Dirichlet series
| L(s) = 1 | − 8·4-s − 16·5-s + 9-s − 12·11-s + 36·16-s + 128·20-s + 4·23-s + 136·25-s + 8·27-s + 4·31-s − 8·36-s + 96·44-s − 16·45-s + 54·49-s − 8·53-s + 192·55-s − 120·64-s − 16·73-s − 576·80-s + 6·81-s − 40·83-s + 80·89-s − 32·92-s − 12·99-s − 1.08e3·100-s − 64·108-s + 32·113-s + ⋯ |
| L(s) = 1 | − 4·4-s − 7.15·5-s + 1/3·9-s − 3.61·11-s + 9·16-s + 28.6·20-s + 0.834·23-s + 27.1·25-s + 1.53·27-s + 0.718·31-s − 4/3·36-s + 14.4·44-s − 2.38·45-s + 54/7·49-s − 1.09·53-s + 25.8·55-s − 15·64-s − 1.87·73-s − 64.3·80-s + 2/3·81-s − 4.39·83-s + 8.47·89-s − 3.33·92-s − 1.20·99-s − 108.·100-s − 6.15·108-s + 3.01·113-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 5^{16} \cdot 23^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 5^{16} \cdot 23^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(2^{16} \cdot 3^{16} \cdot 5^{16} \cdot 23^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(7.21139\times 10^{11}\) |
| Root analytic conductor: | \(2.34727\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 2^{16} \cdot 3^{16} \cdot 5^{16} \cdot 23^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.1999335375\) |
| \(L(\frac12)\) | \(\approx\) | \(0.1999335375\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + T^{2} )^{8} \) |
| 3 | \( 1 - T^{2} - 8 T^{3} - 5 T^{4} + 4 p T^{5} + 17 T^{6} + 20 T^{7} - 56 T^{8} + 20 p T^{9} + 17 p^{2} T^{10} + 4 p^{4} T^{11} - 5 p^{4} T^{12} - 8 p^{5} T^{13} - p^{6} T^{14} + p^{8} T^{16} \) | |
| 5 | \( ( 1 + T )^{16} \) | |
| 23 | \( 1 - 4 T + 28 T^{2} - 4 T^{3} + 468 T^{4} - 1620 T^{5} + 16036 T^{6} + 26604 T^{7} - 81770 T^{8} + 26604 p T^{9} + 16036 p^{2} T^{10} - 1620 p^{3} T^{11} + 468 p^{4} T^{12} - 4 p^{5} T^{13} + 28 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| good | 7 | \( 1 - 54 T^{2} + 1443 T^{4} - 3660 p T^{6} + 341787 T^{8} - 3680164 T^{10} + 33654325 T^{12} - 271748034 T^{14} + 1988748728 T^{16} - 271748034 p^{2} T^{18} + 33654325 p^{4} T^{20} - 3680164 p^{6} T^{22} + 341787 p^{8} T^{24} - 3660 p^{11} T^{26} + 1443 p^{12} T^{28} - 54 p^{14} T^{30} + p^{16} T^{32} \) |
| 11 | \( ( 1 + 6 T + 59 T^{2} + 260 T^{3} + 1567 T^{4} + 5840 T^{5} + 27553 T^{6} + 90402 T^{7} + 354796 T^{8} + 90402 p T^{9} + 27553 p^{2} T^{10} + 5840 p^{3} T^{11} + 1567 p^{4} T^{12} + 260 p^{5} T^{13} + 59 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 13 | \( ( 1 + 25 T^{2} + 84 T^{3} + 633 T^{4} + 1028 T^{5} + 1065 p T^{6} + 26488 T^{7} + 155560 T^{8} + 26488 p T^{9} + 1065 p^{3} T^{10} + 1028 p^{3} T^{11} + 633 p^{4} T^{12} + 84 p^{5} T^{13} + 25 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 17 | \( ( 1 + 45 T^{2} - 84 T^{3} + 1417 T^{4} - 3796 T^{5} + 31817 T^{6} - 95912 T^{7} + 628472 T^{8} - 95912 p T^{9} + 31817 p^{2} T^{10} - 3796 p^{3} T^{11} + 1417 p^{4} T^{12} - 84 p^{5} T^{13} + 45 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 19 | \( 1 - 182 T^{2} + 16543 T^{4} - 995316 T^{6} + 44417439 T^{8} - 1563546664 T^{10} + 45082837165 T^{12} - 1090352697502 T^{14} + 22419124770792 T^{16} - 1090352697502 p^{2} T^{18} + 45082837165 p^{4} T^{20} - 1563546664 p^{6} T^{22} + 44417439 p^{8} T^{24} - 995316 p^{10} T^{26} + 16543 p^{12} T^{28} - 182 p^{14} T^{30} + p^{16} T^{32} \) | |
| 29 | \( 1 - 252 T^{2} + 31652 T^{4} - 2635620 T^{6} + 163970996 T^{8} - 8155970316 T^{10} + 338634762524 T^{12} - 12059925888596 T^{14} + 373937267010262 T^{16} - 12059925888596 p^{2} T^{18} + 338634762524 p^{4} T^{20} - 8155970316 p^{6} T^{22} + 163970996 p^{8} T^{24} - 2635620 p^{10} T^{26} + 31652 p^{12} T^{28} - 252 p^{14} T^{30} + p^{16} T^{32} \) | |
| 31 | \( ( 1 - 2 T + 87 T^{2} - 268 T^{3} + 3447 T^{4} - 13952 T^{5} + 96389 T^{6} - 414902 T^{7} + 2777880 T^{8} - 414902 p T^{9} + 96389 p^{2} T^{10} - 13952 p^{3} T^{11} + 3447 p^{4} T^{12} - 268 p^{5} T^{13} + 87 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 37 | \( 1 - 324 T^{2} + 49844 T^{4} - 4751548 T^{6} + 306732372 T^{8} - 13682395604 T^{10} + 407712129932 T^{12} - 7490332746220 T^{14} + 132047410754582 T^{16} - 7490332746220 p^{2} T^{18} + 407712129932 p^{4} T^{20} - 13682395604 p^{6} T^{22} + 306732372 p^{8} T^{24} - 4751548 p^{10} T^{26} + 49844 p^{12} T^{28} - 324 p^{14} T^{30} + p^{16} T^{32} \) | |
| 41 | \( 1 - 326 T^{2} + 54463 T^{4} - 6272404 T^{6} + 557053871 T^{8} - 40208669592 T^{10} + 2428724881757 T^{12} - 124909303313838 T^{14} + 5518425180737896 T^{16} - 124909303313838 p^{2} T^{18} + 2428724881757 p^{4} T^{20} - 40208669592 p^{6} T^{22} + 557053871 p^{8} T^{24} - 6272404 p^{10} T^{26} + 54463 p^{12} T^{28} - 326 p^{14} T^{30} + p^{16} T^{32} \) | |
| 43 | \( 1 - 164 T^{2} + 18260 T^{4} - 1621980 T^{6} + 116941972 T^{8} - 7441797044 T^{10} + 415840226284 T^{12} - 20767843103372 T^{14} + 943845726667926 T^{16} - 20767843103372 p^{2} T^{18} + 415840226284 p^{4} T^{20} - 7441797044 p^{6} T^{22} + 116941972 p^{8} T^{24} - 1621980 p^{10} T^{26} + 18260 p^{12} T^{28} - 164 p^{14} T^{30} + p^{16} T^{32} \) | |
| 47 | \( 1 - 392 T^{2} + 77656 T^{4} - 10400152 T^{6} + 22547812 p T^{8} - 87327299848 T^{10} + 6019029917288 T^{12} - 353621660643800 T^{14} + 17884944569496262 T^{16} - 353621660643800 p^{2} T^{18} + 6019029917288 p^{4} T^{20} - 87327299848 p^{6} T^{22} + 22547812 p^{9} T^{24} - 10400152 p^{10} T^{26} + 77656 p^{12} T^{28} - 392 p^{14} T^{30} + p^{16} T^{32} \) | |
| 53 | \( ( 1 + 4 T + 224 T^{2} + 412 T^{3} + 22476 T^{4} - 14396 T^{5} + 1364000 T^{6} - 4181892 T^{7} + 69985990 T^{8} - 4181892 p T^{9} + 1364000 p^{2} T^{10} - 14396 p^{3} T^{11} + 22476 p^{4} T^{12} + 412 p^{5} T^{13} + 224 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 59 | \( 1 - 608 T^{2} + 183128 T^{4} - 36356256 T^{6} + 5331002780 T^{8} - 612493377248 T^{10} + 57051853530600 T^{12} - 4395723361812512 T^{14} + 283159177125890310 T^{16} - 4395723361812512 p^{2} T^{18} + 57051853530600 p^{4} T^{20} - 612493377248 p^{6} T^{22} + 5331002780 p^{8} T^{24} - 36356256 p^{10} T^{26} + 183128 p^{12} T^{28} - 608 p^{14} T^{30} + p^{16} T^{32} \) | |
| 61 | \( 1 - 322 T^{2} + 52979 T^{4} - 6397460 T^{6} + 653981107 T^{8} - 57930732628 T^{10} + 4514012542949 T^{12} - 318685089498582 T^{14} + 20468089760249352 T^{16} - 318685089498582 p^{2} T^{18} + 4514012542949 p^{4} T^{20} - 57930732628 p^{6} T^{22} + 653981107 p^{8} T^{24} - 6397460 p^{10} T^{26} + 52979 p^{12} T^{28} - 322 p^{14} T^{30} + p^{16} T^{32} \) | |
| 67 | \( 1 - 588 T^{2} + 174196 T^{4} - 34597556 T^{6} + 5174985044 T^{8} - 619817996988 T^{10} + 61605612719180 T^{12} - 5189985489882628 T^{14} + 374861172443592598 T^{16} - 5189985489882628 p^{2} T^{18} + 61605612719180 p^{4} T^{20} - 619817996988 p^{6} T^{22} + 5174985044 p^{8} T^{24} - 34597556 p^{10} T^{26} + 174196 p^{12} T^{28} - 588 p^{14} T^{30} + p^{16} T^{32} \) | |
| 71 | \( 1 - 522 T^{2} + 151603 T^{4} - 31090692 T^{6} + 4925561555 T^{8} - 632334679292 T^{10} + 67524594224213 T^{12} - 6091399824778054 T^{14} + 468058090950196456 T^{16} - 6091399824778054 p^{2} T^{18} + 67524594224213 p^{4} T^{20} - 632334679292 p^{6} T^{22} + 4925561555 p^{8} T^{24} - 31090692 p^{10} T^{26} + 151603 p^{12} T^{28} - 522 p^{14} T^{30} + p^{16} T^{32} \) | |
| 73 | \( ( 1 + 8 T + 452 T^{2} + 2856 T^{3} + 93060 T^{4} + 472216 T^{5} + 11712252 T^{6} + 48943416 T^{7} + 1013081398 T^{8} + 48943416 p T^{9} + 11712252 p^{2} T^{10} + 472216 p^{3} T^{11} + 93060 p^{4} T^{12} + 2856 p^{5} T^{13} + 452 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 79 | \( 1 - 756 T^{2} + 289604 T^{4} - 74013772 T^{6} + 14085566068 T^{8} - 2114668783076 T^{10} + 259096529587644 T^{12} - 26440183327775196 T^{14} + 2272179968891021846 T^{16} - 26440183327775196 p^{2} T^{18} + 259096529587644 p^{4} T^{20} - 2114668783076 p^{6} T^{22} + 14085566068 p^{8} T^{24} - 74013772 p^{10} T^{26} + 289604 p^{12} T^{28} - 756 p^{14} T^{30} + p^{16} T^{32} \) | |
| 83 | \( ( 1 + 20 T + 398 T^{2} + 4724 T^{3} + 63576 T^{4} + 648020 T^{5} + 7357986 T^{6} + 65080276 T^{7} + 654158446 T^{8} + 65080276 p T^{9} + 7357986 p^{2} T^{10} + 648020 p^{3} T^{11} + 63576 p^{4} T^{12} + 4724 p^{5} T^{13} + 398 p^{6} T^{14} + 20 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 89 | \( ( 1 - 40 T + 952 T^{2} - 14520 T^{3} + 173872 T^{4} - 1766024 T^{5} + 19130568 T^{6} - 207813272 T^{7} + 2146288414 T^{8} - 207813272 p T^{9} + 19130568 p^{2} T^{10} - 1766024 p^{3} T^{11} + 173872 p^{4} T^{12} - 14520 p^{5} T^{13} + 952 p^{6} T^{14} - 40 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 97 | \( 1 - 814 T^{2} + 351339 T^{4} - 103868644 T^{6} + 23306738683 T^{8} - 4182848625620 T^{10} + 618947409793933 T^{12} - 76866067224491178 T^{14} + 8088105369841495608 T^{16} - 76866067224491178 p^{2} T^{18} + 618947409793933 p^{4} T^{20} - 4182848625620 p^{6} T^{22} + 23306738683 p^{8} T^{24} - 103868644 p^{10} T^{26} + 351339 p^{12} T^{28} - 814 p^{14} T^{30} + p^{16} T^{32} \) | |
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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
−2.97718164800625148147050918512, −2.73940417646819242699731901031, −2.67880105240889106641849654961, −2.62151847142217238909736487336, −2.53207719992099672454670099547, −2.40459226654722438141733616657, −2.40451872985110826843879329468, −2.38454817147428606181041048828, −2.35435366157671862246529727811, −2.04972752470908295231638870224, −2.02279856203242805409401885159, −1.77033546211367368329149103878, −1.53067880970679835454150746718, −1.35371379130459133231516563495, −1.30303788680647710818136581720, −1.26196129404947055709915908416, −1.23246340678355620718874745937, −1.16432038828760343343069816884, −0.63110421680519600449659470390, −0.60375957037601511932937144204, −0.59106131199769075195074550953, −0.58018778887774996648560405132, −0.54815533732399856343745047173, −0.31263137503998909847262751653, −0.20313434093206486597765129196, 0.20313434093206486597765129196, 0.31263137503998909847262751653, 0.54815533732399856343745047173, 0.58018778887774996648560405132, 0.59106131199769075195074550953, 0.60375957037601511932937144204, 0.63110421680519600449659470390, 1.16432038828760343343069816884, 1.23246340678355620718874745937, 1.26196129404947055709915908416, 1.30303788680647710818136581720, 1.35371379130459133231516563495, 1.53067880970679835454150746718, 1.77033546211367368329149103878, 2.02279856203242805409401885159, 2.04972752470908295231638870224, 2.35435366157671862246529727811, 2.38454817147428606181041048828, 2.40451872985110826843879329468, 2.40459226654722438141733616657, 2.53207719992099672454670099547, 2.62151847142217238909736487336, 2.67880105240889106641849654961, 2.73940417646819242699731901031, 2.97718164800625148147050918512
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.