Dirichlet series
| L(s) = 1 | − 16·2-s + 144·4-s − 2·7-s − 932·8-s + 2·9-s + 32·14-s + 4.80e3·16-s − 32·18-s + 2·19-s + 20·25-s − 288·28-s − 2.08e4·32-s + 288·36-s − 32·38-s − 14·41-s + 72·47-s − 11·49-s − 320·50-s + 1.86e3·56-s + 46·59-s − 4·63-s + 7.83e4·64-s − 48·67-s + 26·71-s − 1.86e3·72-s + 288·76-s + 15·81-s + ⋯ |
| L(s) = 1 | − 11.3·2-s + 72·4-s − 0.755·7-s − 329.·8-s + 2/3·9-s + 8.55·14-s + 1.20e3·16-s − 7.54·18-s + 0.458·19-s + 4·25-s − 54.4·28-s − 3.67e3·32-s + 48·36-s − 5.19·38-s − 2.18·41-s + 10.5·47-s − 1.57·49-s − 45.2·50-s + 249.·56-s + 5.98·59-s − 0.503·63-s + 9.79e3·64-s − 5.86·67-s + 3.08·71-s − 219.·72-s + 33.0·76-s + 5/3·81-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(31^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(31^{32}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(31^{32}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.44546\times 10^{14}\) |
| Root analytic conductor: | \(2.77013\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 31^{32} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(1.582080959\) |
| \(L(\frac12)\) | \(\approx\) | \(1.582080959\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 31 | \( 1 \) |
| good | 2 | \( ( 1 + p^{2} T + p^{2} T^{2} - 7 T^{3} - 21 T^{4} - 7 p T^{5} + p^{4} T^{6} + p^{5} T^{7} + p^{4} T^{8} )^{4} \) |
| 3 | \( 1 - 2 T^{2} - 11 T^{4} + 14 T^{6} + 4 p T^{8} + 148 T^{10} - 17 T^{12} - 1666 T^{14} + 5251 T^{16} - 1666 p^{2} T^{18} - 17 p^{4} T^{20} + 148 p^{6} T^{22} + 4 p^{9} T^{24} + 14 p^{10} T^{26} - 11 p^{12} T^{28} - 2 p^{14} T^{30} + p^{16} T^{32} \) | |
| 5 | \( ( 1 - p T^{2} )^{8}( 1 + p T^{2} + p^{2} T^{4} )^{4} \) | |
| 7 | \( ( 1 - 4 T + 9 T^{2} - 8 T^{3} - 31 T^{4} - 8 p T^{5} + 9 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2}( 1 + 5 T + 18 T^{2} + 55 T^{3} + 149 T^{4} + 55 p T^{5} + 18 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 11 | \( 1 + 4 T^{2} + p^{2} T^{4} + 1388 T^{6} + 5552 T^{8} - 87176 T^{10} - 7153 p^{2} T^{12} - 30250072 T^{14} - 335359169 T^{16} - 30250072 p^{2} T^{18} - 7153 p^{6} T^{20} - 87176 p^{6} T^{22} + 5552 p^{8} T^{24} + 1388 p^{10} T^{26} + p^{14} T^{28} + 4 p^{14} T^{30} + p^{16} T^{32} \) | |
| 13 | \( 1 - 46 T^{2} + 1069 T^{4} - 1022 p T^{6} + 3812 p T^{8} + 1289852 T^{10} - 27669169 T^{12} + 309797782 T^{14} - 3304974893 T^{16} + 309797782 p^{2} T^{18} - 27669169 p^{4} T^{20} + 1289852 p^{6} T^{22} + 3812 p^{9} T^{24} - 1022 p^{11} T^{26} + 1069 p^{12} T^{28} - 46 p^{14} T^{30} + p^{16} T^{32} \) | |
| 17 | \( 1 + 12 T^{2} - 431 T^{4} - 19344 T^{6} + 396 p T^{8} + 7343112 T^{10} + 127032263 T^{12} - 1311269634 T^{14} - 49701665629 T^{16} - 1311269634 p^{2} T^{18} + 127032263 p^{4} T^{20} + 7343112 p^{6} T^{22} + 396 p^{9} T^{24} - 19344 p^{10} T^{26} - 431 p^{12} T^{28} + 12 p^{14} T^{30} + p^{16} T^{32} \) | |
| 19 | \( ( 1 - 8 T + 45 T^{2} - 208 T^{3} + 809 T^{4} - 208 p T^{5} + 45 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2}( 1 + 7 T + 30 T^{2} + 77 T^{3} - 31 T^{4} + 77 p T^{5} + 30 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 23 | \( ( 1 - 64 T^{2} + 1047 T^{4} + 18958 T^{6} - 981475 T^{8} + 18958 p^{2} T^{10} + 1047 p^{4} T^{12} - 64 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 29 | \( ( 1 - 36 T^{2} + 1895 T^{4} - 82914 T^{6} + 2002669 T^{8} - 82914 p^{2} T^{10} + 1895 p^{4} T^{12} - 36 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 37 | \( ( 1 - 56 T^{2} + 1767 T^{4} - 56 p^{2} T^{6} + p^{4} T^{8} )^{4} \) | |
| 41 | \( ( 1 + 7 T + 21 T^{2} - 582 T^{3} - 5914 T^{4} - 28809 T^{5} + 22514 T^{6} + 1465054 T^{7} + 12207057 T^{8} + 1465054 p T^{9} + 22514 p^{2} T^{10} - 28809 p^{3} T^{11} - 5914 p^{4} T^{12} - 582 p^{5} T^{13} + 21 p^{6} T^{14} + 7 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 43 | \( 1 + 144 T^{2} + 14449 T^{4} + 1103484 T^{6} + 69410196 T^{8} + 3691480752 T^{10} + 177366098591 T^{12} + 7856170271622 T^{14} + 341066001415667 T^{16} + 7856170271622 p^{2} T^{18} + 177366098591 p^{4} T^{20} + 3691480752 p^{6} T^{22} + 69410196 p^{8} T^{24} + 1103484 p^{10} T^{26} + 14449 p^{12} T^{28} + 144 p^{14} T^{30} + p^{16} T^{32} \) | |
| 47 | \( ( 1 - 18 T + 97 T^{2} - 150 T^{3} + 121 T^{4} - 150 p T^{5} + 97 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} )^{4} \) | |
| 53 | \( 1 - 182 T^{2} + 16489 T^{4} - 744586 T^{6} + 2996492 T^{8} + 1819798348 T^{10} - 115551067177 T^{12} + 3445577972924 T^{14} - 84807076726649 T^{16} + 3445577972924 p^{2} T^{18} - 115551067177 p^{4} T^{20} + 1819798348 p^{6} T^{22} + 2996492 p^{8} T^{24} - 744586 p^{10} T^{26} + 16489 p^{12} T^{28} - 182 p^{14} T^{30} + p^{16} T^{32} \) | |
| 59 | \( ( 1 - 23 T + 339 T^{2} - 2664 T^{3} + 11912 T^{4} - 2223 T^{5} + 24758 T^{6} - 4708784 T^{7} + 57456351 T^{8} - 4708784 p T^{9} + 24758 p^{2} T^{10} - 2223 p^{3} T^{11} + 11912 p^{4} T^{12} - 2664 p^{5} T^{13} + 339 p^{6} T^{14} - 23 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 61 | \( ( 1 + 30 T^{2} + 62 T^{4} + 30 p^{2} T^{6} + p^{4} T^{8} )^{4} \) | |
| 67 | \( ( 1 + 6 T - 31 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{8} \) | |
| 71 | \( ( 1 - 13 T + 171 T^{2} - 672 T^{3} + 4136 T^{4} + 34611 T^{5} + 424574 T^{6} - 4281316 T^{7} + 87058227 T^{8} - 4281316 p T^{9} + 424574 p^{2} T^{10} + 34611 p^{3} T^{11} + 4136 p^{4} T^{12} - 672 p^{5} T^{13} + 171 p^{6} T^{14} - 13 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 73 | \( 1 + 128 T^{2} + p^{2} T^{4} - 50816 T^{6} - 6504448 T^{8} + 3073203968 T^{10} + 52407647 p^{2} T^{12} - 1220061975296 T^{14} - 962628024731969 T^{16} - 1220061975296 p^{2} T^{18} + 52407647 p^{6} T^{20} + 3073203968 p^{6} T^{22} - 6504448 p^{8} T^{24} - 50816 p^{10} T^{26} + p^{14} T^{28} + 128 p^{14} T^{30} + p^{16} T^{32} \) | |
| 79 | \( 1 + 230 T^{2} + 39181 T^{4} + 5957350 T^{6} + 748852460 T^{8} + 81706587140 T^{10} + 8313307286759 T^{12} + 752733689468230 T^{14} + 61462185192985699 T^{16} + 752733689468230 p^{2} T^{18} + 8313307286759 p^{4} T^{20} + 81706587140 p^{6} T^{22} + 748852460 p^{8} T^{24} + 5957350 p^{10} T^{26} + 39181 p^{12} T^{28} + 230 p^{14} T^{30} + p^{16} T^{32} \) | |
| 83 | \( 1 + 156 T^{2} + p^{2} T^{4} - 572364 T^{6} - 89288784 T^{8} - 2582536632 T^{10} + 30903599 p^{2} T^{12} + 9475326902808 T^{14} - 774141235300993 T^{16} + 9475326902808 p^{2} T^{18} + 30903599 p^{6} T^{20} - 2582536632 p^{6} T^{22} - 89288784 p^{8} T^{24} - 572364 p^{10} T^{26} + p^{14} T^{28} + 156 p^{14} T^{30} + p^{16} T^{32} \) | |
| 89 | \( ( 1 + 182 T^{2} + 24303 T^{4} + 2899174 T^{6} + 320768105 T^{8} + 2899174 p^{2} T^{10} + 24303 p^{4} T^{12} + 182 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 97 | \( ( 1 - 7 T - 48 T^{2} + 1015 T^{3} - 2449 T^{4} + 1015 p T^{5} - 48 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} )^{4} \) | |
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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
−2.37305809674817352658148156382, −2.32292463352481472496730317059, −2.21392014085475032176319900478, −2.16254666735750405498847529457, −2.13352108831760832180646963230, −2.06971197350000892601212342312, −2.06073665783928280873788286453, −1.92906805704946302242383639179, −1.88894714250206536727615354085, −1.72890435608336599942109772277, −1.58288930365673033583250952737, −1.50585138121979028389730571043, −1.35401583508817817880474944549, −1.34488119921525847075639219176, −1.29547363113886964565482197524, −1.27512625659673273353441279601, −1.16188508262511212115731428530, −1.03639624557693563491376256701, −1.01569173585309782097498974623, −0.72150414002197053348027396057, −0.71154185470442190255215761801, −0.69646965457347143652849594356, −0.68594308637664717334300189952, −0.44321970753655812902979088555, −0.29094015536993313194984623817, 0.29094015536993313194984623817, 0.44321970753655812902979088555, 0.68594308637664717334300189952, 0.69646965457347143652849594356, 0.71154185470442190255215761801, 0.72150414002197053348027396057, 1.01569173585309782097498974623, 1.03639624557693563491376256701, 1.16188508262511212115731428530, 1.27512625659673273353441279601, 1.29547363113886964565482197524, 1.34488119921525847075639219176, 1.35401583508817817880474944549, 1.50585138121979028389730571043, 1.58288930365673033583250952737, 1.72890435608336599942109772277, 1.88894714250206536727615354085, 1.92906805704946302242383639179, 2.06073665783928280873788286453, 2.06971197350000892601212342312, 2.13352108831760832180646963230, 2.16254666735750405498847529457, 2.21392014085475032176319900478, 2.32292463352481472496730317059, 2.37305809674817352658148156382
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.