Dirichlet series
| L(s) = 1 | − 2·2-s − 4-s − 12·5-s + 8·8-s − 10·9-s + 24·10-s − 12·13-s − 8·16-s + 12·17-s + 20·18-s + 12·20-s + 72·25-s + 24·26-s − 8·29-s − 2·32-s − 24·34-s + 10·36-s − 16·37-s − 96·40-s + 48·41-s + 120·45-s + 30·49-s − 144·50-s + 12·52-s − 32·53-s + 16·58-s + 4·61-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1/2·4-s − 5.36·5-s + 2.82·8-s − 3.33·9-s + 7.58·10-s − 3.32·13-s − 2·16-s + 2.91·17-s + 4.71·18-s + 2.68·20-s + 72/5·25-s + 4.70·26-s − 1.48·29-s − 0.353·32-s − 4.11·34-s + 5/3·36-s − 2.63·37-s − 15.1·40-s + 7.49·41-s + 17.8·45-s + 30/7·49-s − 20.3·50-s + 1.66·52-s − 4.39·53-s + 2.10·58-s + 0.512·61-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(2^{32} \cdot 13^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(7.80705\times 10^{-7}\) |
| Root analytic conductor: | \(0.644377\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 2^{32} \cdot 13^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.008532947849\) |
| \(L(\frac12)\) | \(\approx\) | \(0.008532947849\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 + p T + 5 T^{2} + p^{2} T^{3} + 5 T^{4} - p^{3} T^{5} - 3 p^{2} T^{6} - 5 p^{3} T^{7} - 9 p^{2} T^{8} - 5 p^{4} T^{9} - 3 p^{4} T^{10} - p^{6} T^{11} + 5 p^{4} T^{12} + p^{7} T^{13} + 5 p^{6} T^{14} + p^{8} T^{15} + p^{8} T^{16} \) |
| 13 | \( ( 1 + 6 T + 33 T^{2} + 102 T^{3} + 428 T^{4} + 102 p T^{5} + 33 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| good | 3 | \( 1 + 10 T^{2} + 35 T^{4} + 70 T^{6} + 325 T^{8} + 1460 T^{10} + 3530 T^{12} + 10640 T^{14} + 40114 T^{16} + 10640 p^{2} T^{18} + 3530 p^{4} T^{20} + 1460 p^{6} T^{22} + 325 p^{8} T^{24} + 70 p^{10} T^{26} + 35 p^{12} T^{28} + 10 p^{14} T^{30} + p^{16} T^{32} \) |
| 5 | \( ( 1 + 6 T + 18 T^{2} + 48 T^{3} + p^{3} T^{4} + 12 p^{2} T^{5} + 702 T^{6} + 1734 T^{7} + 4164 T^{8} + 1734 p T^{9} + 702 p^{2} T^{10} + 12 p^{5} T^{11} + p^{7} T^{12} + 48 p^{5} T^{13} + 18 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 7 | \( 1 - 30 T^{2} + 459 T^{4} - 4770 T^{6} + 36373 T^{8} - 173580 T^{10} + 5526 p T^{12} + 7584576 T^{14} - 74405166 T^{16} + 7584576 p^{2} T^{18} + 5526 p^{5} T^{20} - 173580 p^{6} T^{22} + 36373 p^{8} T^{24} - 4770 p^{10} T^{26} + 459 p^{12} T^{28} - 30 p^{14} T^{30} + p^{16} T^{32} \) | |
| 11 | \( 1 + 18 T^{2} + 163 T^{4} + 90 p T^{6} + 645 T^{8} + 1548 p T^{10} - 1355254 T^{12} - 46597344 T^{14} - 637928110 T^{16} - 46597344 p^{2} T^{18} - 1355254 p^{4} T^{20} + 1548 p^{7} T^{22} + 645 p^{8} T^{24} + 90 p^{11} T^{26} + 163 p^{12} T^{28} + 18 p^{14} T^{30} + p^{16} T^{32} \) | |
| 17 | \( ( 1 - 6 T + 70 T^{2} - 348 T^{3} + 2521 T^{4} - 11316 T^{5} + 63394 T^{6} - 254970 T^{7} + 1204060 T^{8} - 254970 p T^{9} + 63394 p^{2} T^{10} - 11316 p^{3} T^{11} + 2521 p^{4} T^{12} - 348 p^{5} T^{13} + 70 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 19 | \( 1 - 54 T^{2} + 1423 T^{4} - 24354 T^{6} + 220185 T^{8} - 1422900 T^{10} - 16188226 T^{12} + 1480460448 T^{14} - 33155429110 T^{16} + 1480460448 p^{2} T^{18} - 16188226 p^{4} T^{20} - 1422900 p^{6} T^{22} + 220185 p^{8} T^{24} - 24354 p^{10} T^{26} + 1423 p^{12} T^{28} - 54 p^{14} T^{30} + p^{16} T^{32} \) | |
| 23 | \( 1 - 78 T^{2} + 3507 T^{4} - 101010 T^{6} + 1820485 T^{8} - 11680188 T^{10} - 493303158 T^{12} + 22493168880 T^{14} - 610304342334 T^{16} + 22493168880 p^{2} T^{18} - 493303158 p^{4} T^{20} - 11680188 p^{6} T^{22} + 1820485 p^{8} T^{24} - 101010 p^{10} T^{26} + 3507 p^{12} T^{28} - 78 p^{14} T^{30} + p^{16} T^{32} \) | |
| 29 | \( ( 1 + 4 T - 54 T^{2} + 72 T^{3} + 2333 T^{4} - 6912 T^{5} - 34078 T^{6} + 134500 T^{7} + 265404 T^{8} + 134500 p T^{9} - 34078 p^{2} T^{10} - 6912 p^{3} T^{11} + 2333 p^{4} T^{12} + 72 p^{5} T^{13} - 54 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 31 | \( 1 - 2088 T^{4} + 1355740 T^{8} + 1697054184 T^{12} - 3181864244538 T^{16} + 1697054184 p^{4} T^{20} + 1355740 p^{8} T^{24} - 2088 p^{12} T^{28} + p^{16} T^{32} \) | |
| 37 | \( ( 1 + 4 T + 5 T^{2} - 72 T^{3} - 1516 T^{4} - 72 p T^{5} + 5 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{4} \) | |
| 41 | \( ( 1 - 12 T + 45 T^{2} + 192 T^{3} - 3076 T^{4} + 192 p T^{5} + 45 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} )^{4} \) | |
| 43 | \( 1 - 78 T^{2} + 327 T^{4} + 193302 T^{6} - 6186983 T^{8} - 215409684 T^{10} + 13995362142 T^{12} + 223461915456 T^{14} - 34934336862342 T^{16} + 223461915456 p^{2} T^{18} + 13995362142 p^{4} T^{20} - 215409684 p^{6} T^{22} - 6186983 p^{8} T^{24} + 193302 p^{10} T^{26} + 327 p^{12} T^{28} - 78 p^{14} T^{30} + p^{16} T^{32} \) | |
| 47 | \( 1 - 3464 T^{4} + 13830684 T^{8} - 35168581816 T^{12} + 104685918732614 T^{16} - 35168581816 p^{4} T^{20} + 13830684 p^{8} T^{24} - 3464 p^{12} T^{28} + p^{16} T^{32} \) | |
| 53 | \( ( 1 + 8 T + 209 T^{2} + 1160 T^{3} + 16408 T^{4} + 1160 p T^{5} + 209 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{4} \) | |
| 59 | \( 1 - 6 T^{2} + 7219 T^{4} - 43242 T^{6} + 17569413 T^{8} - 23138460 T^{10} + 73114429418 T^{12} + 563538380448 T^{14} + 405065512248242 T^{16} + 563538380448 p^{2} T^{18} + 73114429418 p^{4} T^{20} - 23138460 p^{6} T^{22} + 17569413 p^{8} T^{24} - 43242 p^{10} T^{26} + 7219 p^{12} T^{28} - 6 p^{14} T^{30} + p^{16} T^{32} \) | |
| 61 | \( ( 1 - 2 T - 182 T^{2} + 44 T^{3} + 19013 T^{4} + 9292 T^{5} - 1418962 T^{6} - 272974 T^{7} + 86637076 T^{8} - 272974 p T^{9} - 1418962 p^{2} T^{10} + 9292 p^{3} T^{11} + 19013 p^{4} T^{12} + 44 p^{5} T^{13} - 182 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 67 | \( 1 - 126 T^{2} + 9895 T^{4} - 579978 T^{6} + 14460249 T^{8} - 98385876 T^{10} + 97862919998 T^{12} - 14003740178160 T^{14} + 1170909629440346 T^{16} - 14003740178160 p^{2} T^{18} + 97862919998 p^{4} T^{20} - 98385876 p^{6} T^{22} + 14460249 p^{8} T^{24} - 579978 p^{10} T^{26} + 9895 p^{12} T^{28} - 126 p^{14} T^{30} + p^{16} T^{32} \) | |
| 71 | \( 1 - 126 T^{2} + 4119 T^{4} + 147798 T^{6} + 588553 T^{8} - 2943385380 T^{10} + 257102153838 T^{12} + 3334539241728 T^{14} - 1294458095032278 T^{16} + 3334539241728 p^{2} T^{18} + 257102153838 p^{4} T^{20} - 2943385380 p^{6} T^{22} + 588553 p^{8} T^{24} + 147798 p^{10} T^{26} + 4119 p^{12} T^{28} - 126 p^{14} T^{30} + p^{16} T^{32} \) | |
| 73 | \( ( 1 - 10 T + 50 T^{2} - 736 T^{3} + 7621 T^{4} - 21700 T^{5} + 106798 T^{6} - 1555650 T^{7} + 22659876 T^{8} - 1555650 p T^{9} + 106798 p^{2} T^{10} - 21700 p^{3} T^{11} + 7621 p^{4} T^{12} - 736 p^{5} T^{13} + 50 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 79 | \( ( 1 - 472 T^{2} + 105948 T^{4} - 14805224 T^{6} + 1404364358 T^{8} - 14805224 p^{2} T^{10} + 105948 p^{4} T^{12} - 472 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 83 | \( 1 - 19968 T^{4} + 187909180 T^{8} - 882419369472 T^{12} + 3838008550507974 T^{16} - 882419369472 p^{4} T^{20} + 187909180 p^{8} T^{24} - 19968 p^{12} T^{28} + p^{16} T^{32} \) | |
| 89 | \( ( 1 + 26 T + 215 T^{2} - 222 T^{3} - 14299 T^{4} - 78956 T^{5} - 60806 T^{6} + 3050412 T^{7} + 47586730 T^{8} + 3050412 p T^{9} - 60806 p^{2} T^{10} - 78956 p^{3} T^{11} - 14299 p^{4} T^{12} - 222 p^{5} T^{13} + 215 p^{6} T^{14} + 26 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 97 | \( ( 1 + 14 T + 323 T^{2} + 4334 T^{3} + 51469 T^{4} + 594788 T^{5} + 5237698 T^{6} + 53022348 T^{7} + 485964162 T^{8} + 53022348 p T^{9} + 5237698 p^{2} T^{10} + 594788 p^{3} T^{11} + 51469 p^{4} T^{12} + 4334 p^{5} T^{13} + 323 p^{6} T^{14} + 14 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
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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
−5.12455908121771661840991352034, −5.12114502386845759393652525324, −4.95410484897495864656367425494, −4.74750394712209040073530888892, −4.65233643092870854138479557057, −4.40548201337442600628230032669, −4.37736992968356998937624781901, −4.37052506787467781274905634462, −4.25406441320956041028244016392, −4.20077156306306909477527504859, −3.83345855683635656007752329412, −3.82459082718375572087543020344, −3.81537264573178636856392011621, −3.71660425719795615591387877204, −3.63876012293311738040768238378, −3.24963357269809124529929716008, −3.15574860445914424757682818840, −2.87863706854200865098661847824, −2.81817418389956831055810737892, −2.77888614299363005415744565685, −2.69858647589604942391039040705, −2.57555361163562246479293078353, −2.14211040022502057497676337930, −1.71285655550302361469242236967, −0.828728017427432260502685804765, 0.828728017427432260502685804765, 1.71285655550302361469242236967, 2.14211040022502057497676337930, 2.57555361163562246479293078353, 2.69858647589604942391039040705, 2.77888614299363005415744565685, 2.81817418389956831055810737892, 2.87863706854200865098661847824, 3.15574860445914424757682818840, 3.24963357269809124529929716008, 3.63876012293311738040768238378, 3.71660425719795615591387877204, 3.81537264573178636856392011621, 3.82459082718375572087543020344, 3.83345855683635656007752329412, 4.20077156306306909477527504859, 4.25406441320956041028244016392, 4.37052506787467781274905634462, 4.37736992968356998937624781901, 4.40548201337442600628230032669, 4.65233643092870854138479557057, 4.74750394712209040073530888892, 4.95410484897495864656367425494, 5.12114502386845759393652525324, 5.12455908121771661840991352034
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.