Dirichlet series
| L(s) = 1 | − 4·4-s − 8·9-s + 9·16-s + 4·19-s − 2·25-s + 28·31-s + 32·36-s − 16·49-s − 12·59-s + 40·61-s − 16·64-s + 12·71-s − 16·76-s − 8·79-s + 31·81-s + 48·89-s + 8·100-s − 36·101-s − 64·109-s − 112·124-s + 127-s + 131-s + 137-s + 139-s − 72·144-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 2·4-s − 8/3·9-s + 9/4·16-s + 0.917·19-s − 2/5·25-s + 5.02·31-s + 16/3·36-s − 2.28·49-s − 1.56·59-s + 5.12·61-s − 2·64-s + 1.42·71-s − 1.83·76-s − 0.900·79-s + 31/9·81-s + 5.08·89-s + 4/5·100-s − 3.58·101-s − 6.13·109-s − 10.0·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 6·144-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{16} \cdot 11^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{16} \cdot 11^{32}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(5^{16} \cdot 11^{32}\) |
| Sign: | $1$ |
| Analytic conductor: | \(8.80075\times 10^{10}\) |
| Root analytic conductor: | \(2.19794\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 5^{16} \cdot 11^{32} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.9283747102\) |
| \(L(\frac12)\) | \(\approx\) | \(0.9283747102\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 5 | \( 1 + 2 T^{2} - 21 T^{4} - 92 T^{6} + 341 T^{8} - 92 p^{2} T^{10} - 21 p^{4} T^{12} + 2 p^{6} T^{14} + p^{8} T^{16} \) |
| 11 | \( 1 \) | |
| good | 2 | \( 1 + p^{2} T^{2} + 7 T^{4} + p^{3} T^{6} + 17 T^{8} + 5 p^{3} T^{10} + 23 T^{12} - 31 p^{2} T^{14} - 335 T^{16} - 31 p^{4} T^{18} + 23 p^{4} T^{20} + 5 p^{9} T^{22} + 17 p^{8} T^{24} + p^{13} T^{26} + 7 p^{12} T^{28} + p^{16} T^{30} + p^{16} T^{32} \) |
| 3 | \( 1 + 8 T^{2} + 11 p T^{4} + 88 T^{6} + 176 T^{8} - 80 T^{10} - 811 p T^{12} - 11440 T^{14} - 36113 T^{16} - 11440 p^{2} T^{18} - 811 p^{5} T^{20} - 80 p^{6} T^{22} + 176 p^{8} T^{24} + 88 p^{10} T^{26} + 11 p^{13} T^{28} + 8 p^{14} T^{30} + p^{16} T^{32} \) | |
| 7 | \( 1 + 16 T^{2} + 121 T^{4} + 80 p T^{6} + 2768 T^{8} + 160 p T^{10} - 207241 T^{12} - 2641504 T^{14} - 20054417 T^{16} - 2641504 p^{2} T^{18} - 207241 p^{4} T^{20} + 160 p^{7} T^{22} + 2768 p^{8} T^{24} + 80 p^{11} T^{26} + 121 p^{12} T^{28} + 16 p^{14} T^{30} + p^{16} T^{32} \) | |
| 13 | \( ( 1 + 8 T^{2} - 105 T^{4} - 2192 T^{6} + 209 T^{8} - 2192 p^{2} T^{10} - 105 p^{4} T^{12} + 8 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 17 | \( 1 + 52 T^{2} + 1498 T^{4} + 30212 T^{6} + 462371 T^{8} + 4441060 T^{10} + 2227172 T^{12} - 814930480 T^{14} - 16939793003 T^{16} - 814930480 p^{2} T^{18} + 2227172 p^{4} T^{20} + 4441060 p^{6} T^{22} + 462371 p^{8} T^{24} + 30212 p^{10} T^{26} + 1498 p^{12} T^{28} + 52 p^{14} T^{30} + p^{16} T^{32} \) | |
| 19 | \( ( 1 - 2 T - 8 T^{2} + 2 T^{3} - 193 T^{4} - 2750 T^{5} + 10628 T^{6} + 18356 T^{7} + 9925 T^{8} + 18356 p T^{9} + 10628 p^{2} T^{10} - 2750 p^{3} T^{11} - 193 p^{4} T^{12} + 2 p^{5} T^{13} - 8 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 23 | \( ( 1 - 40 T^{2} + 1266 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8} )^{4} \) | |
| 29 | \( ( 1 - 10 T^{2} - 741 T^{4} + 15820 T^{6} + 464981 T^{8} + 15820 p^{2} T^{10} - 741 p^{4} T^{12} - 10 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 31 | \( ( 1 - 14 T + 88 T^{2} - 154 T^{3} - 2233 T^{4} + 19390 T^{5} - 48028 T^{6} - 304612 T^{7} + 3182245 T^{8} - 304612 p T^{9} - 48028 p^{2} T^{10} + 19390 p^{3} T^{11} - 2233 p^{4} T^{12} - 154 p^{5} T^{13} + 88 p^{6} T^{14} - 14 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 37 | \( 1 + 100 T^{2} + 5194 T^{4} + 175700 T^{6} + 4423475 T^{8} - 15454700 T^{10} - 8485753084 T^{12} - 498016380400 T^{14} - 19425161477771 T^{16} - 498016380400 p^{2} T^{18} - 8485753084 p^{4} T^{20} - 15454700 p^{6} T^{22} + 4423475 p^{8} T^{24} + 175700 p^{10} T^{26} + 5194 p^{12} T^{28} + 100 p^{14} T^{30} + p^{16} T^{32} \) | |
| 41 | \( ( 1 - 79 T^{2} + 4560 T^{4} - 227441 T^{6} + 10302479 T^{8} - 227441 p^{2} T^{10} + 4560 p^{4} T^{12} - 79 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 43 | \( ( 1 - 16 T^{2} + 2439 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} )^{4} \) | |
| 47 | \( 1 + 40 T^{2} + 49 T^{4} + 28280 T^{6} - 290080 T^{8} + 13207760 p T^{10} + 12467279 p^{2} T^{12} - 239120 p^{3} T^{14} + 2573599 p^{4} T^{16} - 239120 p^{5} T^{18} + 12467279 p^{6} T^{20} + 13207760 p^{7} T^{22} - 290080 p^{8} T^{24} + 28280 p^{10} T^{26} + 49 p^{12} T^{28} + 40 p^{14} T^{30} + p^{16} T^{32} \) | |
| 53 | \( 1 + 64 T^{2} + 2254 T^{4} + 206144 T^{6} + 12656531 T^{8} + 1084259392 T^{10} + 65353870556 T^{12} + 2834207117312 T^{14} + 156065125018597 T^{16} + 2834207117312 p^{2} T^{18} + 65353870556 p^{4} T^{20} + 1084259392 p^{6} T^{22} + 12656531 p^{8} T^{24} + 206144 p^{10} T^{26} + 2254 p^{12} T^{28} + 64 p^{14} T^{30} + p^{16} T^{32} \) | |
| 59 | \( ( 1 + 6 T - 88 T^{2} - 918 T^{3} + 4047 T^{4} + 11130 T^{5} - 453692 T^{6} + 525924 T^{7} + 49265765 T^{8} + 525924 p T^{9} - 453692 p^{2} T^{10} + 11130 p^{3} T^{11} + 4047 p^{4} T^{12} - 918 p^{5} T^{13} - 88 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 61 | \( ( 1 - 20 T + 181 T^{2} - 460 T^{3} - 160 p T^{4} + 117640 T^{5} - 327661 T^{6} - 5591080 T^{7} + 76375519 T^{8} - 5591080 p T^{9} - 327661 p^{2} T^{10} + 117640 p^{3} T^{11} - 160 p^{5} T^{12} - 460 p^{5} T^{13} + 181 p^{6} T^{14} - 20 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 67 | \( ( 1 - 40 T^{2} - 2529 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8} )^{4} \) | |
| 71 | \( ( 1 - 6 T - 88 T^{2} + 846 T^{3} + 3351 T^{4} - 60810 T^{5} + 32548 T^{6} + 1652940 T^{7} - 4940923 T^{8} + 1652940 p T^{9} + 32548 p^{2} T^{10} - 60810 p^{3} T^{11} + 3351 p^{4} T^{12} + 846 p^{5} T^{13} - 88 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 73 | \( ( 1 + 122 T^{2} + 9555 T^{4} + 515572 T^{6} + 11981189 T^{8} + 515572 p^{2} T^{10} + 9555 p^{4} T^{12} + 122 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 79 | \( ( 1 + 4 T - 134 T^{2} - 820 T^{3} + 11843 T^{4} + 36580 T^{5} - 1052956 T^{6} - 838816 T^{7} + 92235253 T^{8} - 838816 p T^{9} - 1052956 p^{2} T^{10} + 36580 p^{3} T^{11} + 11843 p^{4} T^{12} - 820 p^{5} T^{13} - 134 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 83 | \( ( 1 + 68 T^{2} - 2265 T^{4} - 622472 T^{6} - 26724511 T^{8} - 622472 p^{2} T^{10} - 2265 p^{4} T^{12} + 68 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 89 | \( ( 1 - 6 T + 175 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{8} \) | |
| 97 | \( 1 + 304 T^{2} + 52222 T^{4} + 67184 p T^{6} + 663123587 T^{8} + 452906800 p T^{10} + 718925091068 T^{12} - 227423555958784 T^{14} - 31078924898682875 T^{16} - 227423555958784 p^{2} T^{18} + 718925091068 p^{4} T^{20} + 452906800 p^{7} T^{22} + 663123587 p^{8} T^{24} + 67184 p^{11} T^{26} + 52222 p^{12} T^{28} + 304 p^{14} T^{30} + p^{16} T^{32} \) | |
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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
−2.88866077327227936207459266568, −2.84300079159837901788487868483, −2.77636511324926121486000730815, −2.59294545296197340116922306205, −2.54794128241362516756212534666, −2.40174264273507004082442936626, −2.31809930013450127325351682977, −2.29175320930009489186819091427, −2.28357881106101512192902133359, −2.23203190180587544706147177972, −2.04023954911975265092074805484, −1.89633002984233114885036533400, −1.82659809853360574016425756230, −1.62161858143216239269087376701, −1.58138099019158889071061216526, −1.38117254744593153749194629562, −1.37879010460860486108976859641, −1.10046021482210871369833317353, −1.01289171454161089031837884555, −0.945898997790053654127687062827, −0.895582484552506024758037325805, −0.56741193576890193644737118818, −0.53179777966070140744448061264, −0.45019119281285952296496593182, −0.11115922550083756755463854321, 0.11115922550083756755463854321, 0.45019119281285952296496593182, 0.53179777966070140744448061264, 0.56741193576890193644737118818, 0.895582484552506024758037325805, 0.945898997790053654127687062827, 1.01289171454161089031837884555, 1.10046021482210871369833317353, 1.37879010460860486108976859641, 1.38117254744593153749194629562, 1.58138099019158889071061216526, 1.62161858143216239269087376701, 1.82659809853360574016425756230, 1.89633002984233114885036533400, 2.04023954911975265092074805484, 2.23203190180587544706147177972, 2.28357881106101512192902133359, 2.29175320930009489186819091427, 2.31809930013450127325351682977, 2.40174264273507004082442936626, 2.54794128241362516756212534666, 2.59294545296197340116922306205, 2.77636511324926121486000730815, 2.84300079159837901788487868483, 2.88866077327227936207459266568
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.