Dirichlet series
| L(s) = 1 | − 5·9-s − 12·11-s − 8·23-s − 8·25-s + 4·27-s − 8·37-s − 16·47-s − 8·49-s − 32·59-s + 8·61-s + 32·71-s + 48·73-s + 7·81-s − 72·83-s − 8·97-s + 60·99-s − 96·107-s − 4·109-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | − 5/3·9-s − 3.61·11-s − 1.66·23-s − 8/5·25-s + 0.769·27-s − 1.31·37-s − 2.33·47-s − 8/7·49-s − 4.16·59-s + 1.02·61-s + 3.79·71-s + 5.61·73-s + 7/9·81-s − 7.90·83-s − 0.812·97-s + 6.03·99-s − 9.28·107-s − 0.383·109-s − 2·121-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 + 0.0798·157-s + 0.0783·163-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 3^{16} \cdot 5^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 3^{16} \cdot 5^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(2^{64} \cdot 3^{16} \cdot 5^{16} \cdot 7^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.09997\times 10^{18}\) |
| Root analytic conductor: | \(3.66263\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 2^{64} \cdot 3^{16} \cdot 5^{16} \cdot 7^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.1870170105\) |
| \(L(\frac12)\) | \(\approx\) | \(0.1870170105\) |
| \(L(\frac{3}{2})\) | 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 | \( 1 + 5 T^{2} - 4 T^{3} + 2 p^{2} T^{4} - 28 T^{5} + 17 p T^{6} - 136 T^{7} + 154 T^{8} - 136 p T^{9} + 17 p^{3} T^{10} - 28 p^{3} T^{11} + 2 p^{6} T^{12} - 4 p^{5} T^{13} + 5 p^{6} T^{14} + p^{8} T^{16} \) | |
| 5 | \( ( 1 + T^{2} )^{8} \) | |
| 7 | \( ( 1 + T^{2} )^{8} \) | |
| good | 11 | \( ( 1 + 6 T + 65 T^{2} + 294 T^{3} + 1878 T^{4} + 7014 T^{5} + 33991 T^{6} + 108918 T^{7} + 437138 T^{8} + 108918 p T^{9} + 33991 p^{2} T^{10} + 7014 p^{3} T^{11} + 1878 p^{4} T^{12} + 294 p^{5} T^{13} + 65 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} )^{2} \) |
| 13 | \( ( 1 + 61 T^{2} - 4 p T^{3} + 1678 T^{4} - 3252 T^{5} + 28907 T^{6} - 82568 T^{7} + 395394 T^{8} - 82568 p T^{9} + 28907 p^{2} T^{10} - 3252 p^{3} T^{11} + 1678 p^{4} T^{12} - 4 p^{6} T^{13} + 61 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 17 | \( 1 - 150 T^{2} + 11445 T^{4} - 591630 T^{6} + 23144950 T^{8} - 724358034 T^{10} + 18713777787 T^{12} - 406377932490 T^{14} + 7488351070002 T^{16} - 406377932490 p^{2} T^{18} + 18713777787 p^{4} T^{20} - 724358034 p^{6} T^{22} + 23144950 p^{8} T^{24} - 591630 p^{10} T^{26} + 11445 p^{12} T^{28} - 150 p^{14} T^{30} + p^{16} T^{32} \) | |
| 19 | \( 1 - 72 T^{2} + 2776 T^{4} - 84504 T^{6} + 2367132 T^{8} - 60680136 T^{10} + 1389950312 T^{12} - 28983074136 T^{14} + 565630232966 T^{16} - 28983074136 p^{2} T^{18} + 1389950312 p^{4} T^{20} - 60680136 p^{6} T^{22} + 2367132 p^{8} T^{24} - 84504 p^{10} T^{26} + 2776 p^{12} T^{28} - 72 p^{14} T^{30} + p^{16} T^{32} \) | |
| 23 | \( ( 1 + 4 T + 104 T^{2} + 212 T^{3} + 4460 T^{4} - 316 T^{5} + 106584 T^{6} - 257996 T^{7} + 2167782 T^{8} - 257996 p T^{9} + 106584 p^{2} T^{10} - 316 p^{3} T^{11} + 4460 p^{4} T^{12} + 212 p^{5} T^{13} + 104 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 29 | \( 1 - 210 T^{2} + 22157 T^{4} - 1587514 T^{6} + 87205526 T^{8} - 3911701046 T^{10} + 149548849747 T^{12} - 5041826734782 T^{14} + 153341768638450 T^{16} - 5041826734782 p^{2} T^{18} + 149548849747 p^{4} T^{20} - 3911701046 p^{6} T^{22} + 87205526 p^{8} T^{24} - 1587514 p^{10} T^{26} + 22157 p^{12} T^{28} - 210 p^{14} T^{30} + p^{16} T^{32} \) | |
| 31 | \( 1 - 232 T^{2} + 28088 T^{4} - 2380984 T^{6} + 156906268 T^{8} - 8449914472 T^{10} + 382706392072 T^{12} - 14810909530104 T^{14} + 493899807595590 T^{16} - 14810909530104 p^{2} T^{18} + 382706392072 p^{4} T^{20} - 8449914472 p^{6} T^{22} + 156906268 p^{8} T^{24} - 2380984 p^{10} T^{26} + 28088 p^{12} T^{28} - 232 p^{14} T^{30} + p^{16} T^{32} \) | |
| 37 | \( ( 1 + 4 T + 252 T^{2} + 940 T^{3} + 28868 T^{4} + 97556 T^{5} + 1977252 T^{6} + 157100 p T^{7} + 89057110 T^{8} + 157100 p^{2} T^{9} + 1977252 p^{2} T^{10} + 97556 p^{3} T^{11} + 28868 p^{4} T^{12} + 940 p^{5} T^{13} + 252 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 41 | \( 1 - 360 T^{2} + 64024 T^{4} - 7444792 T^{6} + 635762524 T^{8} - 42690799464 T^{10} + 2372747981480 T^{12} - 114078981719288 T^{14} + 4909989659603526 T^{16} - 114078981719288 p^{2} T^{18} + 2372747981480 p^{4} T^{20} - 42690799464 p^{6} T^{22} + 635762524 p^{8} T^{24} - 7444792 p^{10} T^{26} + 64024 p^{12} T^{28} - 360 p^{14} T^{30} + p^{16} T^{32} \) | |
| 43 | \( 1 - 272 T^{2} + 32600 T^{4} - 2363056 T^{6} + 129543580 T^{8} - 6579157008 T^{10} + 314157481960 T^{12} - 13046511672624 T^{14} + 527775035229702 T^{16} - 13046511672624 p^{2} T^{18} + 314157481960 p^{4} T^{20} - 6579157008 p^{6} T^{22} + 129543580 p^{8} T^{24} - 2363056 p^{10} T^{26} + 32600 p^{12} T^{28} - 272 p^{14} T^{30} + p^{16} T^{32} \) | |
| 47 | \( ( 1 + 8 T + 185 T^{2} + 740 T^{3} + 14374 T^{4} + 32412 T^{5} + 925991 T^{6} + 2097280 T^{7} + 52058098 T^{8} + 2097280 p T^{9} + 925991 p^{2} T^{10} + 32412 p^{3} T^{11} + 14374 p^{4} T^{12} + 740 p^{5} T^{13} + 185 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 53 | \( 1 - 488 T^{2} + 117656 T^{4} - 18476664 T^{6} + 2116674204 T^{8} - 189060288360 T^{10} + 13850591597992 T^{12} - 870243792774328 T^{14} + 48584095248442886 T^{16} - 870243792774328 p^{2} T^{18} + 13850591597992 p^{4} T^{20} - 189060288360 p^{6} T^{22} + 2116674204 p^{8} T^{24} - 18476664 p^{10} T^{26} + 117656 p^{12} T^{28} - 488 p^{14} T^{30} + p^{16} T^{32} \) | |
| 59 | \( ( 1 + 16 T + 376 T^{2} + 3760 T^{3} + 52908 T^{4} + 396592 T^{5} + 4543496 T^{6} + 28982416 T^{7} + 298856966 T^{8} + 28982416 p T^{9} + 4543496 p^{2} T^{10} + 396592 p^{3} T^{11} + 52908 p^{4} T^{12} + 3760 p^{5} T^{13} + 376 p^{6} T^{14} + 16 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 61 | \( ( 1 - 4 T + 360 T^{2} - 1516 T^{3} + 61292 T^{4} - 4100 p T^{5} + 6517656 T^{6} - 23929340 T^{7} + 474754630 T^{8} - 23929340 p T^{9} + 6517656 p^{2} T^{10} - 4100 p^{4} T^{11} + 61292 p^{4} T^{12} - 1516 p^{5} T^{13} + 360 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 67 | \( 1 - 552 T^{2} + 159544 T^{4} - 31777080 T^{6} + 4831044444 T^{8} - 590491480872 T^{10} + 59781415398152 T^{12} - 5100034477339896 T^{14} + 370001903221986566 T^{16} - 5100034477339896 p^{2} T^{18} + 59781415398152 p^{4} T^{20} - 590491480872 p^{6} T^{22} + 4831044444 p^{8} T^{24} - 31777080 p^{10} T^{26} + 159544 p^{12} T^{28} - 552 p^{14} T^{30} + p^{16} T^{32} \) | |
| 71 | \( ( 1 - 16 T + 356 T^{2} - 4240 T^{3} + 63556 T^{4} - 653136 T^{5} + 7632860 T^{6} - 65779664 T^{7} + 632479030 T^{8} - 65779664 p T^{9} + 7632860 p^{2} T^{10} - 653136 p^{3} T^{11} + 63556 p^{4} T^{12} - 4240 p^{5} T^{13} + 356 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 73 | \( ( 1 - 24 T + 508 T^{2} - 7336 T^{3} + 96244 T^{4} - 1054392 T^{5} + 10956932 T^{6} - 101956616 T^{7} + 912426198 T^{8} - 101956616 p T^{9} + 10956932 p^{2} T^{10} - 1054392 p^{3} T^{11} + 96244 p^{4} T^{12} - 7336 p^{5} T^{13} + 508 p^{6} T^{14} - 24 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 79 | \( 1 - 594 T^{2} + 184533 T^{4} - 39228210 T^{6} + 6368414326 T^{8} - 837800487486 T^{10} + 92814879160539 T^{12} - 8886659569986366 T^{14} + 747175924197584946 T^{16} - 8886659569986366 p^{2} T^{18} + 92814879160539 p^{4} T^{20} - 837800487486 p^{6} T^{22} + 6368414326 p^{8} T^{24} - 39228210 p^{10} T^{26} + 184533 p^{12} T^{28} - 594 p^{14} T^{30} + p^{16} T^{32} \) | |
| 83 | \( ( 1 + 36 T + 1000 T^{2} + 19652 T^{3} + 334604 T^{4} + 4694084 T^{5} + 58817560 T^{6} + 635287044 T^{7} + 6201424518 T^{8} + 635287044 p T^{9} + 58817560 p^{2} T^{10} + 4694084 p^{3} T^{11} + 334604 p^{4} T^{12} + 19652 p^{5} T^{13} + 1000 p^{6} T^{14} + 36 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 89 | \( 1 - 456 T^{2} + 108952 T^{4} - 18593112 T^{6} + 2519868252 T^{8} - 289190398920 T^{10} + 29769089206568 T^{12} - 2848859672713752 T^{14} + 258861070461300038 T^{16} - 2848859672713752 p^{2} T^{18} + 29769089206568 p^{4} T^{20} - 289190398920 p^{6} T^{22} + 2519868252 p^{8} T^{24} - 18593112 p^{10} T^{26} + 108952 p^{12} T^{28} - 456 p^{14} T^{30} + p^{16} T^{32} \) | |
| 97 | \( ( 1 + 4 T + 453 T^{2} - 260 T^{3} + 86750 T^{4} - 465868 T^{5} + 10080507 T^{6} - 100845748 T^{7} + 965350786 T^{8} - 100845748 p T^{9} + 10080507 p^{2} T^{10} - 465868 p^{3} T^{11} + 86750 p^{4} T^{12} - 260 p^{5} T^{13} + 453 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
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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
−2.44320272196513719125293354670, −2.35385547240578728217377216256, −2.23345774511632199121574628096, −2.18238777586876516885518012490, −2.14272310588373356481757684183, −2.08387407712556531081959910962, −2.02882938655006885018981030358, −1.75915476973289110547559407065, −1.69072051172149092715585950281, −1.52587867950592814860201677301, −1.51090821196247354694046396639, −1.48184258313881287269376638025, −1.43780175250759632910251011978, −1.35025666984157524040372802738, −1.30516760734304269030740032719, −1.22117689398368465323224898875, −1.12913361922391403336187277047, −1.06334559693822195453313319308, −0.926073303433370650809987323853, −0.46309202362904209024487003154, −0.35476896377397141273305704658, −0.31072719696131568099297990998, −0.28665733874260537349290110090, −0.18600301304001046786856779722, −0.10644958304555092035145638260, 0.10644958304555092035145638260, 0.18600301304001046786856779722, 0.28665733874260537349290110090, 0.31072719696131568099297990998, 0.35476896377397141273305704658, 0.46309202362904209024487003154, 0.926073303433370650809987323853, 1.06334559693822195453313319308, 1.12913361922391403336187277047, 1.22117689398368465323224898875, 1.30516760734304269030740032719, 1.35025666984157524040372802738, 1.43780175250759632910251011978, 1.48184258313881287269376638025, 1.51090821196247354694046396639, 1.52587867950592814860201677301, 1.69072051172149092715585950281, 1.75915476973289110547559407065, 2.02882938655006885018981030358, 2.08387407712556531081959910962, 2.14272310588373356481757684183, 2.18238777586876516885518012490, 2.23345774511632199121574628096, 2.35385547240578728217377216256, 2.44320272196513719125293354670
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.