0.984312648078076540740710403171, −0.966163684141777317742881926441, −0.74172284983049771124814873127, −0.66541357790720846029741973637, −0.65274609129038658827933857311, −0.42689994950759106385410891366, −0.33865429473411780854399324967, −0.20433895299113524184049953334, −0.06019500833646774001427569537, 0.06019500833646774001427569537, 0.20433895299113524184049953334, 0.33865429473411780854399324967, 0.42689994950759106385410891366, 0.65274609129038658827933857311, 0.66541357790720846029741973637, 0.74172284983049771124814873127, 0.966163684141777317742881926441, 0.984312648078076540740710403171, 1.06210670258376150423727226397, 1.32635802039883100932476658681, 1.41989455507221031046802955142, 1.42761042218002406715430008486, 1.48290345381601732344853640659, 1.48552753607615977139470773773, 1.60587256575830190171989087926, 1.63410573822088408946638619649, 1.86679387876951253230399295645, 1.93218964070625007688530356390, 1.97908341193160625936515877747, 1.98844458095156198311768873041, 2.15416088526027705291195562830, 2.22541130133437318826005066572, 2.28055663135964647972002781660, 2.34631145853591824246889842973

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

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