Dirichlet series
| L(s) = 1 | + 8·3-s − 4·4-s + 28·9-s − 32·12-s + 6·13-s + 5·16-s − 2·17-s + 24·23-s − 15·25-s + 48·27-s + 24·29-s − 112·36-s + 48·39-s − 76·43-s + 40·48-s + 49-s − 16·51-s − 24·52-s + 10·53-s + 26·61-s + 6·64-s + 8·68-s + 192·69-s − 120·75-s − 10·79-s + 6·81-s + 192·87-s + ⋯ |
| L(s) = 1 | + 4.61·3-s − 2·4-s + 28/3·9-s − 9.23·12-s + 1.66·13-s + 5/4·16-s − 0.485·17-s + 5.00·23-s − 3·25-s + 9.23·27-s + 4.45·29-s − 18.6·36-s + 7.68·39-s − 11.5·43-s + 5.77·48-s + 1/7·49-s − 2.24·51-s − 3.32·52-s + 1.37·53-s + 3.32·61-s + 3/4·64-s + 0.970·68-s + 23.1·69-s − 13.8·75-s − 1.12·79-s + 2/3·81-s + 20.5·87-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{16} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{16} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(3^{16} \cdot 7^{16} \cdot 13^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(260037.\) |
| Root analytic conductor: | \(1.47645\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 3^{16} \cdot 7^{16} \cdot 13^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(2.535307593\) |
| \(L(\frac12)\) | \(\approx\) | \(2.535307593\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( ( 1 - T + T^{2} )^{8} \) |
| 7 | \( 1 - T^{2} + 67 T^{4} + 83 T^{6} + 2797 T^{8} + 83 p^{2} T^{10} + 67 p^{4} T^{12} - p^{6} T^{14} + p^{8} T^{16} \) | |
| 13 | \( ( 1 - 3 T - 12 T^{2} + 3 p T^{3} + 6 p T^{4} + 3 p^{2} T^{5} - 12 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| good | 2 | \( 1 + p^{2} T^{2} + 11 T^{4} + 9 p T^{6} + 21 T^{8} + 9 T^{10} + 145 T^{12} + 67 p^{3} T^{14} + 347 p^{2} T^{16} + 67 p^{5} T^{18} + 145 p^{4} T^{20} + 9 p^{6} T^{22} + 21 p^{8} T^{24} + 9 p^{11} T^{26} + 11 p^{12} T^{28} + p^{16} T^{30} + p^{16} T^{32} \) |
| 5 | \( 1 + 3 p T^{2} + 143 T^{4} + 1058 T^{6} + 1234 p T^{8} + 31282 T^{10} + 137998 T^{12} + 587229 T^{14} + 2794921 T^{16} + 587229 p^{2} T^{18} + 137998 p^{4} T^{20} + 31282 p^{6} T^{22} + 1234 p^{9} T^{24} + 1058 p^{10} T^{26} + 143 p^{12} T^{28} + 3 p^{15} T^{30} + p^{16} T^{32} \) | |
| 11 | \( 1 + 59 T^{2} + 1809 T^{4} + 37652 T^{6} + 602090 T^{8} + 8177698 T^{10} + 102407022 T^{12} + 1228232491 T^{14} + 13967883517 T^{16} + 1228232491 p^{2} T^{18} + 102407022 p^{4} T^{20} + 8177698 p^{6} T^{22} + 602090 p^{8} T^{24} + 37652 p^{10} T^{26} + 1809 p^{12} T^{28} + 59 p^{14} T^{30} + p^{16} T^{32} \) | |
| 17 | \( ( 1 + T - 45 T^{2} - 6 p T^{3} + 1060 T^{4} + 2698 T^{5} - 14366 T^{6} - 24861 T^{7} + 186013 T^{8} - 24861 p T^{9} - 14366 p^{2} T^{10} + 2698 p^{3} T^{11} + 1060 p^{4} T^{12} - 6 p^{6} T^{13} - 45 p^{6} T^{14} + p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 19 | \( 1 + 40 T^{2} + 206 T^{4} - 2176 T^{6} + 155113 T^{8} + 486600 T^{10} - 93205730 T^{12} - 1774284720 T^{14} - 26464419852 T^{16} - 1774284720 p^{2} T^{18} - 93205730 p^{4} T^{20} + 486600 p^{6} T^{22} + 155113 p^{8} T^{24} - 2176 p^{10} T^{26} + 206 p^{12} T^{28} + 40 p^{14} T^{30} + p^{16} T^{32} \) | |
| 23 | \( ( 1 - 12 T + 25 T^{2} + 122 T^{3} + 19 p T^{4} - 3874 T^{5} - 13829 T^{6} - 87666 T^{7} + 1490487 T^{8} - 87666 p T^{9} - 13829 p^{2} T^{10} - 3874 p^{3} T^{11} + 19 p^{5} T^{12} + 122 p^{5} T^{13} + 25 p^{6} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 29 | \( ( 1 - 6 T + 75 T^{2} - 379 T^{3} + 2564 T^{4} - 379 p T^{5} + 75 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{4} \) | |
| 31 | \( 1 + 117 T^{2} + 6910 T^{4} + 217005 T^{6} + 2289747 T^{8} - 101224854 T^{10} - 4410529033 T^{12} - 47536055793 T^{14} + 479480828936 T^{16} - 47536055793 p^{2} T^{18} - 4410529033 p^{4} T^{20} - 101224854 p^{6} T^{22} + 2289747 p^{8} T^{24} + 217005 p^{10} T^{26} + 6910 p^{12} T^{28} + 117 p^{14} T^{30} + p^{16} T^{32} \) | |
| 37 | \( 1 + 90 T^{2} + 2087 T^{4} - 548 T^{6} + 522308 T^{8} - 26698042 T^{10} - 5154920486 T^{12} - 165219374757 T^{14} - 3436387560818 T^{16} - 165219374757 p^{2} T^{18} - 5154920486 p^{4} T^{20} - 26698042 p^{6} T^{22} + 522308 p^{8} T^{24} - 548 p^{10} T^{26} + 2087 p^{12} T^{28} + 90 p^{14} T^{30} + p^{16} T^{32} \) | |
| 41 | \( ( 1 - 116 T^{2} + 8329 T^{4} - 459513 T^{6} + 20665754 T^{8} - 459513 p^{2} T^{10} + 8329 p^{4} T^{12} - 116 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 43 | \( ( 1 + 19 T + 293 T^{2} + 2737 T^{3} + 21669 T^{4} + 2737 p T^{5} + 293 p^{2} T^{6} + 19 p^{3} T^{7} + p^{4} T^{8} )^{4} \) | |
| 47 | \( 1 + 216 T^{2} + 24836 T^{4} + 1840592 T^{6} + 92148170 T^{8} + 2660106664 T^{10} - 18316108400 T^{12} - 7232631680472 T^{14} - 461812139242541 T^{16} - 7232631680472 p^{2} T^{18} - 18316108400 p^{4} T^{20} + 2660106664 p^{6} T^{22} + 92148170 p^{8} T^{24} + 1840592 p^{10} T^{26} + 24836 p^{12} T^{28} + 216 p^{14} T^{30} + p^{16} T^{32} \) | |
| 53 | \( ( 1 - 5 T - 27 T^{2} + 528 T^{3} - 5812 T^{4} + 24090 T^{5} + 2570 T^{6} - 1882121 T^{7} + 26580441 T^{8} - 1882121 p T^{9} + 2570 p^{2} T^{10} + 24090 p^{3} T^{11} - 5812 p^{4} T^{12} + 528 p^{5} T^{13} - 27 p^{6} T^{14} - 5 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 59 | \( 1 + 359 T^{2} + 67743 T^{4} + 9093794 T^{6} + 974334602 T^{8} + 87855580138 T^{10} + 6876020062278 T^{12} + 476817456211861 T^{14} + 29649688253330857 T^{16} + 476817456211861 p^{2} T^{18} + 6876020062278 p^{4} T^{20} + 87855580138 p^{6} T^{22} + 974334602 p^{8} T^{24} + 9093794 p^{10} T^{26} + 67743 p^{12} T^{28} + 359 p^{14} T^{30} + p^{16} T^{32} \) | |
| 61 | \( ( 1 - 13 T + 13 T^{2} - 980 T^{3} + 12580 T^{4} + 14836 T^{5} + 248732 T^{6} - 3114829 T^{7} - 19986519 T^{8} - 3114829 p T^{9} + 248732 p^{2} T^{10} + 14836 p^{3} T^{11} + 12580 p^{4} T^{12} - 980 p^{5} T^{13} + 13 p^{6} T^{14} - 13 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 67 | \( 1 + 84 T^{2} + 1315 T^{4} - 183330 T^{6} - 32623050 T^{8} - 2917096854 T^{10} + 613887836 T^{12} + 12685638348495 T^{14} + 1125042573121574 T^{16} + 12685638348495 p^{2} T^{18} + 613887836 p^{4} T^{20} - 2917096854 p^{6} T^{22} - 32623050 p^{8} T^{24} - 183330 p^{10} T^{26} + 1315 p^{12} T^{28} + 84 p^{14} T^{30} + p^{16} T^{32} \) | |
| 71 | \( ( 1 - 395 T^{2} + 75202 T^{4} - 9130962 T^{6} + 770369492 T^{8} - 9130962 p^{2} T^{10} + 75202 p^{4} T^{12} - 395 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 73 | \( 1 + 501 T^{2} + 1870 p T^{4} + 26117157 T^{6} + 3888373767 T^{8} + 475485817146 T^{10} + 49283306392931 T^{12} + 4411000559858643 T^{14} + 344286885660717104 T^{16} + 4411000559858643 p^{2} T^{18} + 49283306392931 p^{4} T^{20} + 475485817146 p^{6} T^{22} + 3888373767 p^{8} T^{24} + 26117157 p^{10} T^{26} + 1870 p^{13} T^{28} + 501 p^{14} T^{30} + p^{16} T^{32} \) | |
| 79 | \( ( 1 + 5 T + 14 T^{2} + 1209 T^{3} + 2979 T^{4} - 52200 T^{5} + 697849 T^{6} + 3795697 T^{7} - 41353954 T^{8} + 3795697 p T^{9} + 697849 p^{2} T^{10} - 52200 p^{3} T^{11} + 2979 p^{4} T^{12} + 1209 p^{5} T^{13} + 14 p^{6} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 83 | \( ( 1 - 432 T^{2} + 95455 T^{4} - 13529669 T^{6} + 1334072306 T^{8} - 13529669 p^{2} T^{10} + 95455 p^{4} T^{12} - 432 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 89 | \( 1 + 219 T^{2} + 12349 T^{4} - 1234800 T^{6} - 144130086 T^{8} + 11299293930 T^{10} + 1966901053526 T^{12} - 38829677081553 T^{14} - 18396217911938563 T^{16} - 38829677081553 p^{2} T^{18} + 1966901053526 p^{4} T^{20} + 11299293930 p^{6} T^{22} - 144130086 p^{8} T^{24} - 1234800 p^{10} T^{26} + 12349 p^{12} T^{28} + 219 p^{14} T^{30} + p^{16} T^{32} \) | |
| 97 | \( ( 1 - 709 T^{2} + 225578 T^{4} - 42085680 T^{6} + 5036807268 T^{8} - 42085680 p^{2} T^{10} + 225578 p^{4} T^{12} - 709 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
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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
−3.34062328139357157913535977945, −3.23518659255039279977653246740, −3.22675539328352961943907000463, −3.15752184381676238767008287889, −3.11347772500649502753175431082, −3.05055282973581693236653062459, −2.85903853896907827964962056237, −2.78827469068904308508432167909, −2.70378694812219430235124717395, −2.61416709358851542984334031210, −2.53732330226061284962975848492, −2.45122423745897845485936529751, −2.26768468243252895872613710469, −2.13488824929350504936725926476, −1.98427315251744259941396036564, −1.89954046425184566574788998369, −1.82569988403943421046440141618, −1.70422901655639859042344922035, −1.67374415519274240935769788802, −1.39815348508951962501975318259, −1.21836585762299178099166500095, −1.12070913300373515290709604161, −0.846139002071483651440850064442, −0.795806932882091694275170256939, −0.13460339606015733152637379113, 0.13460339606015733152637379113, 0.795806932882091694275170256939, 0.846139002071483651440850064442, 1.12070913300373515290709604161, 1.21836585762299178099166500095, 1.39815348508951962501975318259, 1.67374415519274240935769788802, 1.70422901655639859042344922035, 1.82569988403943421046440141618, 1.89954046425184566574788998369, 1.98427315251744259941396036564, 2.13488824929350504936725926476, 2.26768468243252895872613710469, 2.45122423745897845485936529751, 2.53732330226061284962975848492, 2.61416709358851542984334031210, 2.70378694812219430235124717395, 2.78827469068904308508432167909, 2.85903853896907827964962056237, 3.05055282973581693236653062459, 3.11347772500649502753175431082, 3.15752184381676238767008287889, 3.22675539328352961943907000463, 3.23518659255039279977653246740, 3.34062328139357157913535977945
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.