Dirichlet series
| L(s) = 1 | + 4·4-s + 6·16-s − 32·25-s − 12·29-s + 4·37-s + 4·43-s − 28·67-s − 4·79-s − 9·81-s − 128·100-s + 132·107-s + 28·109-s − 48·116-s + 68·121-s + 127-s + 131-s + 137-s + 139-s + 16·148-s + 149-s + 151-s + 157-s + 163-s + 167-s − 68·169-s + 16·172-s + 173-s + ⋯ |
| L(s) = 1 | + 2·4-s + 3/2·16-s − 6.39·25-s − 2.22·29-s + 0.657·37-s + 0.609·43-s − 3.42·67-s − 0.450·79-s − 81-s − 12.7·100-s + 12.7·107-s + 2.68·109-s − 4.45·116-s + 6.18·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1.31·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 5.23·169-s + 1.21·172-s + 0.0760·173-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{32} \cdot 7^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{32} \cdot 7^{32}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(2^{16} \cdot 3^{32} \cdot 7^{32}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.66379\times 10^{13}\) |
| Root analytic conductor: | \(2.65382\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 2^{16} \cdot 3^{32} \cdot 7^{32} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(8.320313938\) |
| \(L(\frac12)\) | \(\approx\) | \(8.320313938\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - T^{2} + T^{4} )^{4} \) |
| 3 | \( 1 + p^{2} T^{4} + 4 p^{2} T^{6} + 4 p^{2} T^{8} + 4 p^{4} T^{10} + p^{6} T^{12} + p^{8} T^{16} \) | |
| 7 | \( 1 \) | |
| good | 5 | \( ( 1 + 16 T^{2} + 133 T^{4} + 772 T^{6} + 3856 T^{8} + 772 p^{2} T^{10} + 133 p^{4} T^{12} + 16 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 11 | \( ( 1 - 34 T^{2} + 625 T^{4} - 8530 T^{6} + 98932 T^{8} - 8530 p^{2} T^{10} + 625 p^{4} T^{12} - 34 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 13 | \( 1 + 68 T^{2} + 2376 T^{4} + 57352 T^{6} + 1082018 T^{8} + 16951644 T^{10} + 232506496 T^{12} + 2987085740 T^{14} + 38351015667 T^{16} + 2987085740 p^{2} T^{18} + 232506496 p^{4} T^{20} + 16951644 p^{6} T^{22} + 1082018 p^{8} T^{24} + 57352 p^{10} T^{26} + 2376 p^{12} T^{28} + 68 p^{14} T^{30} + p^{16} T^{32} \) | |
| 17 | \( 1 - 94 T^{2} + 4551 T^{4} - 9082 p T^{6} + 4183433 T^{8} - 97483812 T^{10} + 2040829006 T^{12} - 39195392176 T^{14} + 694079215146 T^{16} - 39195392176 p^{2} T^{18} + 2040829006 p^{4} T^{20} - 97483812 p^{6} T^{22} + 4183433 p^{8} T^{24} - 9082 p^{11} T^{26} + 4551 p^{12} T^{28} - 94 p^{14} T^{30} + p^{16} T^{32} \) | |
| 19 | \( 1 + 98 T^{2} + 5058 T^{4} + 175540 T^{6} + 237119 p T^{8} + 87611580 T^{10} + 1299299878 T^{12} + 15453626030 T^{14} + 214883894484 T^{16} + 15453626030 p^{2} T^{18} + 1299299878 p^{4} T^{20} + 87611580 p^{6} T^{22} + 237119 p^{9} T^{24} + 175540 p^{10} T^{26} + 5058 p^{12} T^{28} + 98 p^{14} T^{30} + p^{16} T^{32} \) | |
| 23 | \( ( 1 - 58 T^{2} + 2653 T^{4} - 79666 T^{6} + 2153008 T^{8} - 79666 p^{2} T^{10} + 2653 p^{4} T^{12} - 58 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 29 | \( ( 1 + 6 T + 98 T^{2} + 516 T^{3} + 4846 T^{4} + 25650 T^{5} + 193448 T^{6} + 972210 T^{7} + 6347347 T^{8} + 972210 p T^{9} + 193448 p^{2} T^{10} + 25650 p^{3} T^{11} + 4846 p^{4} T^{12} + 516 p^{5} T^{13} + 98 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 31 | \( 1 + 104 T^{2} + 3888 T^{4} + 96880 T^{6} + 4455362 T^{8} + 148421160 T^{10} + 1870813504 T^{12} + 70884338648 T^{14} + 4079738375235 T^{16} + 70884338648 p^{2} T^{18} + 1870813504 p^{4} T^{20} + 148421160 p^{6} T^{22} + 4455362 p^{8} T^{24} + 96880 p^{10} T^{26} + 3888 p^{12} T^{28} + 104 p^{14} T^{30} + p^{16} T^{32} \) | |
| 37 | \( ( 1 - 2 T - 42 T^{2} - 16 T^{3} + 38 T^{4} + 4854 T^{5} + 32488 T^{6} - 173978 T^{7} - 411165 T^{8} - 173978 p T^{9} + 32488 p^{2} T^{10} + 4854 p^{3} T^{11} + 38 p^{4} T^{12} - 16 p^{5} T^{13} - 42 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 41 | \( 1 - 70 T^{2} - 1569 T^{4} + 150526 T^{6} + 5171081 T^{8} - 280356900 T^{10} - 8583026738 T^{12} + 6221865664 p T^{14} + 9422146980954 T^{16} + 6221865664 p^{3} T^{18} - 8583026738 p^{4} T^{20} - 280356900 p^{6} T^{22} + 5171081 p^{8} T^{24} + 150526 p^{10} T^{26} - 1569 p^{12} T^{28} - 70 p^{14} T^{30} + p^{16} T^{32} \) | |
| 43 | \( ( 1 - 2 T - 3 p T^{2} - 46 T^{3} + 9833 T^{4} + 11184 T^{5} - 521114 T^{6} - 232628 T^{7} + 22298490 T^{8} - 232628 p T^{9} - 521114 p^{2} T^{10} + 11184 p^{3} T^{11} + 9833 p^{4} T^{12} - 46 p^{5} T^{13} - 3 p^{7} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 47 | \( 1 - 136 T^{2} + 8016 T^{4} - 225584 T^{6} - 1533310 T^{8} + 489880632 T^{10} - 30253322048 T^{12} + 1486359308360 T^{14} - 68731587628605 T^{16} + 1486359308360 p^{2} T^{18} - 30253322048 p^{4} T^{20} + 489880632 p^{6} T^{22} - 1533310 p^{8} T^{24} - 225584 p^{10} T^{26} + 8016 p^{12} T^{28} - 136 p^{14} T^{30} + p^{16} T^{32} \) | |
| 53 | \( ( 1 + p T^{2} + p^{2} T^{4} )^{8} \) | |
| 59 | \( 1 - 178 T^{2} + 20643 T^{4} - 1496150 T^{6} + 80456981 T^{8} - 3515943660 T^{10} + 180649052698 T^{12} - 13219132050040 T^{14} + 857467356385554 T^{16} - 13219132050040 p^{2} T^{18} + 180649052698 p^{4} T^{20} - 3515943660 p^{6} T^{22} + 80456981 p^{8} T^{24} - 1496150 p^{10} T^{26} + 20643 p^{12} T^{28} - 178 p^{14} T^{30} + p^{16} T^{32} \) | |
| 61 | \( 1 + 248 T^{2} + 28191 T^{4} + 2052448 T^{6} + 122525357 T^{8} + 7198089144 T^{10} + 457346362462 T^{12} + 32095759051208 T^{14} + 2131627513941198 T^{16} + 32095759051208 p^{2} T^{18} + 457346362462 p^{4} T^{20} + 7198089144 p^{6} T^{22} + 122525357 p^{8} T^{24} + 2052448 p^{10} T^{26} + 28191 p^{12} T^{28} + 248 p^{14} T^{30} + p^{16} T^{32} \) | |
| 67 | \( ( 1 + 14 T + 39 T^{2} - 110 T^{3} - 2659 T^{4} - 53760 T^{5} - 92054 T^{6} + 2669060 T^{7} + 22240746 T^{8} + 2669060 p T^{9} - 92054 p^{2} T^{10} - 53760 p^{3} T^{11} - 2659 p^{4} T^{12} - 110 p^{5} T^{13} + 39 p^{6} T^{14} + 14 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 71 | \( ( 1 - 478 T^{2} + 105553 T^{4} - 13986142 T^{6} + 1213269316 T^{8} - 13986142 p^{2} T^{10} + 105553 p^{4} T^{12} - 478 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 73 | \( 1 + 362 T^{2} + 66999 T^{4} + 8550862 T^{6} + 840900233 T^{8} + 64348718460 T^{10} + 3848101857934 T^{12} + 198223620850304 T^{14} + 11891998605677418 T^{16} + 198223620850304 p^{2} T^{18} + 3848101857934 p^{4} T^{20} + 64348718460 p^{6} T^{22} + 840900233 p^{8} T^{24} + 8550862 p^{10} T^{26} + 66999 p^{12} T^{28} + 362 p^{14} T^{30} + p^{16} T^{32} \) | |
| 79 | \( ( 1 + 2 T - 183 T^{2} + 982 T^{3} + 19715 T^{4} - 144312 T^{5} - 491612 T^{6} + 7480148 T^{7} - 8945118 T^{8} + 7480148 p T^{9} - 491612 p^{2} T^{10} - 144312 p^{3} T^{11} + 19715 p^{4} T^{12} + 982 p^{5} T^{13} - 183 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 83 | \( 1 + 44 T^{2} - 13368 T^{4} - 595880 T^{6} + 87112226 T^{8} + 3596762100 T^{10} - 160886956928 T^{12} - 11841264313180 T^{14} - 673218489607821 T^{16} - 11841264313180 p^{2} T^{18} - 160886956928 p^{4} T^{20} + 3596762100 p^{6} T^{22} + 87112226 p^{8} T^{24} - 595880 p^{10} T^{26} - 13368 p^{12} T^{28} + 44 p^{14} T^{30} + p^{16} T^{32} \) | |
| 89 | \( 1 - 496 T^{2} + 126612 T^{4} - 22679456 T^{6} + 3259241258 T^{8} - 407089119312 T^{10} + 46038542478544 T^{12} - 4754185780844944 T^{14} + 445332554542016595 T^{16} - 4754185780844944 p^{2} T^{18} + 46038542478544 p^{4} T^{20} - 407089119312 p^{6} T^{22} + 3259241258 p^{8} T^{24} - 22679456 p^{10} T^{26} + 126612 p^{12} T^{28} - 496 p^{14} T^{30} + p^{16} T^{32} \) | |
| 97 | \( 1 + 74 T^{2} + 11511 T^{4} - 803858 T^{6} - 150210775 T^{8} - 25062425316 T^{10} - 114467134418 T^{12} + 73367970993632 T^{14} + 27832786456667274 T^{16} + 73367970993632 p^{2} T^{18} - 114467134418 p^{4} T^{20} - 25062425316 p^{6} T^{22} - 150210775 p^{8} T^{24} - 803858 p^{10} T^{26} + 11511 p^{12} T^{28} + 74 p^{14} T^{30} + p^{16} T^{32} \) | |
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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
−2.60646483668053785510554646226, −2.45053870382717861171739034483, −2.41320989662015865961215857271, −2.33940485450743943594760341909, −2.27453625008116255953639916551, −2.24679645297668556221722972454, −2.11152161662867679698948152276, −2.02577881934734665160900868688, −2.00817720661876630546702177348, −1.98864349244398862351956524330, −1.95305801681703617209984083284, −1.69664591407174090970163416479, −1.65987514616079818022778318735, −1.64686520201924475135242736994, −1.56302071255170007141131614629, −1.51598488493588832257744068178, −1.28481814708000719932514373211, −1.16360112053737812405560371471, −1.13495548246155549991736963298, −0.824948009860976761735078269028, −0.68808833248343661893904699838, −0.60802089326901471905585985187, −0.36975928888650739037506355311, −0.36563040532337024616994893489, −0.18370903460143320908635961500, 0.18370903460143320908635961500, 0.36563040532337024616994893489, 0.36975928888650739037506355311, 0.60802089326901471905585985187, 0.68808833248343661893904699838, 0.824948009860976761735078269028, 1.13495548246155549991736963298, 1.16360112053737812405560371471, 1.28481814708000719932514373211, 1.51598488493588832257744068178, 1.56302071255170007141131614629, 1.64686520201924475135242736994, 1.65987514616079818022778318735, 1.69664591407174090970163416479, 1.95305801681703617209984083284, 1.98864349244398862351956524330, 2.00817720661876630546702177348, 2.02577881934734665160900868688, 2.11152161662867679698948152276, 2.24679645297668556221722972454, 2.27453625008116255953639916551, 2.33940485450743943594760341909, 2.41320989662015865961215857271, 2.45053870382717861171739034483, 2.60646483668053785510554646226
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.