Properties

Degree 80
Conductor $ 3^{40} \cdot 5^{40} \cdot 7^{40} $
Sign $1$
Motivic weight 2
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 18·4-s + 6·9-s + 177·16-s − 56·19-s − 3·25-s − 444·31-s + 108·36-s + 78·49-s + 288·61-s + 1.16e3·64-s − 1.00e3·76-s − 340·79-s + 18·81-s − 54·100-s − 880·109-s − 1.39e3·121-s − 7.99e3·124-s + 127-s + 131-s + 137-s + 139-s + 1.06e3·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯
L(s)  = 1  + 9/2·4-s + 2/3·9-s + 11.0·16-s − 2.94·19-s − 0.119·25-s − 14.3·31-s + 3·36-s + 1.59·49-s + 4.72·61-s + 18.2·64-s − 13.2·76-s − 4.30·79-s + 2/9·81-s − 0.539·100-s − 8.07·109-s − 11.5·121-s − 64.4·124-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 59/8·144-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{40} \cdot 5^{40} \cdot 7^{40}\right)^{s/2} \, \Gamma_{\C}(s)^{40} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{40} \cdot 5^{40} \cdot 7^{40}\right)^{s/2} \, \Gamma_{\C}(s+1)^{40} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(80\)
\( N \)  =  \(3^{40} \cdot 5^{40} \cdot 7^{40}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(2\)
character  :  induced by $\chi_{105} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((80,\ 3^{40} \cdot 5^{40} \cdot 7^{40} ,\ ( \ : [1]^{40} ),\ 1 )\)
\(L(\frac{3}{2})\)  \(\approx\)  \(73.1684\)
\(L(\frac12)\)  \(\approx\)  \(73.1684\)
\(L(2)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;5,\;7\}$,\(F_p(T)\) is a polynomial of degree 80. If $p \in \{3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 79.
$p$$F_p(T)$
bad3 \( 1 - 2 p T^{2} + 2 p^{2} T^{4} + 3320 T^{6} - 7811 p T^{8} + 12448 p^{2} T^{10} + 4765156 T^{12} - 1489246 p^{3} T^{14} + 3148757 p^{4} T^{16} + 1712200 p^{7} T^{18} - 1946710 p^{9} T^{20} + 1712200 p^{11} T^{22} + 3148757 p^{12} T^{24} - 1489246 p^{15} T^{26} + 4765156 p^{16} T^{28} + 12448 p^{22} T^{30} - 7811 p^{25} T^{32} + 3320 p^{28} T^{34} + 2 p^{34} T^{36} - 2 p^{37} T^{38} + p^{40} T^{40} \)
5 \( 1 + 3 T^{2} - 1876 T^{4} + 27887 T^{6} + 2020859 T^{8} - 46470294 T^{10} - 822295884 T^{12} + 41970308926 T^{14} - 58620722179 T^{16} - 448293144991 p^{2} T^{18} + 101674281288 p^{5} T^{20} - 448293144991 p^{6} T^{22} - 58620722179 p^{8} T^{24} + 41970308926 p^{12} T^{26} - 822295884 p^{16} T^{28} - 46470294 p^{20} T^{30} + 2020859 p^{24} T^{32} + 27887 p^{28} T^{34} - 1876 p^{32} T^{36} + 3 p^{36} T^{38} + p^{40} T^{40} \)
7 \( ( 1 - 39 T^{2} + 2710 T^{4} - 13369 p T^{6} - 61115 p^{2} T^{8} + 6596 p^{4} T^{10} - 61115 p^{6} T^{12} - 13369 p^{9} T^{14} + 2710 p^{12} T^{16} - 39 p^{16} T^{18} + p^{20} T^{20} )^{2} \)
good2 \( ( 1 - 9 T^{2} + 33 T^{4} - p^{4} T^{6} - 495 T^{8} + 2407 T^{10} - 10223 T^{12} + p^{15} T^{14} + 15889 T^{16} - 957081 T^{18} + 6139553 T^{20} - 957081 p^{4} T^{22} + 15889 p^{8} T^{24} + p^{27} T^{26} - 10223 p^{16} T^{28} + 2407 p^{20} T^{30} - 495 p^{24} T^{32} - p^{32} T^{34} + 33 p^{32} T^{36} - 9 p^{36} T^{38} + p^{40} T^{40} )^{2} \)
11 \( ( 1 + 697 T^{2} + 225553 T^{4} + 51082906 T^{6} + 946812782 p T^{8} + 2012001751114 T^{10} + 347890639418511 T^{12} + 53495214068896647 T^{14} + 7639028381986105503 T^{16} + \)\(10\!\cdots\!92\)\( T^{18} + \)\(13\!\cdots\!88\)\( T^{20} + \)\(10\!\cdots\!92\)\( p^{4} T^{22} + 7639028381986105503 p^{8} T^{24} + 53495214068896647 p^{12} T^{26} + 347890639418511 p^{16} T^{28} + 2012001751114 p^{20} T^{30} + 946812782 p^{25} T^{32} + 51082906 p^{28} T^{34} + 225553 p^{32} T^{36} + 697 p^{36} T^{38} + p^{40} T^{40} )^{2} \)
13 \( ( 1 - 1036 T^{2} + 553818 T^{4} - 194899774 T^{6} + 49808418185 T^{8} - 9619291940612 T^{10} + 49808418185 p^{4} T^{12} - 194899774 p^{8} T^{14} + 553818 p^{12} T^{16} - 1036 p^{16} T^{18} + p^{20} T^{20} )^{4} \)
17 \( ( 1 - 1806 T^{2} + 1664927 T^{4} - 1032072694 T^{6} + 483645099455 T^{8} - 184531955277060 T^{10} + 61191637825972266 T^{12} - 18818205188674099100 T^{14} + \)\(56\!\cdots\!73\)\( T^{16} - \)\(16\!\cdots\!14\)\( T^{18} + \)\(49\!\cdots\!93\)\( T^{20} - \)\(16\!\cdots\!14\)\( p^{4} T^{22} + \)\(56\!\cdots\!73\)\( p^{8} T^{24} - 18818205188674099100 p^{12} T^{26} + 61191637825972266 p^{16} T^{28} - 184531955277060 p^{20} T^{30} + 483645099455 p^{24} T^{32} - 1032072694 p^{28} T^{34} + 1664927 p^{32} T^{36} - 1806 p^{36} T^{38} + p^{40} T^{40} )^{2} \)
19 \( ( 1 + 14 T - 816 T^{2} - 22692 T^{3} + 140027 T^{4} + 9685412 T^{5} + 38187218 T^{6} - 230417254 T^{7} + 25052241233 T^{8} - 346989817504 T^{9} - 25514204384634 T^{10} - 346989817504 p^{2} T^{11} + 25052241233 p^{4} T^{12} - 230417254 p^{6} T^{13} + 38187218 p^{8} T^{14} + 9685412 p^{10} T^{15} + 140027 p^{12} T^{16} - 22692 p^{14} T^{17} - 816 p^{16} T^{18} + 14 p^{18} T^{19} + p^{20} T^{20} )^{4} \)
23 \( ( 1 - 3008 T^{2} + 4385533 T^{4} - 4197762740 T^{6} + 3124299964176 T^{8} - 2138397478363196 T^{10} + 1492393212287537259 T^{12} - \)\(10\!\cdots\!80\)\( T^{14} + \)\(61\!\cdots\!99\)\( T^{16} - \)\(33\!\cdots\!52\)\( T^{18} + \)\(17\!\cdots\!92\)\( T^{20} - \)\(33\!\cdots\!52\)\( p^{4} T^{22} + \)\(61\!\cdots\!99\)\( p^{8} T^{24} - \)\(10\!\cdots\!80\)\( p^{12} T^{26} + 1492393212287537259 p^{16} T^{28} - 2138397478363196 p^{20} T^{30} + 3124299964176 p^{24} T^{32} - 4197762740 p^{28} T^{34} + 4385533 p^{32} T^{36} - 3008 p^{36} T^{38} + p^{40} T^{40} )^{2} \)
29 \( 1 - 3.90e3T^{2} + 1.45e7T^{4} - 3.65e10T^{6} + 8.43e13T^{8} - 1.62e17T^{10} + 2.90e20T^{12} - 4.65e23T^{14} + 7.03e26T^{16} - 9.79e29T^{18} + 1.29e33T^{20} - 1.61e36T^{22} + 1.92e39T^{24} - 2.17e42T^{26} + 2.36e45T^{28} - 2.45e48T^{30} + 2.45e51T^{32} - 2.35e54T^{34} + 2.17e57T^{36} - 1.93e60T^{38}+O(T^{40}) \)
31 \( 1 + 444T + 8.88e4T^{2} + 1.06e7T^{3} + 8.77e8T^{4} + 5.64e10T^{5} + 3.28e12T^{6} + 1.84e14T^{7} + 9.48e15T^{8} + 4.29e17T^{9} + 1.86e19T^{10} + 8.20e20T^{11} + 3.35e22T^{12} + 1.23e24T^{13} + 4.64e25T^{14} + 1.84e27T^{15} + 6.52e28T^{16} + 2.12e30T^{17} + 8.18e31T^{18} + 3.18e33T^{19} + 1.00e35T^{20} + 3.38e36T^{21} + 1.43e38T^{22} + 5.04e39T^{23} + 1.44e41T^{24} + 5.59e42T^{25} + 2.26e44T^{26} + 6.38e45T^{27} + 1.86e47T^{28} + 8.11e48T^{29} + 2.66e50T^{30} + 6.25e51T^{31} + 2.36e53T^{32} + 9.86e54T^{33} + 2.46e56T^{34} + 6.56e57T^{35} + 3.09e59T^{36} + 9.95e60T^{37} + 2.15e62T^{38}+O(T^{39}) \)
37 \( 1 + 1.95e4T^{2} + 1.94e8T^{4} + 1.32e12T^{6} + 6.81e15T^{8} + 2.86e19T^{10} + 1.01e23T^{12} + 3.15e26T^{14} + 8.72e29T^{16} + 2.18e33T^{18} + 5.02e36T^{20} + 1.07e40T^{22} + 2.14e43T^{24} + 4.04e46T^{26} + 7.26e49T^{28} + 1.24e53T^{30} + 2.03e56T^{32} + 3.19e59T^{34} + 4.82e62T^{36}+O(T^{37}) \)
41 \( 1 - 4.14e4T^{2} + 8.39e8T^{4} - 1.10e13T^{6} + 1.06e17T^{8} - 8.00e20T^{10} + 4.91e24T^{12} - 2.54e28T^{14} + 1.13e32T^{16} - 4.43e35T^{18} + 1.55e39T^{20} - 4.91e42T^{22} + 1.42e46T^{24} - 3.82e49T^{26} + 9.54e52T^{28} - 2.22e56T^{30} + 4.86e59T^{32} - 9.99e62T^{34}+O(T^{36}) \)
43 \( 1 - 5.46e4T^{2} + 1.47e9T^{4} - 2.60e13T^{6} + 3.41e17T^{8} - 3.52e21T^{10} + 2.98e25T^{12} - 2.13e29T^{14} + 1.31e33T^{16} - 7.11e36T^{18} + 3.40e40T^{20} - 1.45e44T^{22} + 5.60e47T^{24} - 1.95e51T^{26} + 6.21e54T^{28} - 1.80e58T^{30} + 4.81e61T^{32} - 1.18e65T^{34}+O(T^{36}) \)
47 \( 1 - 2.58e4T^{2} + 3.25e8T^{4} - 2.73e12T^{6} + 1.76e16T^{8} - 9.51e19T^{10} + 4.50e23T^{12} - 1.91e27T^{14} + 7.47e30T^{16} - 2.71e34T^{18} + 9.23e37T^{20} - 2.97e41T^{22} + 9.11e44T^{24} - 2.66e48T^{26} + 7.47e51T^{28} - 2.01e55T^{30} + 5.21e58T^{32} - 1.30e62T^{34}+O(T^{35}) \)
53 \( 1 - 3.02e4T^{2} + 4.62e8T^{4} - 4.74e12T^{6} + 3.65e16T^{8} - 2.23e20T^{10} + 1.12e24T^{12} - 4.63e27T^{14} + 1.55e31T^{16} - 3.96e34T^{18} + 5.32e37T^{20} + 1.44e41T^{22} - 1.43e45T^{24} + 6.63e48T^{26} - 2.22e52T^{28} + 5.62e55T^{30} - 8.91e58T^{32}+O(T^{34}) \)
59 \( 1 + 4.53e4T^{2} + 1.04e9T^{4} + 1.63e13T^{6} + 1.94e17T^{8} + 1.85e21T^{10} + 1.49e25T^{12} + 1.03e29T^{14} + 6.25e32T^{16} + 3.35e36T^{18} + 1.61e40T^{20} + 7.00e43T^{22} + 2.80e47T^{24} + 1.05e51T^{26} + 3.87e54T^{28} + 1.44e58T^{30} + 5.57e61T^{32}+O(T^{33}) \)
61 \( 1 - 288T + 6.45e3T^{2} + 6.48e6T^{3} - 6.72e8T^{4} - 4.77e10T^{5} + 1.10e13T^{6} - 8.51e13T^{7} - 9.66e16T^{8} + 4.77e18T^{9} + 5.31e20T^{10} - 5.20e22T^{11} - 1.54e24T^{12} + 3.64e26T^{13} - 4.05e27T^{14} - 1.79e30T^{15} + 8.66e31T^{16} + 5.49e33T^{17} - 6.36e35T^{18} + 5.46e35T^{19} + 3.04e39T^{20} - 1.38e41T^{21} - 9.02e42T^{22} + 1.07e45T^{23} + 1.61e44T^{24} - 5.19e48T^{25} + 1.93e50T^{26} + 1.64e52T^{27} - 1.42e54T^{28} - 1.90e55T^{29} + 6.51e57T^{30} - 1.63e59T^{31} - 1.96e61T^{32}+O(T^{33}) \)
67 \( 1 + 4.65e4T^{2} + 1.09e9T^{4} + 1.69e13T^{6} + 1.87e17T^{8} + 1.53e21T^{10} + 9.07e24T^{12} + 3.43e28T^{14} + 3.13e31T^{16} - 4.97e35T^{18} - 2.19e39T^{20} + 1.44e43T^{22} + 2.46e47T^{24} + 1.65e51T^{26} + 6.03e54T^{28} + 4.27e57T^{30}+O(T^{32}) \)
71 \( 1 - 1.48e5T^{2} + 1.09e10T^{4} - 5.34e14T^{6} + 1.94e19T^{8} - 5.58e23T^{10} + 1.32e28T^{12} - 2.65e32T^{14} + 4.58e36T^{16} - 6.95e40T^{18} + 9.32e44T^{20} - 1.11e49T^{22} + 1.20e53T^{24} - 1.17e57T^{26} + 1.04e61T^{28} - 8.41e64T^{30}+O(T^{32}) \)
73 \( 1 + 3.26e4T^{2} + 4.21e8T^{4} + 2.58e12T^{6} + 7.75e15T^{8} + 2.80e19T^{10} + 2.05e23T^{12} + 1.54e26T^{14} - 5.52e30T^{16} - 3.96e34T^{18} - 5.94e38T^{20} - 6.21e42T^{22} - 2.93e46T^{24} - 7.08e49T^{26} - 3.23e53T^{28} - 1.56e57T^{30}+O(T^{31}) \)
79 \( 1 + 340T - 5.31e4T^{2} - 2.78e7T^{3} + 1.68e9T^{4} + 1.31e12T^{5} - 4.00e13T^{6} - 4.50e16T^{7} + 9.02e17T^{8} + 1.22e21T^{9} - 2.20e22T^{10} - 2.79e25T^{11} + 5.71e26T^{12} + 5.46e29T^{13} - 1.39e31T^{14} - 9.37e33T^{15} + 3.03e35T^{16} + 1.42e38T^{17} - 5.76e39T^{18} - 1.93e42T^{19} + 9.60e43T^{20} + 2.36e46T^{21} - 1.41e48T^{22} - 2.61e50T^{23} + 1.87e52T^{24} + 2.61e54T^{25} - 2.22e56T^{26} - 2.37e58T^{27} + 2.40e60T^{28} + 1.93e62T^{29} - 2.36e64T^{30}+O(T^{31}) \)
83 \( 1 + 1.52e5T^{2} + 1.18e10T^{4} + 6.26e14T^{6} + 2.50e19T^{8} + 8.11e23T^{10} + 2.20e28T^{12} + 5.13e32T^{14} + 1.04e37T^{16} + 1.90e41T^{18} + 3.10e45T^{20} + 4.56e49T^{22} + 6.13e53T^{24} + 7.52e57T^{26} + 8.49e61T^{28} + 8.84e65T^{30}+O(T^{31}) \)
89 \( 1 + 9.38e4T^{2} + 4.36e9T^{4} + 1.35e14T^{6} + 3.22e18T^{8} + 6.25e22T^{10} + 1.03e27T^{12} + 1.51e31T^{14} + 1.98e35T^{16} + 2.38e39T^{18} + 2.65e43T^{20} + 2.77e47T^{22} + 2.76e51T^{24} + 2.64e55T^{26} + 2.46e59T^{28}+O(T^{30}) \)
97 \( 1 - 1.86e5T^{2} + 1.78e10T^{4} - 1.15e15T^{6} + 5.75e19T^{8} - 2.31e24T^{10} + 7.86e28T^{12} - 2.30e33T^{14} + 5.96e37T^{16} - 1.37e42T^{18} + 2.87e46T^{20} - 5.45e50T^{22} + 9.49e54T^{24} - 1.52e59T^{26} + 2.25e63T^{28}+O(T^{30}) \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{80} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−2.01297505667700787949959751036, −1.97728881988840027011974878174, −1.97234751824795294200281441552, −1.95696981453703377473036784161, −1.88334839087800448039305854121, −1.87537482409708355886494734665, −1.82184123915247958516593818632, −1.79114976302650162503211653291, −1.75998383781690554106826220069, −1.74482848418552191905893574784, −1.67338324119277151891412897142, −1.57613036325070355046080731687, −1.45100669491318578020145093475, −1.38184914916396616131950132354, −1.35917983811930853105099168210, −1.29721520850707136942936865128, −1.27745791688507778411513181386, −0.948914907403487821616535923039, −0.76000732207989964255077787431, −0.62788813843130635130953740108, −0.60341931898030753111712855625, −0.43843016160148215335816713299, −0.36598620097610606500185232684, −0.32223307126540957863418803080, −0.24793524043828209943621306496, 0.24793524043828209943621306496, 0.32223307126540957863418803080, 0.36598620097610606500185232684, 0.43843016160148215335816713299, 0.60341931898030753111712855625, 0.62788813843130635130953740108, 0.76000732207989964255077787431, 0.948914907403487821616535923039, 1.27745791688507778411513181386, 1.29721520850707136942936865128, 1.35917983811930853105099168210, 1.38184914916396616131950132354, 1.45100669491318578020145093475, 1.57613036325070355046080731687, 1.67338324119277151891412897142, 1.74482848418552191905893574784, 1.75998383781690554106826220069, 1.79114976302650162503211653291, 1.82184123915247958516593818632, 1.87537482409708355886494734665, 1.88334839087800448039305854121, 1.95696981453703377473036784161, 1.97234751824795294200281441552, 1.97728881988840027011974878174, 2.01297505667700787949959751036

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

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