Dirichlet series
| L(s) = 1 | + 4·7-s − 18·19-s + 14·25-s + 24·31-s + 16·37-s − 52·43-s + 30·49-s + 6·61-s − 4·67-s − 12·73-s + 34·79-s + 6·103-s − 28·109-s − 19·121-s + 127-s + 131-s − 72·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 4·169-s + 173-s + 56·175-s + ⋯ |
| L(s) = 1 | + 1.51·7-s − 4.12·19-s + 14/5·25-s + 4.31·31-s + 2.63·37-s − 7.92·43-s + 30/7·49-s + 0.768·61-s − 0.488·67-s − 1.40·73-s + 3.82·79-s + 0.591·103-s − 2.68·109-s − 1.72·121-s + 0.0887·127-s + 0.0873·131-s − 6.24·133-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 − 0.307·169-s + 0.0760·173-s + 4.23·175-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{48} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{48} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(2^{48} \cdot 3^{48} \cdot 7^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.03827\times 10^{17}\) |
| Root analytic conductor: | \(3.47467\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 2^{48} \cdot 3^{48} \cdot 7^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.002160753350\) |
| \(L(\frac12)\) | \(\approx\) | \(0.002160753350\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) | |
| 7 | \( ( 1 - 2 T - 9 T^{2} - T^{3} + 116 T^{4} - p T^{5} - 9 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| good | 5 | \( 1 - 14 T^{2} + 61 T^{4} + 8 T^{6} - 116 p T^{8} + 2692 T^{10} - 7864 T^{12} - 231781 T^{14} + 2288086 T^{16} - 231781 p^{2} T^{18} - 7864 p^{4} T^{20} + 2692 p^{6} T^{22} - 116 p^{9} T^{24} + 8 p^{10} T^{26} + 61 p^{12} T^{28} - 14 p^{14} T^{30} + p^{16} T^{32} \) |
| 11 | \( 1 + 19 T^{2} - 23 T^{4} - 4456 T^{6} - 28228 T^{8} + 533860 T^{10} + 7671380 T^{12} - 29265493 T^{14} - 1170886031 T^{16} - 29265493 p^{2} T^{18} + 7671380 p^{4} T^{20} + 533860 p^{6} T^{22} - 28228 p^{8} T^{24} - 4456 p^{10} T^{26} - 23 p^{12} T^{28} + 19 p^{14} T^{30} + p^{16} T^{32} \) | |
| 13 | \( ( 1 + 2 T^{2} + 81 T^{4} - 2123 T^{6} - 31690 T^{8} - 2123 p^{2} T^{10} + 81 p^{4} T^{12} + 2 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 17 | \( 1 - 6 p T^{2} + 5415 T^{4} - 206640 T^{6} + 6384925 T^{8} - 167030250 T^{10} + 3794350455 T^{12} - 76308529980 T^{14} + 1371191367105 T^{16} - 76308529980 p^{2} T^{18} + 3794350455 p^{4} T^{20} - 167030250 p^{6} T^{22} + 6384925 p^{8} T^{24} - 206640 p^{10} T^{26} + 5415 p^{12} T^{28} - 6 p^{15} T^{30} + p^{16} T^{32} \) | |
| 19 | \( ( 1 + 9 T + 77 T^{2} + 450 T^{3} + 2244 T^{4} + 11718 T^{5} + 59482 T^{6} + 301239 T^{7} + 1433825 T^{8} + 301239 p T^{9} + 59482 p^{2} T^{10} + 11718 p^{3} T^{11} + 2244 p^{4} T^{12} + 450 p^{5} T^{13} + 77 p^{6} T^{14} + 9 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 23 | \( 1 + 102 T^{2} + 6023 T^{4} + 223644 T^{6} + 5387737 T^{8} + 54053766 T^{10} - 1721404285 T^{12} - 106929141708 T^{14} - 3095815766783 T^{16} - 106929141708 p^{2} T^{18} - 1721404285 p^{4} T^{20} + 54053766 p^{6} T^{22} + 5387737 p^{8} T^{24} + 223644 p^{10} T^{26} + 6023 p^{12} T^{28} + 102 p^{14} T^{30} + p^{16} T^{32} \) | |
| 29 | \( ( 1 - 154 T^{2} + 12171 T^{4} - 609827 T^{6} + 21113243 T^{8} - 609827 p^{2} T^{10} + 12171 p^{4} T^{12} - 154 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 31 | \( ( 1 - 12 T + 143 T^{2} - 1140 T^{3} + 8619 T^{4} - 60864 T^{5} + 400903 T^{6} - 2550678 T^{7} + 14613095 T^{8} - 2550678 p T^{9} + 400903 p^{2} T^{10} - 60864 p^{3} T^{11} + 8619 p^{4} T^{12} - 1140 p^{5} T^{13} + 143 p^{6} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 37 | \( ( 1 - 8 T - 57 T^{2} + 338 T^{3} + 3173 T^{4} - 3270 T^{5} - 5165 p T^{6} + 39214 T^{7} + 7776297 T^{8} + 39214 p T^{9} - 5165 p^{3} T^{10} - 3270 p^{3} T^{11} + 3173 p^{4} T^{12} + 338 p^{5} T^{13} - 57 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 41 | \( ( 1 + 222 T^{2} + 24117 T^{4} + 40815 p T^{6} + 81144722 T^{8} + 40815 p^{3} T^{10} + 24117 p^{4} T^{12} + 222 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 43 | \( ( 1 + 13 T + 112 T^{2} + 604 T^{3} + 3358 T^{4} + 604 p T^{5} + 112 p^{2} T^{6} + 13 p^{3} T^{7} + p^{4} T^{8} )^{4} \) | |
| 47 | \( 1 - 173 T^{2} + 16633 T^{4} - 1089952 T^{6} + 47343872 T^{8} - 818110040 T^{10} - 64424097160 T^{12} + 7478687522819 T^{14} - 437436851957231 T^{16} + 7478687522819 p^{2} T^{18} - 64424097160 p^{4} T^{20} - 818110040 p^{6} T^{22} + 47343872 p^{8} T^{24} - 1089952 p^{10} T^{26} + 16633 p^{12} T^{28} - 173 p^{14} T^{30} + p^{16} T^{32} \) | |
| 53 | \( 1 + 133 T^{2} + 8944 T^{4} + 599099 T^{6} + 29717867 T^{8} + 797091448 T^{10} + 31263307613 T^{12} + 880034743289 T^{14} - 19324329468344 T^{16} + 880034743289 p^{2} T^{18} + 31263307613 p^{4} T^{20} + 797091448 p^{6} T^{22} + 29717867 p^{8} T^{24} + 599099 p^{10} T^{26} + 8944 p^{12} T^{28} + 133 p^{14} T^{30} + p^{16} T^{32} \) | |
| 59 | \( 1 - 190 T^{2} + 11541 T^{4} - 443000 T^{6} + 56915720 T^{8} - 4100356020 T^{10} + 1624101836 p T^{12} - 9747744998665 T^{14} + 1100644208561934 T^{16} - 9747744998665 p^{2} T^{18} + 1624101836 p^{5} T^{20} - 4100356020 p^{6} T^{22} + 56915720 p^{8} T^{24} - 443000 p^{10} T^{26} + 11541 p^{12} T^{28} - 190 p^{14} T^{30} + p^{16} T^{32} \) | |
| 61 | \( ( 1 - 3 T + 118 T^{2} - 345 T^{3} + 7303 T^{4} - 40824 T^{5} + 56536 T^{6} - 3083880 T^{7} - 8201036 T^{8} - 3083880 p T^{9} + 56536 p^{2} T^{10} - 40824 p^{3} T^{11} + 7303 p^{4} T^{12} - 345 p^{5} T^{13} + 118 p^{6} T^{14} - 3 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 67 | \( ( 1 + 2 T - 179 T^{2} - 908 T^{3} + 16841 T^{4} + 97754 T^{5} - 850057 T^{6} - 3582776 T^{7} + 41096215 T^{8} - 3582776 p T^{9} - 850057 p^{2} T^{10} + 97754 p^{3} T^{11} + 16841 p^{4} T^{12} - 908 p^{5} T^{13} - 179 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 71 | \( ( 1 - p T^{2} + 9904 T^{4} - 812624 T^{6} + 56885380 T^{8} - 812624 p^{2} T^{10} + 9904 p^{4} T^{12} - p^{7} T^{14} + p^{8} T^{16} )^{2} \) | |
| 73 | \( ( 1 + 6 T + 159 T^{2} + 882 T^{3} + 12318 T^{4} + 98346 T^{5} + 584490 T^{6} + 8845941 T^{7} + 32868122 T^{8} + 8845941 p T^{9} + 584490 p^{2} T^{10} + 98346 p^{3} T^{11} + 12318 p^{4} T^{12} + 882 p^{5} T^{13} + 159 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 79 | \( ( 1 - 17 T - 2 T^{2} + 611 T^{3} + 7367 T^{4} - 5816 T^{5} - 1127755 T^{6} + 4949663 T^{7} + 11692942 T^{8} + 4949663 p T^{9} - 1127755 p^{2} T^{10} - 5816 p^{3} T^{11} + 7367 p^{4} T^{12} + 611 p^{5} T^{13} - 2 p^{6} T^{14} - 17 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 83 | \( ( 1 + 429 T^{2} + 94512 T^{4} + 13346472 T^{6} + 1314527120 T^{8} + 13346472 p^{2} T^{10} + 94512 p^{4} T^{12} + 429 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 89 | \( 1 - 459 T^{2} + 117876 T^{4} - 19489923 T^{6} + 2201471017 T^{8} - 146522371824 T^{10} + 290930466582 T^{12} + 1282936732245342 T^{14} - 164379910475968584 T^{16} + 1282936732245342 p^{2} T^{18} + 290930466582 p^{4} T^{20} - 146522371824 p^{6} T^{22} + 2201471017 p^{8} T^{24} - 19489923 p^{10} T^{26} + 117876 p^{12} T^{28} - 459 p^{14} T^{30} + p^{16} T^{32} \) | |
| 97 | \( ( 1 - 252 T^{2} + 22644 T^{4} + 38076 T^{6} - 140127082 T^{8} + 38076 p^{2} T^{10} + 22644 p^{4} T^{12} - 252 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
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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
−2.37233152029697619682938867481, −2.28372189816626534816241061263, −2.27114400691613573005071229127, −2.20082884266139228470455345464, −2.19856283353731933823110216499, −2.07811962276533201447797836998, −2.06113355018976167461200671846, −1.90585204605077152586330384688, −1.67846761410598720103421981674, −1.67106317760324552678087081530, −1.65073786066383727837186435971, −1.58896206271156217597224533514, −1.34716156508035214987460393842, −1.22870079778267974909297418584, −1.21959895547416518119226988599, −1.17953241991792465448134822932, −1.06220907524795126710576117942, −1.05202072642102575827894789252, −0.933307313903469540942372049803, −0.909112696096533178174827264023, −0.804119205230955470950028268094, −0.45660446602191059095705056330, −0.28085603106429371974828858498, −0.02694204584582885286388485527, −0.02318739032165859765057463053, 0.02318739032165859765057463053, 0.02694204584582885286388485527, 0.28085603106429371974828858498, 0.45660446602191059095705056330, 0.804119205230955470950028268094, 0.909112696096533178174827264023, 0.933307313903469540942372049803, 1.05202072642102575827894789252, 1.06220907524795126710576117942, 1.17953241991792465448134822932, 1.21959895547416518119226988599, 1.22870079778267974909297418584, 1.34716156508035214987460393842, 1.58896206271156217597224533514, 1.65073786066383727837186435971, 1.67106317760324552678087081530, 1.67846761410598720103421981674, 1.90585204605077152586330384688, 2.06113355018976167461200671846, 2.07811962276533201447797836998, 2.19856283353731933823110216499, 2.20082884266139228470455345464, 2.27114400691613573005071229127, 2.28372189816626534816241061263, 2.37233152029697619682938867481
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.