Dirichlet series
| L(s) = 1 | − 16·5-s − 8·23-s + 136·25-s − 8·31-s + 36·49-s + 4·53-s + 8·73-s + 20·83-s + 32·89-s − 28·113-s + 128·115-s − 64·121-s − 816·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 128·155-s + 157-s + 163-s + 167-s − 104·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | − 7.15·5-s − 1.66·23-s + 27.1·25-s − 1.43·31-s + 36/7·49-s + 0.549·53-s + 0.936·73-s + 2.19·83-s + 3.39·89-s − 2.63·113-s + 11.9·115-s − 5.81·121-s − 72.9·125-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 + 10.2·155-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 8·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{32} \cdot 5^{16} \cdot 23^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{32} \cdot 5^{16} \cdot 23^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(2^{32} \cdot 3^{32} \cdot 5^{16} \cdot 23^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.03441\times 10^{24}\) |
| Root analytic conductor: | \(5.74961\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 2^{32} \cdot 3^{32} \cdot 5^{16} \cdot 23^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(2.035111774\) |
| \(L(\frac12)\) | \(\approx\) | \(2.035111774\) |
| \(L(\frac{3}{2})\) | 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 | \( 1 \) | |
| 5 | \( ( 1 + T )^{16} \) | |
| 23 | \( 1 + 8 T + 16 T^{2} - 88 T^{3} - 852 T^{4} - 1272 T^{5} + 9520 T^{6} + 85992 T^{7} + 554854 T^{8} + 85992 p T^{9} + 9520 p^{2} T^{10} - 1272 p^{3} T^{11} - 852 p^{4} T^{12} - 88 p^{5} T^{13} + 16 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| good | 7 | \( 1 - 36 T^{2} + 639 T^{4} - 8244 T^{6} + 91521 T^{8} - 127672 p T^{10} + 7711666 T^{12} - 60829656 T^{14} + 444014222 T^{16} - 60829656 p^{2} T^{18} + 7711666 p^{4} T^{20} - 127672 p^{7} T^{22} + 91521 p^{8} T^{24} - 8244 p^{10} T^{26} + 639 p^{12} T^{28} - 36 p^{14} T^{30} + p^{16} T^{32} \) |
| 11 | \( ( 1 + 32 T^{2} - 16 T^{3} + 622 T^{4} - 244 T^{5} + 9652 T^{6} - 3396 T^{7} + 117466 T^{8} - 3396 p T^{9} + 9652 p^{2} T^{10} - 244 p^{3} T^{11} + 622 p^{4} T^{12} - 16 p^{5} T^{13} + 32 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 13 | \( ( 1 + 4 p T^{2} - 36 T^{3} + 1338 T^{4} - 1912 T^{5} + 23424 T^{6} - 45620 T^{7} + 25258 p T^{8} - 45620 p T^{9} + 23424 p^{2} T^{10} - 1912 p^{3} T^{11} + 1338 p^{4} T^{12} - 36 p^{5} T^{13} + 4 p^{7} T^{14} + p^{8} T^{16} )^{2} \) | |
| 17 | \( ( 1 + 63 T^{2} + 48 T^{3} + 7 p^{2} T^{4} + 2012 T^{5} + 49256 T^{6} + 34924 T^{7} + 957854 T^{8} + 34924 p T^{9} + 49256 p^{2} T^{10} + 2012 p^{3} T^{11} + 7 p^{6} T^{12} + 48 p^{5} T^{13} + 63 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 19 | \( 1 - 8 p T^{2} + 12052 T^{4} - 656232 T^{6} + 27287160 T^{8} - 915821608 T^{10} + 25629444940 T^{12} - 609946546072 T^{14} + 12478151631630 T^{16} - 609946546072 p^{2} T^{18} + 25629444940 p^{4} T^{20} - 915821608 p^{6} T^{22} + 27287160 p^{8} T^{24} - 656232 p^{10} T^{26} + 12052 p^{12} T^{28} - 8 p^{15} T^{30} + p^{16} T^{32} \) | |
| 29 | \( 1 - 276 T^{2} + 38915 T^{4} - 126684 p T^{6} + 258602885 T^{8} - 14355457824 T^{10} + 22395451966 p T^{12} - 24421922303216 T^{14} + 771433617407062 T^{16} - 24421922303216 p^{2} T^{18} + 22395451966 p^{5} T^{20} - 14355457824 p^{6} T^{22} + 258602885 p^{8} T^{24} - 126684 p^{11} T^{26} + 38915 p^{12} T^{28} - 276 p^{14} T^{30} + p^{16} T^{32} \) | |
| 31 | \( ( 1 + 4 T + 135 T^{2} + 308 T^{3} + 9357 T^{4} + 15808 T^{5} + 452198 T^{6} + 562720 T^{7} + 15876294 T^{8} + 562720 p T^{9} + 452198 p^{2} T^{10} + 15808 p^{3} T^{11} + 9357 p^{4} T^{12} + 308 p^{5} T^{13} + 135 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 37 | \( 1 - 396 T^{2} + 76679 T^{4} - 9685204 T^{6} + 897029553 T^{8} - 64822622528 T^{10} + 3789984988658 T^{12} - 183153578916544 T^{14} + 7398858100524926 T^{16} - 183153578916544 p^{2} T^{18} + 3789984988658 p^{4} T^{20} - 64822622528 p^{6} T^{22} + 897029553 p^{8} T^{24} - 9685204 p^{10} T^{26} + 76679 p^{12} T^{28} - 396 p^{14} T^{30} + p^{16} T^{32} \) | |
| 41 | \( 1 - 332 T^{2} + 51667 T^{4} - 5071876 T^{6} + 363821093 T^{8} - 21461500032 T^{10} + 1140161051366 T^{12} - 55879308092736 T^{14} + 2448751285807942 T^{16} - 55879308092736 p^{2} T^{18} + 1140161051366 p^{4} T^{20} - 21461500032 p^{6} T^{22} + 363821093 p^{8} T^{24} - 5071876 p^{10} T^{26} + 51667 p^{12} T^{28} - 332 p^{14} T^{30} + p^{16} T^{32} \) | |
| 43 | \( 1 - 416 T^{2} + 85712 T^{4} - 11631360 T^{6} + 1167765724 T^{8} - 92411593664 T^{10} + 5994441257776 T^{12} - 326762968032416 T^{14} + 15183165522474246 T^{16} - 326762968032416 p^{2} T^{18} + 5994441257776 p^{4} T^{20} - 92411593664 p^{6} T^{22} + 1167765724 p^{8} T^{24} - 11631360 p^{10} T^{26} + 85712 p^{12} T^{28} - 416 p^{14} T^{30} + p^{16} T^{32} \) | |
| 47 | \( 1 - 368 T^{2} + 65596 T^{4} - 7780576 T^{6} + 711177032 T^{8} - 54031817488 T^{10} + 3521710853924 T^{12} - 199727597160800 T^{14} + 9974310799133230 T^{16} - 199727597160800 p^{2} T^{18} + 3521710853924 p^{4} T^{20} - 54031817488 p^{6} T^{22} + 711177032 p^{8} T^{24} - 7780576 p^{10} T^{26} + 65596 p^{12} T^{28} - 368 p^{14} T^{30} + p^{16} T^{32} \) | |
| 53 | \( ( 1 - 2 T + 239 T^{2} + 22 T^{3} + 27699 T^{4} + 47188 T^{5} + 2117984 T^{6} + 5594340 T^{7} + 124165162 T^{8} + 5594340 p T^{9} + 2117984 p^{2} T^{10} + 47188 p^{3} T^{11} + 27699 p^{4} T^{12} + 22 p^{5} T^{13} + 239 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 59 | \( 1 - 524 T^{2} + 128555 T^{4} - 19627596 T^{6} + 2105258309 T^{8} - 171924700712 T^{10} + 11544730645014 T^{12} - 695363671022120 T^{14} + 40811494349620854 T^{16} - 695363671022120 p^{2} T^{18} + 11544730645014 p^{4} T^{20} - 171924700712 p^{6} T^{22} + 2105258309 p^{8} T^{24} - 19627596 p^{10} T^{26} + 128555 p^{12} T^{28} - 524 p^{14} T^{30} + p^{16} T^{32} \) | |
| 61 | \( 1 - 232 T^{2} + 38708 T^{4} - 4874648 T^{6} + 512117848 T^{8} - 46396513816 T^{10} + 3703939274540 T^{12} - 265179918103464 T^{14} + 16992106898832462 T^{16} - 265179918103464 p^{2} T^{18} + 3703939274540 p^{4} T^{20} - 46396513816 p^{6} T^{22} + 512117848 p^{8} T^{24} - 4874648 p^{10} T^{26} + 38708 p^{12} T^{28} - 232 p^{14} T^{30} + p^{16} T^{32} \) | |
| 67 | \( 1 - 660 T^{2} + 213103 T^{4} - 45002900 T^{6} + 7020495809 T^{8} - 865402382952 T^{10} + 87737514099218 T^{12} - 7483256305982728 T^{14} + 542822646285896350 T^{16} - 7483256305982728 p^{2} T^{18} + 87737514099218 p^{4} T^{20} - 865402382952 p^{6} T^{22} + 7020495809 p^{8} T^{24} - 45002900 p^{10} T^{26} + 213103 p^{12} T^{28} - 660 p^{14} T^{30} + p^{16} T^{32} \) | |
| 71 | \( 1 - 588 T^{2} + 165787 T^{4} - 30075564 T^{6} + 3981512453 T^{8} - 414714187784 T^{10} + 36045466520822 T^{12} - 2774817613491496 T^{14} + 200577529261368454 T^{16} - 2774817613491496 p^{2} T^{18} + 36045466520822 p^{4} T^{20} - 414714187784 p^{6} T^{22} + 3981512453 p^{8} T^{24} - 30075564 p^{10} T^{26} + 165787 p^{12} T^{28} - 588 p^{14} T^{30} + p^{16} T^{32} \) | |
| 73 | \( ( 1 - 4 T + 188 T^{2} - 324 T^{3} + 22818 T^{4} + 18208 T^{5} + 1686504 T^{6} + 5811000 T^{7} + 125769682 T^{8} + 5811000 p T^{9} + 1686504 p^{2} T^{10} + 18208 p^{3} T^{11} + 22818 p^{4} T^{12} - 324 p^{5} T^{13} + 188 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 79 | \( 1 - 624 T^{2} + 188984 T^{4} - 37675600 T^{6} + 5617926556 T^{8} - 671944769648 T^{10} + 67826853212040 T^{12} - 6052752928356048 T^{14} + 495585621248653382 T^{16} - 6052752928356048 p^{2} T^{18} + 67826853212040 p^{4} T^{20} - 671944769648 p^{6} T^{22} + 5617926556 p^{8} T^{24} - 37675600 p^{10} T^{26} + 188984 p^{12} T^{28} - 624 p^{14} T^{30} + p^{16} T^{32} \) | |
| 83 | \( ( 1 - 10 T + 311 T^{2} - 3130 T^{3} + 44439 T^{4} - 460264 T^{5} + 4448100 T^{6} - 46501664 T^{7} + 386734546 T^{8} - 46501664 p T^{9} + 4448100 p^{2} T^{10} - 460264 p^{3} T^{11} + 44439 p^{4} T^{12} - 3130 p^{5} T^{13} + 311 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 89 | \( ( 1 - 16 T + 376 T^{2} - 4248 T^{3} + 68932 T^{4} - 649688 T^{5} + 8870856 T^{6} - 74566592 T^{7} + 891102646 T^{8} - 74566592 p T^{9} + 8870856 p^{2} T^{10} - 649688 p^{3} T^{11} + 68932 p^{4} T^{12} - 4248 p^{5} T^{13} + 376 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 97 | \( 1 - 760 T^{2} + 279936 T^{4} - 67415656 T^{6} + 12142938172 T^{8} - 1777160115320 T^{10} + 223399256567680 T^{12} - 24990211970608680 T^{14} + 2536722221987987334 T^{16} - 24990211970608680 p^{2} T^{18} + 223399256567680 p^{4} T^{20} - 1777160115320 p^{6} T^{22} + 12142938172 p^{8} T^{24} - 67415656 p^{10} T^{26} + 279936 p^{12} T^{28} - 760 p^{14} T^{30} + p^{16} T^{32} \) | |
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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
−2.05428558299146821695267913899, −2.03489097768691841498081234995, −1.95234172690909986238967361094, −1.87122468268561195978917853844, −1.78851368967311458460261875727, −1.60250357087581464935521305651, −1.43533957843689917329909636909, −1.34986421726498611839058422579, −1.31154349649919659282209823176, −1.30356813971927915074803395408, −1.28277627144128937860442293889, −1.25396423838287920531885403963, −1.23217873065607382059476469012, −0.958726184243134907882934514684, −0.893601690928651512912498094475, −0.860204814972473401826093697240, −0.812676065515661870268405096676, −0.77923770824271514661145831378, −0.60965218417999692610662033277, −0.33502179605785416960881098774, −0.32360102070470655548934459974, −0.31566262551497935031940796783, −0.29219914116126796843763674637, −0.26800471085332038272744085128, −0.17532529439872040794072555064, 0.17532529439872040794072555064, 0.26800471085332038272744085128, 0.29219914116126796843763674637, 0.31566262551497935031940796783, 0.32360102070470655548934459974, 0.33502179605785416960881098774, 0.60965218417999692610662033277, 0.77923770824271514661145831378, 0.812676065515661870268405096676, 0.860204814972473401826093697240, 0.893601690928651512912498094475, 0.958726184243134907882934514684, 1.23217873065607382059476469012, 1.25396423838287920531885403963, 1.28277627144128937860442293889, 1.30356813971927915074803395408, 1.31154349649919659282209823176, 1.34986421726498611839058422579, 1.43533957843689917329909636909, 1.60250357087581464935521305651, 1.78851368967311458460261875727, 1.87122468268561195978917853844, 1.95234172690909986238967361094, 2.03489097768691841498081234995, 2.05428558299146821695267913899
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.