Dirichlet series
| L(s) = 1 | − 4-s − 2·5-s − 5·9-s + 6·19-s + 2·20-s + 14·25-s + 22·29-s − 22·31-s + 5·36-s + 18·41-s + 10·45-s − 24·49-s − 8·59-s + 20·61-s + 6·64-s + 6·71-s − 6·76-s − 22·79-s + 6·81-s + 24·89-s − 12·95-s − 14·100-s + 36·101-s + 20·109-s − 22·116-s + 22·124-s − 10·125-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 0.894·5-s − 5/3·9-s + 1.37·19-s + 0.447·20-s + 14/5·25-s + 4.08·29-s − 3.95·31-s + 5/6·36-s + 2.81·41-s + 1.49·45-s − 3.42·49-s − 1.04·59-s + 2.56·61-s + 3/4·64-s + 0.712·71-s − 0.688·76-s − 2.47·79-s + 2/3·81-s + 2.54·89-s − 1.23·95-s − 7/5·100-s + 3.58·101-s + 1.91·109-s − 2.04·116-s + 1.97·124-s − 0.894·125-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
| \(L(1)\) | \(\approx\) | \(5.425862931\) |
| \(L(\frac12)\) | \(\approx\) | \(5.425862931\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 5 | \( 1 + 2 T - 2 p T^{2} - 38 T^{3} + 19 T^{4} + 298 T^{5} + 84 p T^{6} - 752 T^{7} - 3519 T^{8} - 752 p T^{9} + 84 p^{3} T^{10} + 298 p^{3} T^{11} + 19 p^{4} T^{12} - 38 p^{5} T^{13} - 2 p^{7} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) |
| 11 | \( 1 \) | |
| good | 2 | \( 1 + T^{2} + T^{4} - 5 T^{6} - p T^{8} + 5 T^{10} + 19 T^{12} - 109 T^{14} - 307 T^{16} - 109 p^{2} T^{18} + 19 p^{4} T^{20} + 5 p^{6} T^{22} - p^{9} T^{24} - 5 p^{10} T^{26} + p^{12} T^{28} + p^{14} T^{30} + p^{16} T^{32} \) |
| 3 | \( 1 + 5 T^{2} + 19 T^{4} + 110 T^{6} + 355 T^{8} + 790 T^{10} + 3356 T^{12} + 3035 p T^{14} + 15559 T^{16} + 3035 p^{3} T^{18} + 3356 p^{4} T^{20} + 790 p^{6} T^{22} + 355 p^{8} T^{24} + 110 p^{10} T^{26} + 19 p^{12} T^{28} + 5 p^{14} T^{30} + p^{16} T^{32} \) | |
| 7 | \( 1 + 24 T^{2} + 289 T^{4} + 2139 T^{6} + 1333 p T^{8} + 5107 T^{10} - 249719 T^{12} - 1991908 T^{14} - 11588743 T^{16} - 1991908 p^{2} T^{18} - 249719 p^{4} T^{20} + 5107 p^{6} T^{22} + 1333 p^{9} T^{24} + 2139 p^{10} T^{26} + 289 p^{12} T^{28} + 24 p^{14} T^{30} + p^{16} T^{32} \) | |
| 13 | \( 1 + 17 T^{2} + 235 T^{4} + 1325 T^{6} + 2365 T^{8} - 95024 T^{10} - 1660478 T^{12} - 32759510 T^{14} - 962706325 T^{16} - 32759510 p^{2} T^{18} - 1660478 p^{4} T^{20} - 95024 p^{6} T^{22} + 2365 p^{8} T^{24} + 1325 p^{10} T^{26} + 235 p^{12} T^{28} + 17 p^{14} T^{30} + p^{16} T^{32} \) | |
| 17 | \( 1 + 60 T^{2} + 1839 T^{4} + 46850 T^{6} + 1030530 T^{8} + 20295630 T^{10} + 428422541 T^{12} + 8541377590 T^{14} + 149686339999 T^{16} + 8541377590 p^{2} T^{18} + 428422541 p^{4} T^{20} + 20295630 p^{6} T^{22} + 1030530 p^{8} T^{24} + 46850 p^{10} T^{26} + 1839 p^{12} T^{28} + 60 p^{14} T^{30} + p^{16} T^{32} \) | |
| 19 | \( ( 1 - 3 T - 10 T^{2} + 30 T^{3} + 410 T^{4} - 1569 T^{5} - 4808 T^{6} - 360 p T^{7} + 175795 T^{8} - 360 p^{2} T^{9} - 4808 p^{2} T^{10} - 1569 p^{3} T^{11} + 410 p^{4} T^{12} + 30 p^{5} T^{13} - 10 p^{6} T^{14} - 3 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 23 | \( ( 1 - 85 T^{2} + 3382 T^{4} - 84765 T^{6} + 1856153 T^{8} - 84765 p^{2} T^{10} + 3382 p^{4} T^{12} - 85 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 29 | \( ( 1 - 11 T + 2 T^{2} + 458 T^{3} - 1938 T^{4} + 6475 T^{5} - 39102 T^{6} - 10136 p T^{7} + 4147295 T^{8} - 10136 p^{2} T^{9} - 39102 p^{2} T^{10} + 6475 p^{3} T^{11} - 1938 p^{4} T^{12} + 458 p^{5} T^{13} + 2 p^{6} T^{14} - 11 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 31 | \( ( 1 + 11 T - 13 T^{2} - 511 T^{3} - 679 T^{4} + 7180 T^{5} + 3018 T^{6} + 56950 T^{7} + 1420917 T^{8} + 56950 p T^{9} + 3018 p^{2} T^{10} + 7180 p^{3} T^{11} - 679 p^{4} T^{12} - 511 p^{5} T^{13} - 13 p^{6} T^{14} + 11 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 37 | \( 1 + 75 T^{2} + 3874 T^{4} + 158265 T^{6} + 5594000 T^{8} + 63442080 T^{10} - 5394866374 T^{12} - 375745735260 T^{14} - 15694497490121 T^{16} - 375745735260 p^{2} T^{18} - 5394866374 p^{4} T^{20} + 63442080 p^{6} T^{22} + 5594000 p^{8} T^{24} + 158265 p^{10} T^{26} + 3874 p^{12} T^{28} + 75 p^{14} T^{30} + p^{16} T^{32} \) | |
| 41 | \( ( 1 - 9 T + 48 T^{2} - 679 T^{3} + 5472 T^{4} - 22520 T^{5} + 226802 T^{6} - 1446072 T^{7} + 5872595 T^{8} - 1446072 p T^{9} + 226802 p^{2} T^{10} - 22520 p^{3} T^{11} + 5472 p^{4} T^{12} - 679 p^{5} T^{13} + 48 p^{6} T^{14} - 9 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 43 | \( ( 1 - 171 T^{2} + 15457 T^{4} - 1005398 T^{6} + 49870565 T^{8} - 1005398 p^{2} T^{10} + 15457 p^{4} T^{12} - 171 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 47 | \( 1 - 25 T^{2} + 1117 T^{4} - 198615 T^{6} + 6904433 T^{8} + 115044500 T^{10} + 19824736484 T^{12} - 495801952450 T^{14} - 22620719690945 T^{16} - 495801952450 p^{2} T^{18} + 19824736484 p^{4} T^{20} + 115044500 p^{6} T^{22} + 6904433 p^{8} T^{24} - 198615 p^{10} T^{26} + 1117 p^{12} T^{28} - 25 p^{14} T^{30} + p^{16} T^{32} \) | |
| 53 | \( 1 + 180 T^{2} + 17799 T^{4} + 1527665 T^{6} + 2323005 p T^{8} + 8773109475 T^{10} + 562017285611 T^{12} + 33613598431360 T^{14} + 1866402926312089 T^{16} + 33613598431360 p^{2} T^{18} + 562017285611 p^{4} T^{20} + 8773109475 p^{6} T^{22} + 2323005 p^{9} T^{24} + 1527665 p^{10} T^{26} + 17799 p^{12} T^{28} + 180 p^{14} T^{30} + p^{16} T^{32} \) | |
| 59 | \( ( 1 + 4 T - 159 T^{2} - 725 T^{3} + 6203 T^{4} + 58835 T^{5} + 669309 T^{6} - 1637386 T^{7} - 77814037 T^{8} - 1637386 p T^{9} + 669309 p^{2} T^{10} + 58835 p^{3} T^{11} + 6203 p^{4} T^{12} - 725 p^{5} T^{13} - 159 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 61 | \( ( 1 - 10 T + 91 T^{2} - 260 T^{3} + 4015 T^{4} + 12160 T^{5} - 147431 T^{6} + 3126230 T^{7} - 9403816 T^{8} + 3126230 p T^{9} - 147431 p^{2} T^{10} + 12160 p^{3} T^{11} + 4015 p^{4} T^{12} - 260 p^{5} T^{13} + 91 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 67 | \( ( 1 - 387 T^{2} + 71530 T^{4} - 8300957 T^{6} + 662638709 T^{8} - 8300957 p^{2} T^{10} + 71530 p^{4} T^{12} - 387 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 71 | \( ( 1 - 3 T - 106 T^{2} + 1515 T^{3} + 138 T^{4} - 143130 T^{5} + 1043446 T^{6} + 4304718 T^{7} - 112579267 T^{8} + 4304718 p T^{9} + 1043446 p^{2} T^{10} - 143130 p^{3} T^{11} + 138 p^{4} T^{12} + 1515 p^{5} T^{13} - 106 p^{6} T^{14} - 3 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 73 | \( 1 + 196 T^{2} + 21309 T^{4} + 2516766 T^{6} + 226189156 T^{8} + 14814862678 T^{10} + 1167244718661 T^{12} + 79893370721248 T^{14} + 4620948706020267 T^{16} + 79893370721248 p^{2} T^{18} + 1167244718661 p^{4} T^{20} + 14814862678 p^{6} T^{22} + 226189156 p^{8} T^{24} + 2516766 p^{10} T^{26} + 21309 p^{12} T^{28} + 196 p^{14} T^{30} + p^{16} T^{32} \) | |
| 79 | \( ( 1 + 11 T - 43 T^{2} + 32 T^{3} + 11627 T^{4} + 40600 T^{5} + 107958 T^{6} + 3430939 T^{7} + 26806605 T^{8} + 3430939 p T^{9} + 107958 p^{2} T^{10} + 40600 p^{3} T^{11} + 11627 p^{4} T^{12} + 32 p^{5} T^{13} - 43 p^{6} T^{14} + 11 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 83 | \( 1 + 254 T^{2} + 40689 T^{4} + 5006169 T^{6} + 492199591 T^{8} + 46597831217 T^{10} + 4387443806691 T^{12} + 406499759250242 T^{14} + 36038717946937797 T^{16} + 406499759250242 p^{2} T^{18} + 4387443806691 p^{4} T^{20} + 46597831217 p^{6} T^{22} + 492199591 p^{8} T^{24} + 5006169 p^{10} T^{26} + 40689 p^{12} T^{28} + 254 p^{14} T^{30} + p^{16} T^{32} \) | |
| 89 | \( ( 1 - 6 T + 228 T^{2} - 1116 T^{3} + 26613 T^{4} - 1116 p T^{5} + 228 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{4} \) | |
| 97 | \( 1 + 202 T^{2} + 40183 T^{4} + 6163802 T^{6} + 851282206 T^{8} + 103015707980 T^{10} + 11980528019587 T^{12} + 1255618364764450 T^{14} + 127706802052258067 T^{16} + 1255618364764450 p^{2} T^{18} + 11980528019587 p^{4} T^{20} + 103015707980 p^{6} T^{22} + 851282206 p^{8} T^{24} + 6163802 p^{10} T^{26} + 40183 p^{12} T^{28} + 202 p^{14} T^{30} + p^{16} T^{32} \) | |
| show more | ||
| show less | ||
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.96048409652146450565601600119, −2.74882594616613600080796937183, −2.69960389900650506439907191269, −2.62853898682295476354411822717, −2.54294343402905140654200811401, −2.52435962631446559176119121830, −2.37855466272512933961432159777, −2.31172436826159021395698816411, −2.23394246504390749635200981595, −2.08258192502213500112854929252, −1.89630556805535016587133519743, −1.85689123365581559267247801507, −1.83291250549206385350278451608, −1.72556625823574166102679884195, −1.61229110421777190750484191760, −1.56358083884272239683303847496, −1.37247929252557407110064193124, −1.20461422105080390205688595045, −0.914514931985444580197476536087, −0.821801382694728616425038685287, −0.802606034019632984737393350859, −0.70436860513408337036462160609, −0.66522148712542128584505738348, −0.48665594234849940497550809211, −0.19375455867806498613132707077, 0.19375455867806498613132707077, 0.48665594234849940497550809211, 0.66522148712542128584505738348, 0.70436860513408337036462160609, 0.802606034019632984737393350859, 0.821801382694728616425038685287, 0.914514931985444580197476536087, 1.20461422105080390205688595045, 1.37247929252557407110064193124, 1.56358083884272239683303847496, 1.61229110421777190750484191760, 1.72556625823574166102679884195, 1.83291250549206385350278451608, 1.85689123365581559267247801507, 1.89630556805535016587133519743, 2.08258192502213500112854929252, 2.23394246504390749635200981595, 2.31172436826159021395698816411, 2.37855466272512933961432159777, 2.52435962631446559176119121830, 2.54294343402905140654200811401, 2.62853898682295476354411822717, 2.69960389900650506439907191269, 2.74882594616613600080796937183, 2.96048409652146450565601600119
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.