Properties

Degree 84
Conductor $ 43^{42} $
Sign $1$
Motivic weight 6
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s + 747·4-s − 3·5-s − 147·7-s − 5.72e3·9-s − 4.10e3·11-s − 2.24e3·12-s + 4.84e3·13-s + 9·15-s + 2.69e5·16-s + 3.64e3·17-s − 5.22e3·19-s − 2.24e3·20-s + 441·21-s + 6.57e3·23-s − 1.29e5·25-s + 1.71e4·27-s − 1.09e5·28-s + 9.53e4·29-s − 7.93e4·31-s + 1.23e4·33-s + 441·35-s − 4.27e6·36-s − 6.60e4·37-s − 1.45e4·39-s + 4.06e4·41-s − 3.68e5·43-s + ⋯
L(s)  = 1  − 1/9·3-s + 11.6·4-s − 0.0239·5-s − 3/7·7-s − 7.85·9-s − 3.08·11-s − 1.29·12-s + 2.20·13-s + 0.00266·15-s + 65.6·16-s + 0.741·17-s − 0.762·19-s − 0.280·20-s + 1/21·21-s + 0.540·23-s − 8.30·25-s + 0.872·27-s − 5.00·28-s + 3.91·29-s − 2.66·31-s + 0.342·33-s + 0.0102·35-s − 91.6·36-s − 1.30·37-s − 0.244·39-s + 0.589·41-s − 4.63·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(43^{42}\right)^{s/2} \, \Gamma_{\C}(s)^{42} \, L(s)\cr=\mathstrut & \,\Lambda(7-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(43^{42}\right)^{s/2} \, \Gamma_{\C}(s+3)^{42} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(84\)
\( N \)  =  \(43^{42}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(6\)
character  :  induced by $\chi_{43} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((84,\ 43^{42} ,\ ( \ : [3]^{42} ),\ 1 )\)
\(L(\frac{7}{2})\)  \(\approx\)  \(5.35406\)
\(L(\frac12)\)  \(\approx\)  \(5.35406\)
\(L(4)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \neq 43$,\(F_p(T)\) is a polynomial of degree 84. If $p = 43$, then $F_p(T)$ is a polynomial of degree at most 83.
$p$$F_p(T)$
bad43 \( 1 + 3.68e5T + 5.67e10T^{2} + 2.64e15T^{3} - 4.51e20T^{4} - 8.10e25T^{5} - 3.66e30T^{6} + 2.64e35T^{7} + 2.80e40T^{8} - 1.14e45T^{9} - 1.48e50T^{10} + 2.22e55T^{11} + 3.35e60T^{12}+O(T^{13}) \)
good2 \( 1 - 747 T^{2} + 288939 T^{4} - 38576177 p T^{6} + 15983661675 T^{8} - 2737088806491 T^{10} + 403067794168225 T^{12} - 1638807787290033 p^{5} T^{14} + 384249666040574925 p^{4} T^{16} - 20604542320091743739 p^{5} T^{18} + \)\(20\!\cdots\!79\)\( p^{5} T^{20} - \)\(59\!\cdots\!99\)\( p^{10} T^{22} + \)\(10\!\cdots\!31\)\( p^{9} T^{24} - \)\(17\!\cdots\!55\)\( p^{8} T^{26} + \)\(13\!\cdots\!61\)\( p^{8} T^{28} - \)\(33\!\cdots\!33\)\( p^{13} T^{30} + \)\(31\!\cdots\!21\)\( p^{16} T^{32} - \)\(17\!\cdots\!57\)\( p^{23} T^{34} + \)\(38\!\cdots\!41\)\( p^{28} T^{36} - \)\(10\!\cdots\!59\)\( p^{36} T^{38} + \)\(21\!\cdots\!99\)\( p^{41} T^{40} - \)\(10\!\cdots\!77\)\( p^{48} T^{42} + \)\(21\!\cdots\!99\)\( p^{53} T^{44} - \)\(10\!\cdots\!59\)\( p^{60} T^{46} + \)\(38\!\cdots\!41\)\( p^{64} T^{48} - \)\(17\!\cdots\!57\)\( p^{71} T^{50} + \)\(31\!\cdots\!21\)\( p^{76} T^{52} - \)\(33\!\cdots\!33\)\( p^{85} T^{54} + \)\(13\!\cdots\!61\)\( p^{92} T^{56} - \)\(17\!\cdots\!55\)\( p^{104} T^{58} + \)\(10\!\cdots\!31\)\( p^{117} T^{60} - \)\(59\!\cdots\!99\)\( p^{130} T^{62} + \)\(20\!\cdots\!79\)\( p^{137} T^{64} - 20604542320091743739 p^{149} T^{66} + 384249666040574925 p^{160} T^{68} - 1638807787290033 p^{173} T^{70} + 403067794168225 p^{180} T^{72} - 2737088806491 p^{192} T^{74} + 15983661675 p^{204} T^{76} - 38576177 p^{217} T^{78} + 288939 p^{228} T^{80} - 747 p^{240} T^{82} + p^{252} T^{84} \)
3 \( 1 + 3T + 5.73e3T^{2} + 1.71e4T^{3} + 1.51e7T^{4} + 7.55e7T^{5} + 2.44e10T^{6} + 2.24e11T^{7} + 2.65e13T^{8} + 4.14e14T^{9} + 2.09e16T^{10} + 4.70e17T^{11} + 1.36e19T^{12} + 2.94e20T^{13} + 9.14e21T^{14} + 3.17e22T^{15} + 6.05e24T^{16} - 9.82e25T^{17} + 2.34e27T^{18} - 5.91e28T^{19} - 6.33e29T^{20} + 1.39e31T^{21} - 9.07e32T^{22} + 2.22e34T^{23} + 5.98e35T^{24} - 8.68e36T^{25} + 1.45e39T^{26} - 2.05e40T^{27} + 8.55e41T^{28} - 7.25e42T^{29} - 1.10e44T^{30} + 3.53e45T^{31} - 4.15e47T^{32} - 9.36e46T^{33} - 1.43e50T^{34} - 7.05e51T^{35} + 1.15e53T^{36} - 6.52e54T^{37} + 1.05e56T^{38} - 6.07e56T^{39}+O(T^{40}) \)
5 \( 1 + 3T + 1.29e5T^{2} + 3.89e5T^{3} + 8.52e9T^{4} + 6.40e10T^{5} + 3.69e14T^{6} + 6.47e15T^{7} + 1.17e19T^{8} + 4.10e20T^{9} + 2.92e23T^{10} + 1.82e25T^{11} + 5.95e27T^{12} + 6.12e29T^{13} + 1.08e32T^{14} + 1.62e34T^{15} + 2.00e36T^{16} + 3.56e38T^{17} + 4.06e40T^{18} + 6.77e42T^{19} + 8.60e44T^{20} + 1.18e47T^{21} + 1.72e49T^{22} + 2.06e51T^{23} + 3.16e53T^{24} + 3.65e55T^{25} + 5.37e57T^{26} + 6.46e59T^{27}+O(T^{28}) \)
7 \( 1 + 147T + 1.25e6T^{2} + 1.83e8T^{3} + 7.61e11T^{4} + 1.23e14T^{5} + 3.00e17T^{6} + 5.83e19T^{7} + 8.90e22T^{8} + 2.12e25T^{9} + 2.17e28T^{10} + 6.21e30T^{11} + 4.67e33T^{12} + 1.50e36T^{13} + 9.27e38T^{14} + 3.18e41T^{15} + 1.73e44T^{16} + 6.08e46T^{17} + 3.07e49T^{18} + 1.08e52T^{19} + 5.19e54T^{20} + 1.84e57T^{21} + 8.41e59T^{22} + 2.98e62T^{23}+O(T^{24}) \)
11 \( 1 + 4.10e3T + 4.45e7T^{2} + 1.52e11T^{3} + 9.53e14T^{4} + 2.85e18T^{5} + 1.32e22T^{6} + 3.56e25T^{7} + 1.37e29T^{8} + 3.35e32T^{9} + 1.12e36T^{10} + 2.54e39T^{11} + 7.60e42T^{12} + 1.61e46T^{13} + 4.40e49T^{14} + 8.80e52T^{15} + 2.22e56T^{16} + 4.21e59T^{17} + 1.00e63T^{18}+O(T^{19}) \)
13 \( 1 - 4.84e3T - 3.71e7T^{2} + 1.71e11T^{3} + 8.15e14T^{4} - 2.88e18T^{5} - 1.42e22T^{6} + 3.04e25T^{7} + 2.04e29T^{8} - 2.03e32T^{9} - 2.36e36T^{10} + 3.20e38T^{11} + 2.26e43T^{12} + 1.35e46T^{13} - 1.83e50T^{14} - 2.39e53T^{15} + 1.27e57T^{16} + 2.64e60T^{17}+O(T^{18}) \)
17 \( 1 - 3.64e3T - 2.15e8T^{2} + 1.15e12T^{3} + 2.24e16T^{4} - 1.64e20T^{5} - 1.38e24T^{6} + 1.44e28T^{7} + 4.72e31T^{8} - 8.73e35T^{9} - 2.28e37T^{10} + 3.74e43T^{11} - 1.02e47T^{12} - 1.07e51T^{13} + 6.94e54T^{14} + 1.36e58T^{15} - 2.74e62T^{16}+O(T^{17}) \)
19 \( 1 + 5.22e3T + 5.18e8T^{2} + 2.66e12T^{3} + 1.33e17T^{4} + 6.82e20T^{5} + 2.28e25T^{6} + 1.18e29T^{7} + 2.94e33T^{8} + 1.58e37T^{9} + 3.07e41T^{10} + 1.75e45T^{11} + 2.73e49T^{12} + 1.66e53T^{13} + 2.13e57T^{14} + 1.38e61T^{15}+O(T^{16}) \)
23 \( 1 - 6.57e3T - 1.72e9T^{2} + 2.50e12T^{3} + 1.54e18T^{4} + 4.74e21T^{5} - 9.07e26T^{6} - 6.50e30T^{7} + 3.80e35T^{8} + 4.26e39T^{9} - 1.17e44T^{10} - 1.85e48T^{11} + 2.72e52T^{12} + 5.92e56T^{13} - 4.67e60T^{14}+O(T^{15}) \)
29 \( 1 - 9.53e4T + 1.07e10T^{2} - 7.33e14T^{3} + 5.01e19T^{4} - 2.70e24T^{5} + 1.40e29T^{6} - 6.28e33T^{7} + 2.68e38T^{8} - 1.02e43T^{9} + 3.73e47T^{10} - 1.23e52T^{11} + 3.89e56T^{12} - 1.12e61T^{13}+O(T^{14}) \)
31 \( 1 + 7.93e4T - 3.49e9T^{2} - 4.25e14T^{3} + 4.79e18T^{4} + 1.21e24T^{5} + 1.74e27T^{6} - 2.41e33T^{7} - 2.31e37T^{8} + 3.55e42T^{9} + 6.01e46T^{10} - 3.86e51T^{11} - 1.00e56T^{12} + 2.68e60T^{13}+O(T^{14}) \)
37 \( 1 + 6.60e4T + 2.54e10T^{2} + 1.58e15T^{3} + 3.21e20T^{4} + 1.81e25T^{5} + 2.60e30T^{6} + 1.27e35T^{7} + 1.49e40T^{8} + 5.93e44T^{9} + 6.21e49T^{10} + 1.69e54T^{11} + 1.87e59T^{12}+O(T^{13}) \)
41 \( 1 - 4.06e4T + 9.92e10T^{2} - 4.02e15T^{3} + 4.95e21T^{4} - 1.99e26T^{5} + 1.65e32T^{6} - 6.63e36T^{7} + 4.16e42T^{8} - 1.66e47T^{9} + 8.41e52T^{10} - 3.34e57T^{11} + 1.42e63T^{12}+O(T^{13}) \)
47 \( 1 + 2.47e5T + 2.56e11T^{2} + 6.20e16T^{3} + 3.32e22T^{4} + 7.75e27T^{5} + 2.90e33T^{6} + 6.47e38T^{7} + 1.91e44T^{8} + 4.05e49T^{9} + 1.01e55T^{10} + 2.03e60T^{11} + 4.48e65T^{12}+O(T^{13}) \)
53 \( 1 - 1.16e4T - 2.31e11T^{2} + 4.90e14T^{3} + 2.70e22T^{4} + 1.49e26T^{5} - 2.09e33T^{6} - 2.34e37T^{7} + 1.19e44T^{8} + 1.73e48T^{9} - 5.25e54T^{10} - 8.05e58T^{11}+O(T^{12}) \)
59 \( 1 + 6.49e5T + 1.00e12T^{2} + 5.51e17T^{3} + 4.84e23T^{4} + 2.35e29T^{5} + 1.53e35T^{6} + 6.76e40T^{7} + 3.63e46T^{8} + 1.47e52T^{9} + 6.83e57T^{10} + 2.57e63T^{11}+O(T^{12}) \)
61 \( 1 + 5.22e4T + 4.72e11T^{2} + 2.46e16T^{3} + 1.10e23T^{4} + 6.66e27T^{5} + 1.71e34T^{6} + 1.42e39T^{7} + 2.03e45T^{8} + 2.50e50T^{9} + 1.99e56T^{10} + 3.59e61T^{11}+O(T^{12}) \)
67 \( 1 + 1.73e5T - 1.00e12T^{2} - 1.34e17T^{3} + 5.05e23T^{4} + 4.83e28T^{5} - 1.69e35T^{6} - 1.05e40T^{7} + 4.31e46T^{8} + 1.51e51T^{9} - 8.85e57T^{10} - 1.39e62T^{11}+O(T^{12}) \)
71 \( 1 - 1.67e6T + 2.57e12T^{2} - 2.74e18T^{3} + 2.64e24T^{4} - 2.14e30T^{5} + 1.59e36T^{6} - 1.06e42T^{7} + 6.60e47T^{8} - 3.78e53T^{9} + 2.04e59T^{10} - 1.03e65T^{11}+O(T^{12}) \)
73 \( 1 + 2.64e6T + 5.16e12T^{2} + 7.48e18T^{3} + 9.22e24T^{4} + 9.86e30T^{5} + 9.48e36T^{6} + 8.33e42T^{7} + 6.78e48T^{8} + 5.17e54T^{9} + 3.72e60T^{10} + 2.54e66T^{11}+O(T^{12}) \)
79 \( 1 - 1.89e6T + 1.46e11T^{2} + 2.38e18T^{3} - 1.81e24T^{4} - 7.89e29T^{5} + 1.68e36T^{6} - 4.58e41T^{7} - 6.30e47T^{8} + 5.34e53T^{9} - 3.47e57T^{10}+O(T^{11}) \)
83 \( 1 - 1.58e6T - 2.86e12T^{2} + 5.58e18T^{3} + 4.04e24T^{4} - 1.00e31T^{5} - 3.80e36T^{6} + 1.23e43T^{7} + 2.72e48T^{8} - 1.16e55T^{9} - 1.63e60T^{10}+O(T^{11}) \)
89 \( 1 - 2.73e6T + 8.42e12T^{2} - 1.62e19T^{3} + 3.06e25T^{4} - 4.67e31T^{5} + 6.82e37T^{6} - 8.79e43T^{7} + 1.08e50T^{8} - 1.22e56T^{9} + 1.32e62T^{10}+O(T^{11}) \)
97 \( 1 - 5.26e6T + 2.54e13T^{2} - 7.85e19T^{3} + 2.25e26T^{4} - 5.08e32T^{5} + 1.10e39T^{6} - 2.00e45T^{7} + 3.61e51T^{8} - 5.70e57T^{9} + 9.09e63T^{10}+O(T^{11}) \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{84} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−1.62527753598556205534325938878, −1.61527687417267778772885296975, −1.59013634620543436638165611597, −1.52694279763287481352335941273, −1.49574881477229280027304285094, −1.45075063364234644928626121968, −1.33349574776233432371727580568, −1.33323539579376744681146282779, −1.24781001965846931696284210368, −1.07942562087223403732503570781, −1.02356687059889190350785519298, −0.910556586961999959624838230924, −0.906726618899109818000942011911, −0.76102889027898227518027293444, −0.59965362851439279646405698690, −0.54889944266192950816339370142, −0.44637450956478186238126633273, −0.34601058867296315927194144876, −0.32223682412904537949064935509, −0.32020147128093427364800200643, −0.31307186102944976767628802142, −0.18033646965567431426527912959, −0.14678293376653745041513948080, −0.13848413660799481942449066795, −0.04485297255182404206344950036, 0.04485297255182404206344950036, 0.13848413660799481942449066795, 0.14678293376653745041513948080, 0.18033646965567431426527912959, 0.31307186102944976767628802142, 0.32020147128093427364800200643, 0.32223682412904537949064935509, 0.34601058867296315927194144876, 0.44637450956478186238126633273, 0.54889944266192950816339370142, 0.59965362851439279646405698690, 0.76102889027898227518027293444, 0.906726618899109818000942011911, 0.910556586961999959624838230924, 1.02356687059889190350785519298, 1.07942562087223403732503570781, 1.24781001965846931696284210368, 1.33323539579376744681146282779, 1.33349574776233432371727580568, 1.45075063364234644928626121968, 1.49574881477229280027304285094, 1.52694279763287481352335941273, 1.59013634620543436638165611597, 1.61527687417267778772885296975, 1.62527753598556205534325938878

Graph of the $Z$-function along the critical line

Plot not available for L-functions of degree greater than 10.