Properties

Label 32-72e16-1.1-c1e16-0-1
Degree $32$
Conductor $5.216\times 10^{29}$
Sign $1$
Analytic cond. $0.000142479$
Root an. cond. $0.758236$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 6·7-s − 8-s + 9-s + 6·14-s − 3·16-s − 28·17-s + 18-s − 10·23-s − 19·25-s − 10·31-s − 28·34-s − 8·41-s − 10·46-s + 6·47-s + 55·49-s − 19·50-s − 6·56-s − 10·62-s + 6·63-s + 7·64-s + 72·71-s − 72-s − 44·73-s − 30·79-s + 3·81-s − 8·82-s + ⋯
L(s)  = 1  + 0.707·2-s + 2.26·7-s − 0.353·8-s + 1/3·9-s + 1.60·14-s − 3/4·16-s − 6.79·17-s + 0.235·18-s − 2.08·23-s − 3.79·25-s − 1.79·31-s − 4.80·34-s − 1.24·41-s − 1.47·46-s + 0.875·47-s + 55/7·49-s − 2.68·50-s − 0.801·56-s − 1.27·62-s + 0.755·63-s + 7/8·64-s + 8.54·71-s − 0.117·72-s − 5.14·73-s − 3.37·79-s + 1/3·81-s − 0.883·82-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{32}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(32\)
Conductor: \(2^{48} \cdot 3^{32}\)
Sign: $1$
Analytic conductor: \(0.000142479\)
Root analytic conductor: \(0.758236\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{72} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((32,\ 2^{48} \cdot 3^{32} ,\ ( \ : [1/2]^{16} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.3417673320\)
\(L(\frac12)\) \(\approx\) \(0.3417673320\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T + T^{2} + p T^{4} - p^{2} T^{5} - p^{3} T^{7} + p^{2} T^{8} - p^{4} T^{9} - p^{5} T^{11} + p^{5} T^{12} + p^{6} T^{14} - p^{7} T^{15} + p^{8} T^{16} \)
3 \( 1 - T^{2} - 2 T^{4} + p^{2} T^{6} + 2 p^{2} T^{8} + p^{4} T^{10} - 2 p^{4} T^{12} - p^{6} T^{14} + p^{8} T^{16} \)
good5 \( 1 + 19 T^{2} + 176 T^{4} + 1031 T^{6} + 3893 T^{8} + 4928 T^{10} - 61926 T^{12} - 631122 T^{14} - 3717344 T^{16} - 631122 p^{2} T^{18} - 61926 p^{4} T^{20} + 4928 p^{6} T^{22} + 3893 p^{8} T^{24} + 1031 p^{10} T^{26} + 176 p^{12} T^{28} + 19 p^{14} T^{30} + p^{16} T^{32} \)
7 \( ( 1 - 3 T - 2 p T^{2} + 39 T^{3} + 139 T^{4} - 36 p T^{5} - 1208 T^{6} + 666 T^{7} + 9424 T^{8} + 666 p T^{9} - 1208 p^{2} T^{10} - 36 p^{4} T^{11} + 139 p^{4} T^{12} + 39 p^{5} T^{13} - 2 p^{7} T^{14} - 3 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
11 \( 1 + 48 T^{2} + 1090 T^{4} + 17592 T^{6} + 242041 T^{8} + 2632140 T^{10} + 21031138 T^{12} + 158095260 T^{14} + 1558598596 T^{16} + 158095260 p^{2} T^{18} + 21031138 p^{4} T^{20} + 2632140 p^{6} T^{22} + 242041 p^{8} T^{24} + 17592 p^{10} T^{26} + 1090 p^{12} T^{28} + 48 p^{14} T^{30} + p^{16} T^{32} \)
13 \( 1 + 51 T^{2} + 1288 T^{4} + 16911 T^{6} + 5665 p T^{8} - 1059264 T^{10} - 5456606 T^{12} + 411982806 T^{14} + 9084740848 T^{16} + 411982806 p^{2} T^{18} - 5456606 p^{4} T^{20} - 1059264 p^{6} T^{22} + 5665 p^{9} T^{24} + 16911 p^{10} T^{26} + 1288 p^{12} T^{28} + 51 p^{14} T^{30} + p^{16} T^{32} \)
17 \( ( 1 + 7 T + 66 T^{2} + 309 T^{3} + 1702 T^{4} + 309 p T^{5} + 66 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} )^{4} \)
19 \( ( 1 - 69 T^{2} + 2306 T^{4} - 53763 T^{6} + 1069146 T^{8} - 53763 p^{2} T^{10} + 2306 p^{4} T^{12} - 69 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
23 \( ( 1 + 5 T - 32 T^{2} + 45 T^{3} + 1385 T^{4} - 3040 T^{5} - 11142 T^{6} + 48640 T^{7} - 123716 T^{8} + 48640 p T^{9} - 11142 p^{2} T^{10} - 3040 p^{3} T^{11} + 1385 p^{4} T^{12} + 45 p^{5} T^{13} - 32 p^{6} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
29 \( 1 + 123 T^{2} + 7912 T^{4} + 340647 T^{6} + 10067965 T^{8} + 144056448 T^{10} - 3705324014 T^{12} - 328820871018 T^{14} - 12137186779472 T^{16} - 328820871018 p^{2} T^{18} - 3705324014 p^{4} T^{20} + 144056448 p^{6} T^{22} + 10067965 p^{8} T^{24} + 340647 p^{10} T^{26} + 7912 p^{12} T^{28} + 123 p^{14} T^{30} + p^{16} T^{32} \)
31 \( ( 1 + 5 T - 48 T^{2} - 5 p T^{3} + 1121 T^{4} - 2040 T^{5} - 44678 T^{6} + 79040 T^{7} + 1805724 T^{8} + 79040 p T^{9} - 44678 p^{2} T^{10} - 2040 p^{3} T^{11} + 1121 p^{4} T^{12} - 5 p^{6} T^{13} - 48 p^{6} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
37 \( ( 1 - 144 T^{2} + 12668 T^{4} - 733824 T^{6} + 31784838 T^{8} - 733824 p^{2} T^{10} + 12668 p^{4} T^{12} - 144 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
41 \( ( 1 + 4 T - 74 T^{2} + 288 T^{3} + 4517 T^{4} - 556 p T^{5} - 11538 T^{6} + 738968 T^{7} - 2462804 T^{8} + 738968 p T^{9} - 11538 p^{2} T^{10} - 556 p^{4} T^{11} + 4517 p^{4} T^{12} + 288 p^{5} T^{13} - 74 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
43 \( 1 + 324 T^{2} + 58258 T^{4} + 7305024 T^{6} + 705172105 T^{8} + 54954910764 T^{10} + 3558033934834 T^{12} + 194436327001344 T^{14} + 9048929543300068 T^{16} + 194436327001344 p^{2} T^{18} + 3558033934834 p^{4} T^{20} + 54954910764 p^{6} T^{22} + 705172105 p^{8} T^{24} + 7305024 p^{10} T^{26} + 58258 p^{12} T^{28} + 324 p^{14} T^{30} + p^{16} T^{32} \)
47 \( ( 1 - 3 T - 98 T^{2} + 219 T^{3} + 3523 T^{4} + 1116 T^{5} - 253856 T^{6} - 226218 T^{7} + 18909448 T^{8} - 226218 p T^{9} - 253856 p^{2} T^{10} + 1116 p^{3} T^{11} + 3523 p^{4} T^{12} + 219 p^{5} T^{13} - 98 p^{6} T^{14} - 3 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
53 \( ( 1 - 264 T^{2} + 35516 T^{4} - 3123720 T^{6} + 194863110 T^{8} - 3123720 p^{2} T^{10} + 35516 p^{4} T^{12} - 264 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
59 \( 1 + 364 T^{2} + 70130 T^{4} + 9549968 T^{6} + 1028964713 T^{8} + 92991430988 T^{10} + 7291973853618 T^{12} + 506327334867240 T^{14} + 31497451193778532 T^{16} + 506327334867240 p^{2} T^{18} + 7291973853618 p^{4} T^{20} + 92991430988 p^{6} T^{22} + 1028964713 p^{8} T^{24} + 9549968 p^{10} T^{26} + 70130 p^{12} T^{28} + 364 p^{14} T^{30} + p^{16} T^{32} \)
61 \( 1 + 323 T^{2} + 52152 T^{4} + 6035887 T^{6} + 584536925 T^{8} + 49803739968 T^{10} + 3801157073266 T^{12} + 264102902456582 T^{14} + 16823476283802768 T^{16} + 264102902456582 p^{2} T^{18} + 3801157073266 p^{4} T^{20} + 49803739968 p^{6} T^{22} + 584536925 p^{8} T^{24} + 6035887 p^{10} T^{26} + 52152 p^{12} T^{28} + 323 p^{14} T^{30} + p^{16} T^{32} \)
67 \( 1 + 296 T^{2} + 39450 T^{4} + 3796936 T^{6} + 362561921 T^{8} + 33090030780 T^{10} + 2631719645962 T^{12} + 193036342550180 T^{14} + 13423375489686516 T^{16} + 193036342550180 p^{2} T^{18} + 2631719645962 p^{4} T^{20} + 33090030780 p^{6} T^{22} + 362561921 p^{8} T^{24} + 3796936 p^{10} T^{26} + 39450 p^{12} T^{28} + 296 p^{14} T^{30} + p^{16} T^{32} \)
71 \( ( 1 - 18 T + 332 T^{2} - 3582 T^{3} + 36198 T^{4} - 3582 p T^{5} + 332 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} )^{4} \)
73 \( ( 1 + 11 T + 278 T^{2} + 2325 T^{3} + 29966 T^{4} + 2325 p T^{5} + 278 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} )^{4} \)
79 \( ( 1 + 15 T - 80 T^{2} - 1305 T^{3} + 13273 T^{4} + 59400 T^{5} - 1907150 T^{6} - 913560 T^{7} + 190569148 T^{8} - 913560 p T^{9} - 1907150 p^{2} T^{10} + 59400 p^{3} T^{11} + 13273 p^{4} T^{12} - 1305 p^{5} T^{13} - 80 p^{6} T^{14} + 15 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
83 \( 1 + 559 T^{2} + 168260 T^{4} + 35967119 T^{6} + 6055752221 T^{8} + 843016361960 T^{10} + 99713806040442 T^{12} + 10182388433060610 T^{14} + 904920089638581976 T^{16} + 10182388433060610 p^{2} T^{18} + 99713806040442 p^{4} T^{20} + 843016361960 p^{6} T^{22} + 6055752221 p^{8} T^{24} + 35967119 p^{10} T^{26} + 168260 p^{12} T^{28} + 559 p^{14} T^{30} + p^{16} T^{32} \)
89 \( ( 1 - 16 T + 408 T^{2} - 4116 T^{3} + 56278 T^{4} - 4116 p T^{5} + 408 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} )^{4} \)
97 \( ( 1 - 254 T^{2} + 1560 T^{3} + 35209 T^{4} - 273780 T^{5} - 2055806 T^{6} + 15575820 T^{7} + 104325124 T^{8} + 15575820 p T^{9} - 2055806 p^{2} T^{10} - 273780 p^{3} T^{11} + 35209 p^{4} T^{12} + 1560 p^{5} T^{13} - 254 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−4.70070231611120625295778626878, −4.50058612110268657776741812499, −4.49583117848105205069057465377, −4.43438005215861515822575523786, −4.40993694174807581063971495976, −4.38263627593294020065297729947, −3.93334731717163109013865403712, −3.90102024084730398680336707285, −3.78299947695915591829500263405, −3.76707217298851479536211762504, −3.71141571055986324987796352922, −3.70635646021338764309467466835, −3.68930295527200324136076743947, −3.48748889216191682597875390263, −2.82368472877987340325548906309, −2.79090460979376617458335126158, −2.61346717802520765215091249341, −2.57014684954820428635206915791, −2.38219152355092295704020305647, −2.20081231374687686700906514321, −2.08195674293928121016472794343, −2.04116080153799008088038296487, −1.90477263585986738383773588277, −1.89339069970918461618636920027, −1.39370703819009206755234613233, 1.39370703819009206755234613233, 1.89339069970918461618636920027, 1.90477263585986738383773588277, 2.04116080153799008088038296487, 2.08195674293928121016472794343, 2.20081231374687686700906514321, 2.38219152355092295704020305647, 2.57014684954820428635206915791, 2.61346717802520765215091249341, 2.79090460979376617458335126158, 2.82368472877987340325548906309, 3.48748889216191682597875390263, 3.68930295527200324136076743947, 3.70635646021338764309467466835, 3.71141571055986324987796352922, 3.76707217298851479536211762504, 3.78299947695915591829500263405, 3.90102024084730398680336707285, 3.93334731717163109013865403712, 4.38263627593294020065297729947, 4.40993694174807581063971495976, 4.43438005215861515822575523786, 4.49583117848105205069057465377, 4.50058612110268657776741812499, 4.70070231611120625295778626878

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

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