Properties

Degree 56
Conductor $ 43^{28} $
Sign $1$
Motivic weight 8
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.44e3·4-s + 5.14e4·9-s + 4.53e3·11-s + 2.20e4·13-s + 8.96e5·16-s − 1.35e5·17-s − 1.84e5·23-s + 4.14e6·25-s − 1.10e5·31-s + 7.42e7·36-s + 1.30e6·41-s + 2.47e6·43-s + 6.54e6·44-s + 1.98e6·47-s + 7.29e7·49-s + 3.18e7·52-s + 2.39e7·53-s − 5.55e6·59-s + 2.87e8·64-s − 1.30e8·67-s − 1.94e8·68-s + 7.38e6·79-s + 1.17e9·81-s − 4.26e7·83-s − 2.65e8·92-s − 3.18e8·97-s + 2.33e8·99-s + ⋯
L(s)  = 1  + 5.63·4-s + 7.84·9-s + 0.309·11-s + 0.773·13-s + 13.6·16-s − 1.61·17-s − 0.659·23-s + 10.6·25-s − 0.119·31-s + 44.1·36-s + 0.460·41-s + 0.723·43-s + 1.74·44-s + 0.406·47-s + 12.6·49-s + 4.35·52-s + 3.03·53-s − 0.458·59-s + 17.1·64-s − 6.49·67-s − 9.11·68-s + 0.189·79-s + 27.3·81-s − 0.897·83-s − 3.71·92-s − 3.60·97-s + 2.43·99-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(43^{28}\right)^{s/2} \, \Gamma_{\C}(s)^{28} \, L(s)\cr=\mathstrut & \,\Lambda(9-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(43^{28}\right)^{s/2} \, \Gamma_{\C}(s+4)^{28} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(56\)
\( N \)  =  \(43^{28}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(8\)
character  :  induced by $\chi_{43} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((56,\ 43^{28} ,\ ( \ : [4]^{28} ),\ 1 )\)
\(L(\frac{9}{2})\)  \(\approx\)  \(433.341\)
\(L(\frac12)\)  \(\approx\)  \(433.341\)
\(L(5)\)   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 56. If $p = 43$, then $F_p(T)$ is a polynomial of degree at most 55.
$p$$F_p(T)$
bad43 \( 1 - 2.47e6T + 1.23e13T^{2} - 7.87e19T^{3} + 1.85e26T^{4} - 9.47e32T^{5} + 3.04e39T^{6} - 1.18e46T^{7} + 4.38e52T^{8} - 8.39e58T^{9}+O(T^{10}) \)
good2 \( 1 - 721 p T^{2} + 1182405 T^{4} - 87446783 p^{3} T^{6} + 2567662221 p^{7} T^{8} - 3970652584683 p^{5} T^{10} + 2593131978297831 p^{4} T^{12} - 91450005897116683 p^{7} T^{14} + 2905251387945617193 p^{10} T^{16} - 89919085930630669323 p^{13} T^{18} + \)\(30\!\cdots\!95\)\( p^{16} T^{20} - \)\(11\!\cdots\!01\)\( p^{19} T^{22} + \)\(47\!\cdots\!87\)\( p^{22} T^{24} - \)\(18\!\cdots\!25\)\( p^{25} T^{26} + \)\(30\!\cdots\!15\)\( p^{29} T^{28} - \)\(18\!\cdots\!25\)\( p^{41} T^{30} + \)\(47\!\cdots\!87\)\( p^{54} T^{32} - \)\(11\!\cdots\!01\)\( p^{67} T^{34} + \)\(30\!\cdots\!95\)\( p^{80} T^{36} - 89919085930630669323 p^{93} T^{38} + 2905251387945617193 p^{106} T^{40} - 91450005897116683 p^{119} T^{42} + 2593131978297831 p^{132} T^{44} - 3970652584683 p^{149} T^{46} + 2567662221 p^{167} T^{48} - 87446783 p^{179} T^{50} + 1182405 p^{192} T^{52} - 721 p^{209} T^{54} + p^{224} T^{56} \)
3 \( 1 - 17159 p T^{2} + 1471797764 T^{4} - 30307288988912 T^{6} + 166208488362704131 p T^{8} - \)\(25\!\cdots\!14\)\( p^{3} T^{10} + \)\(31\!\cdots\!99\)\( p^{3} T^{12} - \)\(11\!\cdots\!76\)\( p^{4} T^{14} + \)\(12\!\cdots\!71\)\( p^{6} T^{16} - \)\(11\!\cdots\!20\)\( p^{6} T^{18} + \)\(31\!\cdots\!92\)\( p^{7} T^{20} - \)\(27\!\cdots\!57\)\( p^{9} T^{22} + \)\(25\!\cdots\!61\)\( p^{13} T^{24} - \)\(22\!\cdots\!82\)\( p^{17} T^{26} + \)\(18\!\cdots\!22\)\( p^{21} T^{28} - \)\(22\!\cdots\!82\)\( p^{33} T^{30} + \)\(25\!\cdots\!61\)\( p^{45} T^{32} - \)\(27\!\cdots\!57\)\( p^{57} T^{34} + \)\(31\!\cdots\!92\)\( p^{71} T^{36} - \)\(11\!\cdots\!20\)\( p^{86} T^{38} + \)\(12\!\cdots\!71\)\( p^{102} T^{40} - \)\(11\!\cdots\!76\)\( p^{116} T^{42} + \)\(31\!\cdots\!99\)\( p^{131} T^{44} - \)\(25\!\cdots\!14\)\( p^{147} T^{46} + 166208488362704131 p^{161} T^{48} - 30307288988912 p^{176} T^{50} + 1471797764 p^{192} T^{52} - 17159 p^{209} T^{54} + p^{224} T^{56} \)
5 \( 1 - 4.14e6T^{2} + 8.81e12T^{4} - 1.26e19T^{6} + 1.37e25T^{8} - 1.19e31T^{10} + 8.74e36T^{12} - 5.53e42T^{14} + 3.13e48T^{16} - 1.62e54T^{18} + 7.90e59T^{20}+O(T^{22}) \)
7 \( 1 - 7.29e7T^{2} + 2.70e15T^{4} - 6.76e22T^{6} + 1.28e30T^{8} - 1.97e37T^{10} + 2.54e44T^{12} - 2.85e51T^{14} + 2.83e58T^{16}+O(T^{18}) \)
11 \( 1 - 4.53e3T + 2.78e9T^{2} - 1.46e13T^{3} + 3.92e18T^{4} - 2.27e22T^{5} + 3.75e27T^{6} - 2.33e31T^{7} + 2.74e36T^{8} - 1.78e40T^{9} + 1.63e45T^{10} - 1.09e49T^{11} + 8.24e53T^{12} - 5.63e57T^{13} + 3.61e62T^{14}+O(T^{15}) \)
13 \( 1 - 2.20e4T + 1.36e10T^{2} - 2.83e14T^{3} + 9.28e19T^{4} - 1.84e24T^{5} + 4.22e29T^{6} - 8.03e33T^{7} + 1.44e39T^{8} - 2.64e43T^{9} + 3.94e48T^{10} - 6.95e52T^{11} + 8.95e57T^{12} - 1.52e62T^{13}+O(T^{14}) \)
17 \( 1 + 1.35e5T + 9.16e10T^{2} + 1.06e16T^{3} + 4.08e21T^{4} + 4.08e26T^{5} + 1.17e32T^{6} + 9.99e36T^{7} + 2.44e42T^{8} + 1.72e47T^{9} + 3.92e52T^{10} + 2.16e57T^{11} + 5.05e62T^{12}+O(T^{13}) \)
19 \( 1 - 2.18e11T^{2} + 2.47e22T^{4} - 1.92e33T^{6} + 1.14e44T^{8} - 5.56e54T^{10} + 2.28e65T^{12}+O(T^{13}) \)
23 \( 1 + 1.84e5T + 1.02e12T^{2} + 1.73e17T^{3} + 5.48e23T^{4} + 8.61e28T^{5} + 2.01e35T^{6} + 2.95e40T^{7} + 5.65e46T^{8} + 7.82e51T^{9} + 1.29e58T^{10} + 1.69e63T^{11}+O(T^{12}) \)
29 \( 1 - 5.56e12T^{2} + 1.54e25T^{4} - 2.87e37T^{6} + 4.06e49T^{8} - 4.66e61T^{10}+O(T^{11}) \)
31 \( 1 + 1.10e5T + 1.85e13T^{2} + 1.69e18T^{3} + 1.70e26T^{4} + 1.27e31T^{5} + 1.01e39T^{6} + 6.17e43T^{7} + 4.49e51T^{8} + 2.16e56T^{9} + 1.55e64T^{10}+O(T^{11}) \)
37 \( 1 - 4.53e13T^{2} + 1.02e27T^{4} - 1.52e40T^{6} + 1.68e53T^{8} - 1.46e66T^{10}+O(T^{11}) \)
41 \( 1 - 1.30e6T + 9.14e13T^{2} - 1.10e20T^{3} + 4.05e27T^{4} - 4.86e33T^{5} + 1.17e41T^{6} - 1.51e47T^{7} + 2.50e54T^{8} - 3.74e60T^{9}+O(T^{10}) \)
47 \( 1 - 1.98e6T + 4.22e14T^{2} - 7.36e20T^{3} + 8.74e28T^{4} - 1.28e35T^{5} + 1.18e43T^{6} - 1.38e49T^{7} + 1.18e57T^{8} - 9.86e62T^{9}+O(T^{10}) \)
53 \( 1 - 2.39e7T + 1.31e15T^{2} - 2.74e22T^{3} + 8.49e29T^{4} - 1.56e37T^{5} + 3.56e44T^{6} - 5.87e51T^{7} + 1.09e59T^{8} - 1.63e66T^{9}+O(T^{10}) \)
59 \( 1 + 5.55e6T + 1.75e15T^{2} + 1.11e22T^{3} + 1.58e30T^{4} + 1.08e37T^{5} + 9.90e44T^{6} + 7.00e51T^{7} + 4.76e59T^{8} + 3.42e66T^{9}+O(T^{10}) \)
61 \( 1 - 3.72e15T^{2} + 6.89e30T^{4} - 8.45e45T^{6} + 7.69e60T^{8}+O(T^{9}) \)
67 \( 1 + 1.30e8T + 1.33e16T^{2} + 9.80e23T^{3} + 6.26e31T^{4} + 3.43e39T^{5} + 1.70e47T^{6} + 7.72e54T^{7} + 3.25e62T^{8}+O(T^{9}) \)
71 \( 1 - 8.44e15T^{2} + 3.52e31T^{4} - 9.74e46T^{6} + 2.01e62T^{8}+O(T^{9}) \)
73 \( 1 - 1.17e16T^{2} + 6.77e31T^{4} - 2.56e47T^{6} + 7.20e62T^{8}+O(T^{9}) \)
79 \( 1 - 7.38e6T + 2.39e16T^{2} + 5.29e22T^{3} + 2.84e32T^{4} + 3.18e39T^{5} + 2.25e48T^{6} + 4.36e55T^{7} + 1.35e64T^{8}+O(T^{9}) \)
83 \( 1 + 4.26e7T + 3.00e16T^{2} + 9.02e23T^{3} + 4.54e32T^{4} + 9.12e39T^{5} + 4.67e48T^{6} + 5.61e55T^{7} + 3.69e64T^{8}+O(T^{9}) \)
89 \( 1 - 7.53e16T^{2} + 2.79e33T^{4} - 6.84e49T^{6} + 1.23e66T^{8}+O(T^{9}) \)
97 \( 1 + 3.18e8T + 1.57e17T^{2} + 3.90e25T^{3} + 1.14e34T^{4} + 2.35e42T^{5} + 5.24e50T^{6} + 9.31e58T^{7} + 1.72e67T^{8}+O(T^{9}) \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{56} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−1.88422791563667698349543030388, −1.72088461779461490496802889897, −1.67974681237816226234701169223, −1.54982406239232026318982965904, −1.47419040653978984525823255733, −1.46616431157382584197890701007, −1.44396101608817631505358528992, −1.43238974412245003418600575797, −1.30201666516492542882992296176, −1.16312553609002665318239927140, −1.14107291358834451033293268753, −1.09346899731175003901002504101, −1.09069196321194895350632066452, −1.08888187820332121352924683995, −1.06562537123398876859868930986, −0.906506023504202229375683455935, −0.813803040489193088465429346494, −0.796167258052677394494734866817, −0.69651122987050327627663698659, −0.59520557299264531433888982469, −0.40260477446746573872087862472, −0.36134767044291127805143051293, −0.29083841340393805562481991794, −0.18068095920229381643747090808, −0.03167890074263516998636272734, 0.03167890074263516998636272734, 0.18068095920229381643747090808, 0.29083841340393805562481991794, 0.36134767044291127805143051293, 0.40260477446746573872087862472, 0.59520557299264531433888982469, 0.69651122987050327627663698659, 0.796167258052677394494734866817, 0.813803040489193088465429346494, 0.906506023504202229375683455935, 1.06562537123398876859868930986, 1.08888187820332121352924683995, 1.09069196321194895350632066452, 1.09346899731175003901002504101, 1.14107291358834451033293268753, 1.16312553609002665318239927140, 1.30201666516492542882992296176, 1.43238974412245003418600575797, 1.44396101608817631505358528992, 1.46616431157382584197890701007, 1.47419040653978984525823255733, 1.54982406239232026318982965904, 1.67974681237816226234701169223, 1.72088461779461490496802889897, 1.88422791563667698349543030388

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

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