Properties

Degree 68
Conductor $ 43^{34} $
Sign $1$
Motivic weight 10
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 9.33e3·4-s + 6.08e5·9-s − 2.18e5·11-s − 1.91e5·13-s + 3.99e7·16-s − 1.07e6·17-s + 8.91e6·23-s + 1.24e8·25-s − 5.50e7·31-s + 5.67e9·36-s + 8.03e7·41-s + 1.58e6·43-s − 2.03e9·44-s − 5.44e8·47-s + 3.56e9·49-s − 1.78e9·52-s − 9.15e8·53-s + 3.39e9·59-s + 1.02e11·64-s + 3.40e9·67-s − 9.98e9·68-s − 1.38e10·79-s + 1.78e11·81-s + 1.13e8·83-s + 8.31e10·92-s + 1.57e10·97-s − 1.32e11·99-s + ⋯
L(s)  = 1  + 9.11·4-s + 10.3·9-s − 1.35·11-s − 0.514·13-s + 38.0·16-s − 0.754·17-s + 1.38·23-s + 12.7·25-s − 1.92·31-s + 93.8·36-s + 0.693·41-s + 0.0107·43-s − 12.3·44-s − 2.37·47-s + 12.6·49-s − 4.68·52-s − 2.19·53-s + 4.74·59-s + 95.7·64-s + 2.52·67-s − 6.87·68-s − 4.50·79-s + 51.2·81-s + 0.0289·83-s + 12.6·92-s + 1.83·97-s − 13.9·99-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(43^{34}\right)^{s/2} \, \Gamma_{\C}(s)^{34} \, L(s)\cr=\mathstrut & \,\Lambda(11-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(43^{34}\right)^{s/2} \, \Gamma_{\C}(s+5)^{34} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(68\)
\( N \)  =  \(43^{34}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(10\)
character  :  induced by $\chi_{43} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((68,\ 43^{34} ,\ ( \ : [5]^{34} ),\ 1 )\)
\(L(\frac{11}{2})\)  \(\approx\)  \(20501.4\)
\(L(\frac12)\)  \(\approx\)  \(20501.4\)
\(L(6)\)   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 68. If $p = 43$, then $F_p(T)$ is a polynomial of degree at most 67.
$p$$F_p(T)$
bad43 \( 1 - 1.58e6T + 6.68e16T^{2} + 4.01e24T^{3} + 2.70e33T^{4} + 2.50e41T^{5} + 7.90e49T^{6} + 9.19e57T^{7}+O(T^{8}) \)
good2 \( 1 - 4665 p T^{2} + 47104309 T^{4} - 21203536681 p^{3} T^{6} + 7591817730651 p^{6} T^{8} - 73348803752593147 p^{4} T^{10} + \)\(15\!\cdots\!95\)\( p^{4} T^{12} - \)\(73\!\cdots\!89\)\( p^{6} T^{14} + \)\(15\!\cdots\!69\)\( p^{9} T^{16} - \)\(15\!\cdots\!33\)\( p^{13} T^{18} + \)\(74\!\cdots\!39\)\( p^{18} T^{20} - \)\(32\!\cdots\!15\)\( p^{23} T^{22} + \)\(10\!\cdots\!57\)\( p^{25} T^{24} - \)\(86\!\cdots\!25\)\( p^{29} T^{26} + \)\(16\!\cdots\!65\)\( p^{35} T^{28} - \)\(29\!\cdots\!75\)\( p^{41} T^{30} + \)\(99\!\cdots\!27\)\( p^{46} T^{32} - \)\(13\!\cdots\!43\)\( p^{49} T^{34} + \)\(99\!\cdots\!27\)\( p^{66} T^{36} - \)\(29\!\cdots\!75\)\( p^{81} T^{38} + \)\(16\!\cdots\!65\)\( p^{95} T^{40} - \)\(86\!\cdots\!25\)\( p^{109} T^{42} + \)\(10\!\cdots\!57\)\( p^{125} T^{44} - \)\(32\!\cdots\!15\)\( p^{143} T^{46} + \)\(74\!\cdots\!39\)\( p^{158} T^{48} - \)\(15\!\cdots\!33\)\( p^{173} T^{50} + \)\(15\!\cdots\!69\)\( p^{189} T^{52} - \)\(73\!\cdots\!89\)\( p^{206} T^{54} + \)\(15\!\cdots\!95\)\( p^{224} T^{56} - 73348803752593147 p^{244} T^{58} + 7591817730651 p^{266} T^{60} - 21203536681 p^{283} T^{62} + 47104309 p^{300} T^{64} - 4665 p^{321} T^{66} + p^{340} T^{68} \)
3 \( 1 - 6.08e5T^{2} + 1.91e11T^{4} - 4.18e16T^{6} + 7.09e21T^{8} - 9.96e26T^{10} + 1.20e32T^{12} - 1.27e37T^{14} + 1.20e42T^{16} - 1.03e47T^{18} + 8.12e51T^{20} - 5.84e56T^{22} + 3.89e61T^{24}+O(T^{25}) \)
5 \( 1 - 1.24e8T^{2} + 7.99e15T^{4} - 3.53e23T^{6} + 1.21e31T^{8} - 3.42e38T^{10} + 8.33e45T^{12} - 1.78e53T^{14} + 3.43e60T^{16}+O(T^{18}) \)
7 \( 1 - 3.56e9T^{2} + 6.50e18T^{4} - 8.11e27T^{6} + 7.77e36T^{8} - 6.09e45T^{10} + 4.07e54T^{12} - 2.38e63T^{14}+O(T^{15}) \)
11 \( 1 + 2.18e5T + 4.69e11T^{2} + 9.73e16T^{3} + 1.09e23T^{4} + 2.17e28T^{5} + 1.68e34T^{6} + 3.24e39T^{7} + 1.94e45T^{8} + 3.62e50T^{9} + 1.77e56T^{10} + 3.25e61T^{11}+O(T^{12}) \)
13 \( 1 + 1.91e5T + 2.00e12T^{2} + 4.57e17T^{3} + 2.08e24T^{4} + 5.17e29T^{5} + 1.49e36T^{6} + 3.81e41T^{7} + 8.14e47T^{8} + 2.09e53T^{9} + 3.62e59T^{10} + 9.18e64T^{11}+O(T^{12}) \)
17 \( 1 + 1.07e6T + 4.37e13T^{2} + 5.70e19T^{3} + 9.43e26T^{4} + 1.42e33T^{5} + 1.34e40T^{6} + 2.25e46T^{7} + 1.42e53T^{8} + 2.57e59T^{9} + 1.19e66T^{10}+O(T^{11}) \)
19 \( 1 - 9.85e13T^{2} + 4.99e27T^{4} - 1.72e41T^{6} + 4.54e54T^{8}+O(T^{10}) \)
23 \( 1 - 8.91e6T + 6.31e14T^{2} - 5.49e21T^{3} + 2.01e29T^{4} - 1.72e36T^{5} + 4.35e43T^{6} - 3.67e50T^{7} + 7.16e57T^{8} - 5.95e64T^{9}+O(T^{10}) \)
29 \( 1 - 6.82e15T^{2} + 2.34e31T^{4} - 5.37e46T^{6} + 9.27e61T^{8}+O(T^{9}) \)
31 \( 1 + 5.50e7T + 1.33e16T^{2} + 6.77e23T^{3} + 8.83e31T^{4} + 4.09e39T^{5} + 3.82e47T^{6} + 1.61e55T^{7} + 1.21e63T^{8}+O(T^{9}) \)
37 \( 1 - 7.60e16T^{2} + 2.91e33T^{4} - 7.49e49T^{6} + 1.45e66T^{8}+O(T^{9}) \)
41 \( 1 - 8.03e7T + 3.33e17T^{2} - 2.77e25T^{3} + 5.51e34T^{4} - 4.70e42T^{5} + 6.01e51T^{6} - 5.22e59T^{7} + 4.86e68T^{8}+O(T^{9}) \)
47 \( 1 + 5.44e8T + 1.18e18T^{2} + 6.05e26T^{3} + 6.97e35T^{4} + 3.34e44T^{5} + 2.70e53T^{6} + 1.21e62T^{7}+O(T^{8}) \)
53 \( 1 + 9.15e8T + 3.93e18T^{2} + 3.14e27T^{3} + 7.54e36T^{4} + 5.39e45T^{5} + 9.44e54T^{6} + 6.13e63T^{7}+O(T^{8}) \)
59 \( 1 - 3.39e9T + 1.47e19T^{2} - 3.48e28T^{3} + 9.08e37T^{4} - 1.68e47T^{5} + 3.32e56T^{6} - 5.16e65T^{7}+O(T^{8}) \)
61 \( 1 - 1.17e19T^{2} + 6.97e37T^{4} - 2.80e56T^{6}+O(T^{8}) \)
67 \( 1 - 3.40e9T + 4.75e19T^{2} - 1.45e29T^{3} + 1.10e39T^{4} - 3.10e48T^{5} + 1.69e58T^{6} - 4.36e67T^{7}+O(T^{8}) \)
71 \( 1 - 5.45e19T^{2} + 1.49e39T^{4} - 2.73e58T^{6}+O(T^{8}) \)
73 \( 1 - 8.28e19T^{2} + 3.42e39T^{4} - 9.46e58T^{6}+O(T^{8}) \)
79 \( 1 + 1.38e10T + 2.51e20T^{2} + 2.58e30T^{3} + 2.84e40T^{4} + 2.37e50T^{5} + 2.00e60T^{6}+O(T^{7}) \)
83 \( 1 - 1.13e8T + 1.90e20T^{2} + 1.21e28T^{3} + 1.84e40T^{4} + 3.29e48T^{5} + 1.20e60T^{6}+O(T^{7}) \)
89 \( 1 - 5.64e20T^{2} + 1.62e41T^{4} - 3.17e61T^{6}+O(T^{7}) \)
97 \( 1 - 1.57e10T + 1.43e21T^{2} - 2.18e31T^{3} + 1.04e42T^{4} - 1.53e52T^{5} + 5.16e62T^{6}+O(T^{7}) \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{68} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−1.37064642788171625998842932140, −1.32816977273461670570634966865, −1.26567612365102684636483348264, −1.24155269505815859623675883538, −1.21772605194688622356920837890, −1.21141277153675673249217649135, −1.16827822551596270692899609020, −1.15458822181859401286521677534, −1.02630456561437440512285416711, −0.965747897241147854764484274353, −0.910357235244336164511921072597, −0.877861647262700336472932672793, −0.876609248419893901335776567495, −0.810480869031902552764870608221, −0.69888453517419502166610947746, −0.69484646391007882738646138004, −0.68326096846780782324351477062, −0.64804307152826066463629428214, −0.33414145984390626745334166683, −0.33292563962522141714429183426, −0.32949994621304945916876958639, −0.32192040582889467752120179020, −0.20026816111082491605668673573, −0.18892030775990726682886710679, −0.03694832313173052287741712882, 0.03694832313173052287741712882, 0.18892030775990726682886710679, 0.20026816111082491605668673573, 0.32192040582889467752120179020, 0.32949994621304945916876958639, 0.33292563962522141714429183426, 0.33414145984390626745334166683, 0.64804307152826066463629428214, 0.68326096846780782324351477062, 0.69484646391007882738646138004, 0.69888453517419502166610947746, 0.810480869031902552764870608221, 0.876609248419893901335776567495, 0.877861647262700336472932672793, 0.910357235244336164511921072597, 0.965747897241147854764484274353, 1.02630456561437440512285416711, 1.15458822181859401286521677534, 1.16827822551596270692899609020, 1.21141277153675673249217649135, 1.21772605194688622356920837890, 1.24155269505815859623675883538, 1.26567612365102684636483348264, 1.32816977273461670570634966865, 1.37064642788171625998842932140

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

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