Dirichlet series
| L(s) = 1 | + 6·5-s − 9·9-s − 8·23-s − 3·25-s − 2·31-s + 12·37-s − 48·43-s − 54·45-s − 62·49-s − 32·59-s + 6·61-s − 34·73-s + 45·81-s + 12·83-s + 34·103-s + 92·107-s − 18·113-s − 48·115-s − 82·121-s − 106·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 12·155-s + ⋯ |
| L(s) = 1 | + 2.68·5-s − 3·9-s − 1.66·23-s − 3/5·25-s − 0.359·31-s + 1.97·37-s − 7.31·43-s − 8.04·45-s − 8.85·49-s − 4.16·59-s + 0.768·61-s − 3.97·73-s + 5·81-s + 1.31·83-s + 3.35·103-s + 8.89·107-s − 1.69·113-s − 4.47·115-s − 7.45·121-s − 9.48·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s − 0.963·155-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 41^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 41^{32}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(2^{32} \cdot 41^{32}\) |
| Sign: | $1$ |
| Analytic conductor: | \(4.76954\times 10^{27}\) |
| Root analytic conductor: | \(7.32744\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 2^{32} \cdot 41^{32} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(8.919608033\) |
| \(L(\frac12)\) | \(\approx\) | \(8.919608033\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 41 | \( 1 \) | |
| good | 3 | \( 1 + p^{2} T^{2} + 4 p^{2} T^{4} + 5 p^{3} T^{6} + 653 T^{8} + 305 p^{2} T^{10} + 352 p^{3} T^{12} + 3311 p^{2} T^{14} + 89788 T^{16} + 3311 p^{4} T^{18} + 352 p^{7} T^{20} + 305 p^{8} T^{22} + 653 p^{8} T^{24} + 5 p^{13} T^{26} + 4 p^{14} T^{28} + p^{16} T^{30} + p^{16} T^{32} \) |
| 5 | \( ( 1 - 3 T + 3 p T^{2} - 28 T^{3} + 94 T^{4} - 187 T^{5} + 24 p^{2} T^{6} - 1502 T^{7} + 3811 T^{8} - 1502 p T^{9} + 24 p^{4} T^{10} - 187 p^{3} T^{11} + 94 p^{4} T^{12} - 28 p^{5} T^{13} + 3 p^{7} T^{14} - 3 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 7 | \( 1 + 62 T^{2} + 1867 T^{4} + 36803 T^{6} + 540847 T^{8} + 6388365 T^{10} + 63498738 T^{12} + 544682214 T^{14} + 4076330190 T^{16} + 544682214 p^{2} T^{18} + 63498738 p^{4} T^{20} + 6388365 p^{6} T^{22} + 540847 p^{8} T^{24} + 36803 p^{10} T^{26} + 1867 p^{12} T^{28} + 62 p^{14} T^{30} + p^{16} T^{32} \) | |
| 11 | \( 1 + 82 T^{2} + 3283 T^{4} + 84631 T^{6} + 1563427 T^{8} + 21912645 T^{10} + 245100162 T^{12} + 2399177442 T^{14} + 24414412830 T^{16} + 2399177442 p^{2} T^{18} + 245100162 p^{4} T^{20} + 21912645 p^{6} T^{22} + 1563427 p^{8} T^{24} + 84631 p^{10} T^{26} + 3283 p^{12} T^{28} + 82 p^{14} T^{30} + p^{16} T^{32} \) | |
| 13 | \( 1 + 138 T^{2} + 9417 T^{4} + 422317 T^{6} + 13938122 T^{8} + 358994050 T^{10} + 7463029608 T^{12} + 127705222886 T^{14} + 1817463938735 T^{16} + 127705222886 p^{2} T^{18} + 7463029608 p^{4} T^{20} + 358994050 p^{6} T^{22} + 13938122 p^{8} T^{24} + 422317 p^{10} T^{26} + 9417 p^{12} T^{28} + 138 p^{14} T^{30} + p^{16} T^{32} \) | |
| 17 | \( 1 + 11 p T^{2} + 17053 T^{4} + 1011182 T^{6} + 43809086 T^{8} + 1475051755 T^{10} + 40017309722 T^{12} + 893688857030 T^{14} + 16619064223537 T^{16} + 893688857030 p^{2} T^{18} + 40017309722 p^{4} T^{20} + 1475051755 p^{6} T^{22} + 43809086 p^{8} T^{24} + 1011182 p^{10} T^{26} + 17053 p^{12} T^{28} + 11 p^{15} T^{30} + p^{16} T^{32} \) | |
| 19 | \( 1 + 109 T^{2} + 7308 T^{4} + 352904 T^{6} + 13567372 T^{8} + 431657335 T^{10} + 615391113 p T^{12} + 273252722072 T^{14} + 5560046245200 T^{16} + 273252722072 p^{2} T^{18} + 615391113 p^{5} T^{20} + 431657335 p^{6} T^{22} + 13567372 p^{8} T^{24} + 352904 p^{10} T^{26} + 7308 p^{12} T^{28} + 109 p^{14} T^{30} + p^{16} T^{32} \) | |
| 23 | \( ( 1 + 4 T + 74 T^{2} + 215 T^{3} + 3211 T^{4} + 9172 T^{5} + 4675 p T^{6} + 252073 T^{7} + 2673422 T^{8} + 252073 p T^{9} + 4675 p^{3} T^{10} + 9172 p^{3} T^{11} + 3211 p^{4} T^{12} + 215 p^{5} T^{13} + 74 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 29 | \( 1 + 263 T^{2} + 35553 T^{4} + 3250024 T^{6} + 223541982 T^{8} + 12218829910 T^{10} + 547572763772 T^{12} + 20487187532343 T^{14} + 646136671737760 T^{16} + 20487187532343 p^{2} T^{18} + 547572763772 p^{4} T^{20} + 12218829910 p^{6} T^{22} + 223541982 p^{8} T^{24} + 3250024 p^{10} T^{26} + 35553 p^{12} T^{28} + 263 p^{14} T^{30} + p^{16} T^{32} \) | |
| 31 | \( ( 1 + T + 63 T^{2} + 71 T^{3} + 2302 T^{4} + 5335 T^{5} + 80577 T^{6} + 9743 p T^{7} + 3047490 T^{8} + 9743 p^{2} T^{9} + 80577 p^{2} T^{10} + 5335 p^{3} T^{11} + 2302 p^{4} T^{12} + 71 p^{5} T^{13} + 63 p^{6} T^{14} + p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 37 | \( ( 1 - 6 T + 165 T^{2} - 1136 T^{3} + 14034 T^{4} - 96098 T^{5} + 796816 T^{6} - 5133462 T^{7} + 33445201 T^{8} - 5133462 p T^{9} + 796816 p^{2} T^{10} - 96098 p^{3} T^{11} + 14034 p^{4} T^{12} - 1136 p^{5} T^{13} + 165 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 43 | \( ( 1 + 24 T + 494 T^{2} + 6975 T^{3} + 86951 T^{4} + 880272 T^{5} + 8027625 T^{6} + 62408883 T^{7} + 440547662 T^{8} + 62408883 p T^{9} + 8027625 p^{2} T^{10} + 880272 p^{3} T^{11} + 86951 p^{4} T^{12} + 6975 p^{5} T^{13} + 494 p^{6} T^{14} + 24 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 47 | \( 1 + 411 T^{2} + 85281 T^{4} + 11835195 T^{6} + 1233214813 T^{8} + 102751093440 T^{10} + 7110951798474 T^{12} + 418183317981666 T^{14} + 21152702402009358 T^{16} + 418183317981666 p^{2} T^{18} + 7110951798474 p^{4} T^{20} + 102751093440 p^{6} T^{22} + 1233214813 p^{8} T^{24} + 11835195 p^{10} T^{26} + 85281 p^{12} T^{28} + 411 p^{14} T^{30} + p^{16} T^{32} \) | |
| 53 | \( 1 + 452 T^{2} + 100602 T^{4} + 14872828 T^{6} + 1658401227 T^{8} + 149390662180 T^{10} + 11308669085468 T^{12} + 736056577002384 T^{14} + 41717756523171565 T^{16} + 736056577002384 p^{2} T^{18} + 11308669085468 p^{4} T^{20} + 149390662180 p^{6} T^{22} + 1658401227 p^{8} T^{24} + 14872828 p^{10} T^{26} + 100602 p^{12} T^{28} + 452 p^{14} T^{30} + p^{16} T^{32} \) | |
| 59 | \( ( 1 + 16 T + 352 T^{2} + 4587 T^{3} + 61327 T^{4} + 656030 T^{5} + 6546883 T^{6} + 58180359 T^{7} + 468246930 T^{8} + 58180359 p T^{9} + 6546883 p^{2} T^{10} + 656030 p^{3} T^{11} + 61327 p^{4} T^{12} + 4587 p^{5} T^{13} + 352 p^{6} T^{14} + 16 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 61 | \( ( 1 - 3 T + 169 T^{2} - 1480 T^{3} + 19068 T^{4} - 189665 T^{5} + 1966756 T^{6} - 14157352 T^{7} + 150050573 T^{8} - 14157352 p T^{9} + 1966756 p^{2} T^{10} - 189665 p^{3} T^{11} + 19068 p^{4} T^{12} - 1480 p^{5} T^{13} + 169 p^{6} T^{14} - 3 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 67 | \( 1 + 529 T^{2} + 133272 T^{4} + 20761148 T^{6} + 2148145192 T^{8} + 142240087195 T^{10} + 4242959687523 T^{12} - 210983787032944 T^{14} - 29829027147585720 T^{16} - 210983787032944 p^{2} T^{18} + 4242959687523 p^{4} T^{20} + 142240087195 p^{6} T^{22} + 2148145192 p^{8} T^{24} + 20761148 p^{10} T^{26} + 133272 p^{12} T^{28} + 529 p^{14} T^{30} + p^{16} T^{32} \) | |
| 71 | \( 1 + 612 T^{2} + 190218 T^{4} + 39688831 T^{6} + 6222862057 T^{8} + 779399006130 T^{10} + 81003284494537 T^{12} + 7159764679029907 T^{14} + 545983357835495910 T^{16} + 7159764679029907 p^{2} T^{18} + 81003284494537 p^{4} T^{20} + 779399006130 p^{6} T^{22} + 6222862057 p^{8} T^{24} + 39688831 p^{10} T^{26} + 190218 p^{12} T^{28} + 612 p^{14} T^{30} + p^{16} T^{32} \) | |
| 73 | \( ( 1 + 17 T + 485 T^{2} + 5868 T^{3} + 100534 T^{4} + 986446 T^{5} + 12875724 T^{6} + 106236101 T^{7} + 1127476896 T^{8} + 106236101 p T^{9} + 12875724 p^{2} T^{10} + 986446 p^{3} T^{11} + 100534 p^{4} T^{12} + 5868 p^{5} T^{13} + 485 p^{6} T^{14} + 17 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 79 | \( 1 + 737 T^{2} + 270304 T^{4} + 65719015 T^{6} + 11895746573 T^{8} + 1706613469745 T^{10} + 201629278078916 T^{12} + 20096005908122503 T^{14} + 1712905283658948508 T^{16} + 20096005908122503 p^{2} T^{18} + 201629278078916 p^{4} T^{20} + 1706613469745 p^{6} T^{22} + 11895746573 p^{8} T^{24} + 65719015 p^{10} T^{26} + 270304 p^{12} T^{28} + 737 p^{14} T^{30} + p^{16} T^{32} \) | |
| 83 | \( ( 1 - 6 T + 471 T^{2} - 3054 T^{3} + 107213 T^{4} - 687396 T^{5} + 15437769 T^{6} - 89913372 T^{7} + 1529216388 T^{8} - 89913372 p T^{9} + 15437769 p^{2} T^{10} - 687396 p^{3} T^{11} + 107213 p^{4} T^{12} - 3054 p^{5} T^{13} + 471 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 89 | \( 1 + 815 T^{2} + 322545 T^{4} + 83707780 T^{6} + 16210711354 T^{8} + 2513994731630 T^{10} + 325103023012860 T^{12} + 35888626783284595 T^{14} + 3426762152446002456 T^{16} + 35888626783284595 p^{2} T^{18} + 325103023012860 p^{4} T^{20} + 2513994731630 p^{6} T^{22} + 16210711354 p^{8} T^{24} + 83707780 p^{10} T^{26} + 322545 p^{12} T^{28} + 815 p^{14} T^{30} + p^{16} T^{32} \) | |
| 97 | \( 1 + 1092 T^{2} + 589002 T^{4} + 207766748 T^{6} + 53536518987 T^{8} + 10669108983620 T^{10} + 1699173287685388 T^{12} + 220440178694853264 T^{14} + 23533940563544404285 T^{16} + 220440178694853264 p^{2} T^{18} + 1699173287685388 p^{4} T^{20} + 10669108983620 p^{6} T^{22} + 53536518987 p^{8} T^{24} + 207766748 p^{10} T^{26} + 589002 p^{12} T^{28} + 1092 p^{14} T^{30} + p^{16} T^{32} \) | |
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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
−1.85201501040115832272182463342, −1.82101772314869430273489095213, −1.72149409478512351040594327503, −1.71265855059223819553259358534, −1.70053876731515393909839068178, −1.64637305727580727211527766901, −1.52309486925452046464234549084, −1.51679745250505785513784463444, −1.47229087457161443061834329943, −1.42470560319373982481184185620, −1.38064626554865823151605138671, −1.33965531368946215437538843023, −1.12080735525493652463391508964, −0.951889243809969101355786804915, −0.945490432367873547813678288798, −0.64238301391094371486607423248, −0.64094322613360223064163388825, −0.64074605440332971996230686270, −0.48145079064166907315511313796, −0.44892007654201318976522735472, −0.41514691781965539648353853905, −0.36055638956735653338131038452, −0.28206512925190298450399467622, −0.21141238111107290605333225610, −0.093953081181131649391013884997, 0.093953081181131649391013884997, 0.21141238111107290605333225610, 0.28206512925190298450399467622, 0.36055638956735653338131038452, 0.41514691781965539648353853905, 0.44892007654201318976522735472, 0.48145079064166907315511313796, 0.64074605440332971996230686270, 0.64094322613360223064163388825, 0.64238301391094371486607423248, 0.945490432367873547813678288798, 0.951889243809969101355786804915, 1.12080735525493652463391508964, 1.33965531368946215437538843023, 1.38064626554865823151605138671, 1.42470560319373982481184185620, 1.47229087457161443061834329943, 1.51679745250505785513784463444, 1.52309486925452046464234549084, 1.64637305727580727211527766901, 1.70053876731515393909839068178, 1.71265855059223819553259358534, 1.72149409478512351040594327503, 1.82101772314869430273489095213, 1.85201501040115832272182463342
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.