Dirichlet series
| L(s) = 1 | + 4·4-s − 9-s + 20·11-s + 6·16-s − 34·29-s + 44·31-s − 4·36-s + 14·41-s + 80·44-s + 12·49-s − 28·59-s − 18·61-s + 28·71-s + 34·79-s − 81-s − 28·89-s − 20·99-s − 52·101-s + 32·109-s − 136·116-s + 154·121-s + 176·124-s + 127-s + 131-s + 137-s + 139-s − 6·144-s + ⋯ |
| L(s) = 1 | + 2·4-s − 1/3·9-s + 6.03·11-s + 3/2·16-s − 6.31·29-s + 7.90·31-s − 2/3·36-s + 2.18·41-s + 12.0·44-s + 12/7·49-s − 3.64·59-s − 2.30·61-s + 3.32·71-s + 3.82·79-s − 1/9·81-s − 2.96·89-s − 2.01·99-s − 5.17·101-s + 3.06·109-s − 12.6·116-s + 14·121-s + 15.8·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1/2·144-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{32} \cdot 19^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{32} \cdot 19^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(2^{16} \cdot 5^{32} \cdot 19^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.20229\times 10^{14}\) |
| Root analytic conductor: | \(2.75423\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 2^{16} \cdot 5^{32} \cdot 19^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(1.005055645\) |
| \(L(\frac12)\) | \(\approx\) | \(1.005055645\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - T^{2} + T^{4} )^{4} \) |
| 5 | \( 1 \) | |
| 19 | \( ( 1 + 35 T^{2} + 36 p T^{4} + 35 p^{2} T^{6} + p^{4} T^{8} )^{2} \) | |
| good | 3 | \( 1 + T^{2} + 2 T^{4} - 13 p T^{6} - 106 T^{8} - 8 p T^{10} + 116 T^{12} + 2662 T^{14} - 689 T^{16} + 2662 p^{2} T^{18} + 116 p^{4} T^{20} - 8 p^{7} T^{22} - 106 p^{8} T^{24} - 13 p^{11} T^{26} + 2 p^{12} T^{28} + p^{14} T^{30} + p^{16} T^{32} \) |
| 7 | \( ( 1 - 6 T^{2} + 93 T^{4} - 206 T^{6} + 4320 T^{8} - 206 p^{2} T^{10} + 93 p^{4} T^{12} - 6 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 11 | \( ( 1 - 5 T + 24 T^{2} - 42 T^{3} + 169 T^{4} - 42 p T^{5} + 24 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} )^{4} \) | |
| 13 | \( 1 + 25 T^{2} - 30 T^{4} - 3511 T^{6} + 2006 p T^{8} + 474432 T^{10} - 6844676 T^{12} - 78003374 T^{14} + 87222807 T^{16} - 78003374 p^{2} T^{18} - 6844676 p^{4} T^{20} + 474432 p^{6} T^{22} + 2006 p^{9} T^{24} - 3511 p^{10} T^{26} - 30 p^{12} T^{28} + 25 p^{14} T^{30} + p^{16} T^{32} \) | |
| 17 | \( 1 + 55 T^{2} + 1211 T^{4} + 18410 T^{6} + 294555 T^{8} + 2099530 T^{10} - 64906786 T^{12} - 2130682215 T^{14} - 38496400226 T^{16} - 2130682215 p^{2} T^{18} - 64906786 p^{4} T^{20} + 2099530 p^{6} T^{22} + 294555 p^{8} T^{24} + 18410 p^{10} T^{26} + 1211 p^{12} T^{28} + 55 p^{14} T^{30} + p^{16} T^{32} \) | |
| 23 | \( ( 1 + 11 T^{2} - 623 T^{4} - 3454 T^{6} + 209686 T^{8} - 3454 p^{2} T^{10} - 623 p^{4} T^{12} + 11 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 29 | \( ( 1 + 17 T + 94 T^{2} + 357 T^{3} + 3704 T^{4} + 19244 T^{5} - 3426 T^{6} + 1700 p T^{7} + 2210431 T^{8} + 1700 p^{2} T^{9} - 3426 p^{2} T^{10} + 19244 p^{3} T^{11} + 3704 p^{4} T^{12} + 357 p^{5} T^{13} + 94 p^{6} T^{14} + 17 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 31 | \( ( 1 - 11 T + 83 T^{2} - 663 T^{3} + 4342 T^{4} - 663 p T^{5} + 83 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} )^{4} \) | |
| 37 | \( ( 1 - 60 T^{2} + 6056 T^{4} - 234372 T^{6} + 12677454 T^{8} - 234372 p^{2} T^{10} + 6056 p^{4} T^{12} - 60 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 41 | \( ( 1 - 7 T - 15 T^{2} - 808 T^{3} + 5909 T^{4} + 11158 T^{5} + 334620 T^{6} - 2279715 T^{7} - 3979470 T^{8} - 2279715 p T^{9} + 334620 p^{2} T^{10} + 11158 p^{3} T^{11} + 5909 p^{4} T^{12} - 808 p^{5} T^{13} - 15 p^{6} T^{14} - 7 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 43 | \( 1 + 177 T^{2} + 15186 T^{4} + 831497 T^{6} + 32561262 T^{8} + 966493352 T^{10} + 17881176460 T^{12} - 4650911646 p T^{14} - 14572237145 p^{2} T^{16} - 4650911646 p^{3} T^{18} + 17881176460 p^{4} T^{20} + 966493352 p^{6} T^{22} + 32561262 p^{8} T^{24} + 831497 p^{10} T^{26} + 15186 p^{12} T^{28} + 177 p^{14} T^{30} + p^{16} T^{32} \) | |
| 47 | \( 1 + 286 T^{2} + 42815 T^{4} + 4590986 T^{6} + 396114069 T^{8} + 28849918756 T^{10} + 1821823289450 T^{12} + 101591548315800 T^{14} + 5050522168124170 T^{16} + 101591548315800 p^{2} T^{18} + 1821823289450 p^{4} T^{20} + 28849918756 p^{6} T^{22} + 396114069 p^{8} T^{24} + 4590986 p^{10} T^{26} + 42815 p^{12} T^{28} + 286 p^{14} T^{30} + p^{16} T^{32} \) | |
| 53 | \( 1 + 334 T^{2} + 59015 T^{4} + 7444634 T^{6} + 749115189 T^{8} + 63155461204 T^{10} + 4585690904810 T^{12} + 291814252810920 T^{14} + 16423762091035210 T^{16} + 291814252810920 p^{2} T^{18} + 4585690904810 p^{4} T^{20} + 63155461204 p^{6} T^{22} + 749115189 p^{8} T^{24} + 7444634 p^{10} T^{26} + 59015 p^{12} T^{28} + 334 p^{14} T^{30} + p^{16} T^{32} \) | |
| 59 | \( ( 1 + 14 T - 78 T^{2} - 796 T^{3} + 17885 T^{4} + 77752 T^{5} - 1427730 T^{6} - 1292118 T^{7} + 103988436 T^{8} - 1292118 p T^{9} - 1427730 p^{2} T^{10} + 77752 p^{3} T^{11} + 17885 p^{4} T^{12} - 796 p^{5} T^{13} - 78 p^{6} T^{14} + 14 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 61 | \( ( 1 + 9 T - 126 T^{2} - 1175 T^{3} + 11382 T^{4} + 86764 T^{5} - 669848 T^{6} - 2353650 T^{7} + 39432547 T^{8} - 2353650 p T^{9} - 669848 p^{2} T^{10} + 86764 p^{3} T^{11} + 11382 p^{4} T^{12} - 1175 p^{5} T^{13} - 126 p^{6} T^{14} + 9 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 67 | \( 1 + 486 T^{2} + 130395 T^{4} + 24355586 T^{6} + 3501946569 T^{8} + 408233730596 T^{10} + 39839988216910 T^{12} + 3320243100698640 T^{14} + 238946980809693010 T^{16} + 3320243100698640 p^{2} T^{18} + 39839988216910 p^{4} T^{20} + 408233730596 p^{6} T^{22} + 3501946569 p^{8} T^{24} + 24355586 p^{10} T^{26} + 130395 p^{12} T^{28} + 486 p^{14} T^{30} + p^{16} T^{32} \) | |
| 71 | \( ( 1 - 14 T - 45 T^{2} + 622 T^{3} + 7661 T^{4} + 5168 T^{5} - 974214 T^{6} + 1505364 T^{7} + 42583530 T^{8} + 1505364 p T^{9} - 974214 p^{2} T^{10} + 5168 p^{3} T^{11} + 7661 p^{4} T^{12} + 622 p^{5} T^{13} - 45 p^{6} T^{14} - 14 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 73 | \( 1 + 357 T^{2} + 59446 T^{4} + 7561617 T^{6} + 917567182 T^{8} + 96572382732 T^{10} + 8638583986780 T^{12} + 729714952955562 T^{14} + 57124404714729175 T^{16} + 729714952955562 p^{2} T^{18} + 8638583986780 p^{4} T^{20} + 96572382732 p^{6} T^{22} + 917567182 p^{8} T^{24} + 7561617 p^{10} T^{26} + 59446 p^{12} T^{28} + 357 p^{14} T^{30} + p^{16} T^{32} \) | |
| 79 | \( ( 1 - 17 T + 2 T^{2} + 1139 T^{3} + 1002 T^{4} - 74018 T^{5} - 76090 T^{6} + 6700686 T^{7} - 70425071 T^{8} + 6700686 p T^{9} - 76090 p^{2} T^{10} - 74018 p^{3} T^{11} + 1002 p^{4} T^{12} + 1139 p^{5} T^{13} + 2 p^{6} T^{14} - 17 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 83 | \( ( 1 - 397 T^{2} + 79967 T^{4} - 10572039 T^{6} + 1015759768 T^{8} - 10572039 p^{2} T^{10} + 79967 p^{4} T^{12} - 397 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 89 | \( ( 1 + 14 T - 116 T^{2} - 888 T^{3} + 23783 T^{4} + 24296 T^{5} - 2897328 T^{6} - 4851182 T^{7} + 179834968 T^{8} - 4851182 p T^{9} - 2897328 p^{2} T^{10} + 24296 p^{3} T^{11} + 23783 p^{4} T^{12} - 888 p^{5} T^{13} - 116 p^{6} T^{14} + 14 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 97 | \( 1 + 507 T^{2} + 148339 T^{4} + 27423570 T^{6} + 3408990739 T^{8} + 227205437346 T^{10} - 6067988715890 T^{12} - 3793249192621227 T^{14} - 500798636389358498 T^{16} - 3793249192621227 p^{2} T^{18} - 6067988715890 p^{4} T^{20} + 227205437346 p^{6} T^{22} + 3408990739 p^{8} T^{24} + 27423570 p^{10} T^{26} + 148339 p^{12} T^{28} + 507 p^{14} T^{30} + p^{16} T^{32} \) | |
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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
−2.54979730017363580390852252635, −2.45493181662175115890462265523, −2.38364795762281673638693138083, −2.37083586449909622893095571210, −2.36835567274701515608560255620, −2.23140782310884523113670489478, −2.19114671704699852432850642621, −2.01176933074450823862815219446, −1.92117771926075669215304282182, −1.78561566062354957624069080558, −1.76480272336940546531895623386, −1.73422961887703739972556874581, −1.70384857161331009263451780483, −1.45244346206368166916246875120, −1.35248002451110378295912532316, −1.27897762587160084145662048947, −1.27740375149558987112806532348, −1.12905750900718771636746293056, −1.01797480147208732508748464324, −1.01709953471674070569707844103, −0.913790865588669623641307549448, −0.72826884029468695304364096955, −0.66943547917489216750678027427, −0.24988469118070087061733729794, −0.03350274156682594418347010020, 0.03350274156682594418347010020, 0.24988469118070087061733729794, 0.66943547917489216750678027427, 0.72826884029468695304364096955, 0.913790865588669623641307549448, 1.01709953471674070569707844103, 1.01797480147208732508748464324, 1.12905750900718771636746293056, 1.27740375149558987112806532348, 1.27897762587160084145662048947, 1.35248002451110378295912532316, 1.45244346206368166916246875120, 1.70384857161331009263451780483, 1.73422961887703739972556874581, 1.76480272336940546531895623386, 1.78561566062354957624069080558, 1.92117771926075669215304282182, 2.01176933074450823862815219446, 2.19114671704699852432850642621, 2.23140782310884523113670489478, 2.36835567274701515608560255620, 2.37083586449909622893095571210, 2.38364795762281673638693138083, 2.45493181662175115890462265523, 2.54979730017363580390852252635
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.