Dirichlet series
| L(s) = 1 | − 3·4-s − 11·9-s − 8·11-s + 6·16-s − 10·19-s − 2·29-s + 33·36-s + 16·41-s + 24·44-s + 76·49-s − 10·59-s − 37·64-s − 40·71-s + 30·76-s + 34·79-s + 56·81-s + 22·89-s + 88·99-s + 24·101-s + 68·109-s + 6·116-s − 36·121-s + 127-s + 131-s + 137-s + 139-s − 66·144-s + ⋯ |
| L(s) = 1 | − 3/2·4-s − 3.66·9-s − 2.41·11-s + 3/2·16-s − 2.29·19-s − 0.371·29-s + 11/2·36-s + 2.49·41-s + 3.61·44-s + 76/7·49-s − 1.30·59-s − 4.62·64-s − 4.74·71-s + 3.44·76-s + 3.82·79-s + 56/9·81-s + 2.33·89-s + 8.84·99-s + 2.38·101-s + 6.51·109-s + 0.557·116-s − 3.27·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 5.5·144-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{32} \cdot 19^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{32} \cdot 19^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(5^{32} \cdot 19^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.83455\times 10^{9}\) |
| Root analytic conductor: | \(1.94753\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 5^{32} \cdot 19^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.01189704072\) |
| \(L(\frac12)\) | \(\approx\) | \(0.01189704072\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 5 | \( 1 \) |
| 19 | \( ( 1 + 5 T + 31 T^{2} + 67 T^{3} + 395 T^{4} + 67 p T^{5} + 31 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| good | 2 | \( 1 + 3 T^{2} + 3 T^{4} + 7 p^{2} T^{6} + 67 T^{8} + 23 p T^{10} + 373 T^{12} + 25 p^{5} T^{14} + 119 p^{2} T^{16} + 25 p^{7} T^{18} + 373 p^{4} T^{20} + 23 p^{7} T^{22} + 67 p^{8} T^{24} + 7 p^{12} T^{26} + 3 p^{12} T^{28} + 3 p^{14} T^{30} + p^{16} T^{32} \) |
| 3 | \( 1 + 11 T^{2} + 65 T^{4} + 256 T^{6} + 8 p^{4} T^{8} + 524 T^{10} - 4796 T^{12} - 3565 p^{2} T^{14} - 117695 T^{16} - 3565 p^{4} T^{18} - 4796 p^{4} T^{20} + 524 p^{6} T^{22} + 8 p^{12} T^{24} + 256 p^{10} T^{26} + 65 p^{12} T^{28} + 11 p^{14} T^{30} + p^{16} T^{32} \) | |
| 7 | \( ( 1 - 38 T^{2} + 103 p T^{4} - 8709 T^{6} + 72598 T^{8} - 8709 p^{2} T^{10} + 103 p^{5} T^{12} - 38 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 11 | \( ( 1 + 2 T + 19 T^{2} + 47 T^{3} + 179 T^{4} + 47 p T^{5} + 19 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{4} \) | |
| 13 | \( 1 + 69 T^{2} + 2420 T^{4} + 59165 T^{6} + 1162929 T^{8} + 20027248 T^{10} + 316942670 T^{12} + 4666821142 T^{14} + 63383135416 T^{16} + 4666821142 p^{2} T^{18} + 316942670 p^{4} T^{20} + 20027248 p^{6} T^{22} + 1162929 p^{8} T^{24} + 59165 p^{10} T^{26} + 2420 p^{12} T^{28} + 69 p^{14} T^{30} + p^{16} T^{32} \) | |
| 17 | \( 1 + 75 T^{2} + 2673 T^{4} + 66400 T^{6} + 1391488 T^{8} + 24630100 T^{10} + 361092652 T^{12} + 5116517075 T^{14} + 81992679281 T^{16} + 5116517075 p^{2} T^{18} + 361092652 p^{4} T^{20} + 24630100 p^{6} T^{22} + 1391488 p^{8} T^{24} + 66400 p^{10} T^{26} + 2673 p^{12} T^{28} + 75 p^{14} T^{30} + p^{16} T^{32} \) | |
| 23 | \( 1 + 146 T^{2} + 11475 T^{4} + 627416 T^{6} + 26564421 T^{8} + 927103250 T^{10} + 27834137647 T^{12} + 1401496660 p^{2} T^{14} + 17866621536721 T^{16} + 1401496660 p^{4} T^{18} + 27834137647 p^{4} T^{20} + 927103250 p^{6} T^{22} + 26564421 p^{8} T^{24} + 627416 p^{10} T^{26} + 11475 p^{12} T^{28} + 146 p^{14} T^{30} + p^{16} T^{32} \) | |
| 29 | \( ( 1 + T - 52 T^{2} - 509 T^{3} + 1277 T^{4} + 20540 T^{5} + 93655 T^{6} - 443301 T^{7} - 3745740 T^{8} - 443301 p T^{9} + 93655 p^{2} T^{10} + 20540 p^{3} T^{11} + 1277 p^{4} T^{12} - 509 p^{5} T^{13} - 52 p^{6} T^{14} + p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 31 | \( ( 1 + 57 T^{2} - 5 T^{3} + 2675 T^{4} - 5 p T^{5} + 57 p^{2} T^{6} + p^{4} T^{8} )^{4} \) | |
| 37 | \( ( 1 - 230 T^{2} + 24897 T^{4} - 1653561 T^{6} + 73761510 T^{8} - 1653561 p^{2} T^{10} + 24897 p^{4} T^{12} - 230 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 41 | \( ( 1 - 8 T - 13 T^{2} - 590 T^{3} + 5111 T^{4} + 9890 T^{5} + 199801 T^{6} - 1514064 T^{7} - 3467655 T^{8} - 1514064 p T^{9} + 199801 p^{2} T^{10} + 9890 p^{3} T^{11} + 5111 p^{4} T^{12} - 590 p^{5} T^{13} - 13 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 43 | \( 1 + 147 T^{2} + 9077 T^{4} + 268292 T^{6} + 2865720 T^{8} - 2600732 T^{10} + 8574833828 T^{12} + 1057190362423 T^{14} + 61960491520177 T^{16} + 1057190362423 p^{2} T^{18} + 8574833828 p^{4} T^{20} - 2600732 p^{6} T^{22} + 2865720 p^{8} T^{24} + 268292 p^{10} T^{26} + 9077 p^{12} T^{28} + 147 p^{14} T^{30} + p^{16} T^{32} \) | |
| 47 | \( 1 + 122 T^{2} + 3971 T^{4} - 26712 T^{6} - 304687 T^{8} - 8762854 T^{10} - 14307721309 T^{12} - 105470794440 T^{14} + 33554443899897 T^{16} - 105470794440 p^{2} T^{18} - 14307721309 p^{4} T^{20} - 8762854 p^{6} T^{22} - 304687 p^{8} T^{24} - 26712 p^{10} T^{26} + 3971 p^{12} T^{28} + 122 p^{14} T^{30} + p^{16} T^{32} \) | |
| 53 | \( 1 + 227 T^{2} + 25965 T^{4} + 1761884 T^{6} + 72474060 T^{8} + 1771806188 T^{10} + 69358283668 T^{12} + 148947795827 p T^{14} + 572490005849953 T^{16} + 148947795827 p^{3} T^{18} + 69358283668 p^{4} T^{20} + 1771806188 p^{6} T^{22} + 72474060 p^{8} T^{24} + 1761884 p^{10} T^{26} + 25965 p^{12} T^{28} + 227 p^{14} T^{30} + p^{16} T^{32} \) | |
| 59 | \( ( 1 + 5 T - 86 T^{2} - 215 T^{3} + 2385 T^{4} - 14660 T^{5} - 81789 T^{6} + 1000355 T^{7} + 10693034 T^{8} + 1000355 p T^{9} - 81789 p^{2} T^{10} - 14660 p^{3} T^{11} + 2385 p^{4} T^{12} - 215 p^{5} T^{13} - 86 p^{6} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 61 | \( ( 1 - 114 T^{2} + 176 T^{3} + 3481 T^{4} - 15400 T^{5} - 228578 T^{6} + 386584 T^{7} + 27858884 T^{8} + 386584 p T^{9} - 228578 p^{2} T^{10} - 15400 p^{3} T^{11} + 3481 p^{4} T^{12} + 176 p^{5} T^{13} - 114 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 67 | \( 1 + 448 T^{2} + 109156 T^{4} + 18529024 T^{6} + 2430785066 T^{8} + 260970967552 T^{10} + 23818650659472 T^{12} + 1899140950603456 T^{14} + 134560163149521651 T^{16} + 1899140950603456 p^{2} T^{18} + 23818650659472 p^{4} T^{20} + 260970967552 p^{6} T^{22} + 2430785066 p^{8} T^{24} + 18529024 p^{10} T^{26} + 109156 p^{12} T^{28} + 448 p^{14} T^{30} + p^{16} T^{32} \) | |
| 71 | \( ( 1 + 20 T + 25 T^{2} - 830 T^{3} + 14400 T^{4} + 140590 T^{5} - 883200 T^{6} + 1962095 T^{7} + 162929144 T^{8} + 1962095 p T^{9} - 883200 p^{2} T^{10} + 140590 p^{3} T^{11} + 14400 p^{4} T^{12} - 830 p^{5} T^{13} + 25 p^{6} T^{14} + 20 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 73 | \( 1 + 374 T^{2} + 68311 T^{4} + 9149436 T^{6} + 1066994445 T^{8} + 108508523862 T^{10} + 9672722258967 T^{12} + 795957878893760 T^{14} + 60851482185815185 T^{16} + 795957878893760 p^{2} T^{18} + 9672722258967 p^{4} T^{20} + 108508523862 p^{6} T^{22} + 1066994445 p^{8} T^{24} + 9149436 p^{10} T^{26} + 68311 p^{12} T^{28} + 374 p^{14} T^{30} + p^{16} T^{32} \) | |
| 79 | \( ( 1 - 17 T - 99 T^{2} + 18 p T^{3} + 33753 T^{4} - 218511 T^{5} - 3340534 T^{6} + 1230695 T^{7} + 402442002 T^{8} + 1230695 p T^{9} - 3340534 p^{2} T^{10} - 218511 p^{3} T^{11} + 33753 p^{4} T^{12} + 18 p^{6} T^{13} - 99 p^{6} T^{14} - 17 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 83 | \( ( 1 - 539 T^{2} + 135328 T^{4} - 20610540 T^{6} + 2078404156 T^{8} - 20610540 p^{2} T^{10} + 135328 p^{4} T^{12} - 539 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 89 | \( ( 1 - 11 T - 145 T^{2} + 172 T^{3} + 26027 T^{4} + 59285 T^{5} - 2268638 T^{6} - 481263 T^{7} + 100591392 T^{8} - 481263 p T^{9} - 2268638 p^{2} T^{10} + 59285 p^{3} T^{11} + 26027 p^{4} T^{12} + 172 p^{5} T^{13} - 145 p^{6} T^{14} - 11 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 97 | \( 1 + 243 T^{2} + 21773 T^{4} + 1295660 T^{6} + 72989916 T^{8} - 267547628 T^{10} - 1128181621108 T^{12} - 269037036737465 T^{14} - 34800431293496975 T^{16} - 269037036737465 p^{2} T^{18} - 1128181621108 p^{4} T^{20} - 267547628 p^{6} T^{22} + 72989916 p^{8} T^{24} + 1295660 p^{10} T^{26} + 21773 p^{12} T^{28} + 243 p^{14} T^{30} + p^{16} T^{32} \) | |
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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
−2.97196213585712680948659303555, −2.91391673133943453371331947510, −2.86252484843042626590007242381, −2.82352522189151544923548610558, −2.68598670088307410929299433322, −2.62233553175053946952169566869, −2.49593730552223371356570747096, −2.32975020296729982922550012618, −2.32622309478224459092833906682, −2.29452645865666748987171706570, −2.14958811639758940628369202044, −2.07521284210434386943903486288, −2.06365640218063522582199525100, −1.84672717579735951432355452135, −1.81795457569367428111751856794, −1.68472625871532679793876505187, −1.54590032027215215792444074912, −1.23783326218780237854781286195, −0.961766580424582837667672865557, −0.878416261434917805901092019380, −0.873098489761392680247066256778, −0.72687419741158034102610185750, −0.60758512182353977284587055371, −0.32675783309284023612900439178, −0.02205667776929365591382112343, 0.02205667776929365591382112343, 0.32675783309284023612900439178, 0.60758512182353977284587055371, 0.72687419741158034102610185750, 0.873098489761392680247066256778, 0.878416261434917805901092019380, 0.961766580424582837667672865557, 1.23783326218780237854781286195, 1.54590032027215215792444074912, 1.68472625871532679793876505187, 1.81795457569367428111751856794, 1.84672717579735951432355452135, 2.06365640218063522582199525100, 2.07521284210434386943903486288, 2.14958811639758940628369202044, 2.29452645865666748987171706570, 2.32622309478224459092833906682, 2.32975020296729982922550012618, 2.49593730552223371356570747096, 2.62233553175053946952169566869, 2.68598670088307410929299433322, 2.82352522189151544923548610558, 2.86252484843042626590007242381, 2.91391673133943453371331947510, 2.97196213585712680948659303555
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.