Dirichlet series
| L(s) = 1 | + 8·4-s + 24·9-s − 16·11-s + 24·16-s + 240·29-s − 144·31-s + 192·36-s − 128·44-s − 24·49-s − 96·59-s + 144·61-s + 128·71-s + 576·79-s + 494·81-s + 264·89-s − 384·99-s − 504·101-s + 240·109-s + 1.92e3·116-s + 648·121-s − 1.15e3·124-s + 127-s + 131-s + 137-s + 139-s + 576·144-s + 149-s + ⋯ |
| L(s) = 1 | + 2·4-s + 8/3·9-s − 1.45·11-s + 3/2·16-s + 8.27·29-s − 4.64·31-s + 16/3·36-s − 2.90·44-s − 0.489·49-s − 1.62·59-s + 2.36·61-s + 1.80·71-s + 7.29·79-s + 6.09·81-s + 2.96·89-s − 3.87·99-s − 4.99·101-s + 2.20·109-s + 16.5·116-s + 5.35·121-s − 9.29·124-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 4·144-s + 0.00671·149-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{32} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{32} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(2^{16} \cdot 5^{32} \cdot 7^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(4.68219\times 10^{15}\) |
| Root analytic conductor: | \(3.08817\) |
| Motivic weight: | \(2\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 2^{16} \cdot 5^{32} \cdot 7^{16} ,\ ( \ : [1]^{16} ),\ 1 )\) |
Particular Values
| \(L(\frac{3}{2})\) | \(\approx\) | \(16.58018706\) |
| \(L(\frac12)\) | \(\approx\) | \(16.58018706\) |
| \(L(2)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - p T^{2} + p^{2} T^{4} )^{4} \) |
| 5 | \( 1 \) | |
| 7 | \( 1 + 24 T^{2} + 2830 T^{4} - 3456 p^{2} T^{6} - 123 p^{5} T^{8} - 3456 p^{6} T^{10} + 2830 p^{8} T^{12} + 24 p^{12} T^{14} + p^{16} T^{16} \) | |
| good | 3 | \( 1 - 8 p T^{2} + 82 T^{4} + 16 p^{3} T^{6} + 9995 p T^{8} - 37360 p^{2} T^{10} - 610862 T^{12} - 2669816 p T^{14} + 345299524 T^{16} - 2669816 p^{5} T^{18} - 610862 p^{8} T^{20} - 37360 p^{14} T^{22} + 9995 p^{17} T^{24} + 16 p^{23} T^{26} + 82 p^{24} T^{28} - 8 p^{29} T^{30} + p^{32} T^{32} \) |
| 11 | \( ( 1 + 8 T - 228 T^{2} - 4688 T^{3} + 15770 T^{4} + 788088 T^{5} + 5396656 T^{6} - 54408328 T^{7} - 1014586317 T^{8} - 54408328 p^{2} T^{9} + 5396656 p^{4} T^{10} + 788088 p^{6} T^{11} + 15770 p^{8} T^{12} - 4688 p^{10} T^{13} - 228 p^{12} T^{14} + 8 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 13 | \( ( 1 + 296 T^{2} + 98076 T^{4} + 18130072 T^{6} + 3958421126 T^{8} + 18130072 p^{4} T^{10} + 98076 p^{8} T^{12} + 296 p^{12} T^{14} + p^{16} T^{16} )^{2} \) | |
| 17 | \( 1 - 1384 T^{2} + 964452 T^{4} - 455154608 T^{6} + 165719843210 T^{8} - 49965097508760 T^{10} + 12995011595054608 T^{12} - 3097329425143970968 T^{14} + \)\(80\!\cdots\!79\)\( T^{16} - 3097329425143970968 p^{4} T^{18} + 12995011595054608 p^{8} T^{20} - 49965097508760 p^{12} T^{22} + 165719843210 p^{16} T^{24} - 455154608 p^{20} T^{26} + 964452 p^{24} T^{28} - 1384 p^{28} T^{30} + p^{32} T^{32} \) | |
| 19 | \( ( 1 + 484 T^{2} + 193050 T^{4} - 2952000 T^{5} + 85792976 T^{6} - 988944000 T^{7} + 23497895219 T^{8} - 988944000 p^{2} T^{9} + 85792976 p^{4} T^{10} - 2952000 p^{6} T^{11} + 193050 p^{8} T^{12} + 484 p^{12} T^{14} + p^{16} T^{16} )^{2} \) | |
| 23 | \( 1 + 2088 T^{2} + 2390066 T^{4} + 1814786352 T^{6} + 902600966401 T^{8} + 170768598704784 T^{10} - 181274817006561166 T^{12} - \)\(23\!\cdots\!64\)\( T^{14} - \)\(15\!\cdots\!68\)\( T^{16} - \)\(23\!\cdots\!64\)\( p^{4} T^{18} - 181274817006561166 p^{8} T^{20} + 170768598704784 p^{12} T^{22} + 902600966401 p^{16} T^{24} + 1814786352 p^{20} T^{26} + 2390066 p^{24} T^{28} + 2088 p^{28} T^{30} + p^{32} T^{32} \) | |
| 29 | \( ( 1 - 60 T + 4458 T^{2} - 157200 T^{3} + 6089363 T^{4} - 157200 p^{2} T^{5} + 4458 p^{4} T^{6} - 60 p^{6} T^{7} + p^{8} T^{8} )^{4} \) | |
| 31 | \( ( 1 + 72 T + 3940 T^{2} + 159264 T^{3} + 5902026 T^{4} + 166723272 T^{5} + 3604087568 T^{6} + 80807933736 T^{7} + 2044410282323 T^{8} + 80807933736 p^{2} T^{9} + 3604087568 p^{4} T^{10} + 166723272 p^{6} T^{11} + 5902026 p^{8} T^{12} + 159264 p^{10} T^{13} + 3940 p^{12} T^{14} + 72 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 37 | \( 1 + 2568 T^{2} - 479644 T^{4} - 6542099088 T^{6} - 1820594205494 T^{8} + 7114863559599864 T^{10} + 394112005073079824 T^{12} + \)\(19\!\cdots\!96\)\( T^{14} + \)\(18\!\cdots\!47\)\( T^{16} + \)\(19\!\cdots\!96\)\( p^{4} T^{18} + 394112005073079824 p^{8} T^{20} + 7114863559599864 p^{12} T^{22} - 1820594205494 p^{16} T^{24} - 6542099088 p^{20} T^{26} - 479644 p^{24} T^{28} + 2568 p^{28} T^{30} + p^{32} T^{32} \) | |
| 41 | \( ( 1 - 3716 T^{2} + 10727274 T^{4} - 24786880336 T^{6} + 46146959629235 T^{8} - 24786880336 p^{4} T^{10} + 10727274 p^{8} T^{12} - 3716 p^{12} T^{14} + p^{16} T^{16} )^{2} \) | |
| 43 | \( ( 1 - 4424 T^{2} + 8444814 T^{4} - 9406036864 T^{6} + 10147485567875 T^{8} - 9406036864 p^{4} T^{10} + 8444814 p^{8} T^{12} - 4424 p^{12} T^{14} + p^{16} T^{16} )^{2} \) | |
| 47 | \( 1 - 3056 T^{2} - 10504100 T^{4} + 20193448928 T^{6} + 125665123657418 T^{8} - 115594758171979472 T^{10} - \)\(89\!\cdots\!72\)\( T^{12} + \)\(18\!\cdots\!20\)\( T^{14} + \)\(51\!\cdots\!87\)\( T^{16} + \)\(18\!\cdots\!20\)\( p^{4} T^{18} - \)\(89\!\cdots\!72\)\( p^{8} T^{20} - 115594758171979472 p^{12} T^{22} + 125665123657418 p^{16} T^{24} + 20193448928 p^{20} T^{26} - 10504100 p^{24} T^{28} - 3056 p^{28} T^{30} + p^{32} T^{32} \) | |
| 53 | \( 1 + 8120 T^{2} + 28255972 T^{4} + 54254817040 T^{6} + 29331001951946 T^{8} - 249001868039210680 T^{10} - \)\(14\!\cdots\!48\)\( T^{12} - \)\(54\!\cdots\!00\)\( T^{14} - \)\(16\!\cdots\!01\)\( T^{16} - \)\(54\!\cdots\!00\)\( p^{4} T^{18} - \)\(14\!\cdots\!48\)\( p^{8} T^{20} - 249001868039210680 p^{12} T^{22} + 29331001951946 p^{16} T^{24} + 54254817040 p^{20} T^{26} + 28255972 p^{24} T^{28} + 8120 p^{28} T^{30} + p^{32} T^{32} \) | |
| 59 | \( ( 1 + 48 T + 8612 T^{2} + 376512 T^{3} + 30329050 T^{4} + 1530261168 T^{5} + 120119465936 T^{6} + 6323967202032 T^{7} + 540612754864243 T^{8} + 6323967202032 p^{2} T^{9} + 120119465936 p^{4} T^{10} + 1530261168 p^{6} T^{11} + 30329050 p^{8} T^{12} + 376512 p^{10} T^{13} + 8612 p^{12} T^{14} + 48 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 61 | \( ( 1 - 72 T + 14578 T^{2} - 925200 T^{3} + 115490913 T^{4} - 7140338640 T^{5} + 10774109210 p T^{6} - 36982785415128 T^{7} + 2781811232132132 T^{8} - 36982785415128 p^{2} T^{9} + 10774109210 p^{5} T^{10} - 7140338640 p^{6} T^{11} + 115490913 p^{8} T^{12} - 925200 p^{10} T^{13} + 14578 p^{12} T^{14} - 72 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 67 | \( 1 + 15848 T^{2} + 131399250 T^{4} + 697062062512 T^{6} + 2204853654610721 T^{8} + 911591061628947600 T^{10} - \)\(47\!\cdots\!42\)\( T^{12} - \)\(42\!\cdots\!20\)\( T^{14} - \)\(22\!\cdots\!24\)\( T^{16} - \)\(42\!\cdots\!20\)\( p^{4} T^{18} - \)\(47\!\cdots\!42\)\( p^{8} T^{20} + 911591061628947600 p^{12} T^{22} + 2204853654610721 p^{16} T^{24} + 697062062512 p^{20} T^{26} + 131399250 p^{24} T^{28} + 15848 p^{28} T^{30} + p^{32} T^{32} \) | |
| 71 | \( ( 1 - 32 T + 8292 T^{2} + 81632 T^{3} + 28299158 T^{4} + 81632 p^{2} T^{5} + 8292 p^{4} T^{6} - 32 p^{6} T^{7} + p^{8} T^{8} )^{4} \) | |
| 73 | \( 1 - 33384 T^{2} + 596206436 T^{4} - 7449345901488 T^{6} + 72588435482668042 T^{8} - \)\(58\!\cdots\!64\)\( T^{10} + \)\(40\!\cdots\!52\)\( T^{12} - \)\(25\!\cdots\!28\)\( T^{14} + \)\(14\!\cdots\!71\)\( T^{16} - \)\(25\!\cdots\!28\)\( p^{4} T^{18} + \)\(40\!\cdots\!52\)\( p^{8} T^{20} - \)\(58\!\cdots\!64\)\( p^{12} T^{22} + 72588435482668042 p^{16} T^{24} - 7449345901488 p^{20} T^{26} + 596206436 p^{24} T^{28} - 33384 p^{28} T^{30} + p^{32} T^{32} \) | |
| 79 | \( ( 1 - 288 T + 30300 T^{2} - 2390592 T^{3} + 343035706 T^{4} - 36542007840 T^{5} + 2371926750768 T^{6} - 227910022136160 T^{7} + 23724159497509971 T^{8} - 227910022136160 p^{2} T^{9} + 2371926750768 p^{4} T^{10} - 36542007840 p^{6} T^{11} + 343035706 p^{8} T^{12} - 2390592 p^{10} T^{13} + 30300 p^{12} T^{14} - 288 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 83 | \( ( 1 + 49400 T^{2} + 1103892174 T^{4} + 14541036476800 T^{6} + 123368183239590851 T^{8} + 14541036476800 p^{4} T^{10} + 1103892174 p^{8} T^{12} + 49400 p^{12} T^{14} + p^{16} T^{16} )^{2} \) | |
| 89 | \( ( 1 - 132 T + 36502 T^{2} - 4051608 T^{3} + 716519265 T^{4} - 69860823192 T^{5} + 9283830649286 T^{6} - 786348344536908 T^{7} + 86046231197013188 T^{8} - 786348344536908 p^{2} T^{9} + 9283830649286 p^{4} T^{10} - 69860823192 p^{6} T^{11} + 716519265 p^{8} T^{12} - 4051608 p^{10} T^{13} + 36502 p^{12} T^{14} - 132 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 97 | \( ( 1 + 5880 T^{2} - 9876452 T^{4} + 849078891720 T^{6} + 12359117115903558 T^{8} + 849078891720 p^{4} T^{10} - 9876452 p^{8} T^{12} + 5880 p^{12} T^{14} + p^{16} T^{16} )^{2} \) | |
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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
−2.86566293829859243732706023490, −2.81821378180541288185718575001, −2.66262457604544823326272681495, −2.52634732183419397212451722097, −2.47571244716517223427524959822, −2.38158025422405719003735300059, −2.32900391326526425640107229125, −2.26977112348981070847760353843, −2.26004915055756421089629610010, −2.19878659632745835910576886060, −2.05944045376889521400682951163, −1.78070638924825637068740361358, −1.74999175877336005774895064520, −1.71626765746358717965740169653, −1.61145210644580424204135211979, −1.39290258270393203737487002257, −1.17298542043265438526344726022, −1.14298990716477432762930943855, −1.03716621297997148896736014560, −0.900122258463975655749099769394, −0.853044783783241402374009305457, −0.842772160355125727006001021027, −0.45958938295333924348742091038, −0.33502434469305831221515690635, −0.11368856219218936295824019241, 0.11368856219218936295824019241, 0.33502434469305831221515690635, 0.45958938295333924348742091038, 0.842772160355125727006001021027, 0.853044783783241402374009305457, 0.900122258463975655749099769394, 1.03716621297997148896736014560, 1.14298990716477432762930943855, 1.17298542043265438526344726022, 1.39290258270393203737487002257, 1.61145210644580424204135211979, 1.71626765746358717965740169653, 1.74999175877336005774895064520, 1.78070638924825637068740361358, 2.05944045376889521400682951163, 2.19878659632745835910576886060, 2.26004915055756421089629610010, 2.26977112348981070847760353843, 2.32900391326526425640107229125, 2.38158025422405719003735300059, 2.47571244716517223427524959822, 2.52634732183419397212451722097, 2.66262457604544823326272681495, 2.81821378180541288185718575001, 2.86566293829859243732706023490
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.