Properties

Degree 48
Conductor $ 13^{24} $
Sign $1$
Motivic weight 6
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 12·2-s − 2·3-s + 69·4-s − 114·5-s + 24·6-s + 316·7-s − 334·8-s + 3.40e3·9-s + 1.36e3·10-s + 228·11-s − 138·12-s − 1.55e3·13-s − 3.79e3·14-s + 228·15-s − 4.01e3·16-s + 2.08e4·17-s − 4.08e4·18-s − 1.40e4·19-s − 7.86e3·20-s − 632·21-s − 2.73e3·22-s + 2.96e4·23-s + 668·24-s + 6.49e3·25-s + 1.86e4·26-s − 4.34e4·27-s + 2.18e4·28-s + ⋯
L(s)  = 1  − 3/2·2-s − 0.0740·3-s + 1.07·4-s − 0.911·5-s + 1/9·6-s + 0.921·7-s − 0.652·8-s + 4.66·9-s + 1.36·10-s + 0.171·11-s − 0.0798·12-s − 0.706·13-s − 1.38·14-s + 0.0675·15-s − 0.980·16-s + 4.25·17-s − 7.00·18-s − 2.04·19-s − 0.983·20-s − 0.0682·21-s − 0.256·22-s + 2.43·23-s + 0.0483·24-s + 0.415·25-s + 1.05·26-s − 2.20·27-s + 0.993·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(13^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{24} \, L(s)\cr=\mathstrut & \,\Lambda(7-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(13^{24}\right)^{s/2} \, \Gamma_{\C}(s+3)^{24} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(48\)
\( N \)  =  \(13^{24}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(6\)
character  :  induced by $\chi_{13} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(48,\ 13^{24} ,\ ( \ : [3]^{24} ),\ 1 )$
$L(\frac{7}{2})$  $\approx$  $1.58393$
$L(\frac12)$  $\approx$  $1.58393$
$L(4)$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \neq 13$,\(F_p(T)\) is a polynomial of degree 48. If $p = 13$, then $F_p(T)$ is a polynomial of degree at most 47.
$p$$F_p(T)$
bad13 \( 1 + 1.55e3T + 8.95e5T^{2} + 4.14e8T^{3} - 2.56e13T^{4} - 6.15e16T^{5} - 6.44e19T^{6} + 7.01e22T^{7} + 8.01e25T^{8} + 8.03e28T^{9} + 1.68e33T^{10} + 3.22e36T^{11} + 4.34e39T^{12} + 1.55e43T^{13} + 3.92e46T^{14} + 9.03e48T^{15} + 4.35e52T^{16} + 1.83e56T^{17} - 8.15e59T^{18}+O(T^{19}) \)
good2 \( 1 + 3 p^{2} T + 75 T^{2} + 203 p T^{3} + 7723 T^{4} + 21063 p T^{5} + 21325 p T^{6} - 584213 p^{2} T^{7} - 2186621 p^{3} T^{8} - 451107 p^{3} T^{9} + 233889547 p^{3} T^{10} + 1004037725 p^{4} T^{11} + 5161458111 p^{5} T^{12} + 72748200675 p^{5} T^{13} + 711705439691 p^{5} T^{14} + 1966701913993 p^{6} T^{15} + 3985654245793 p^{6} T^{16} - 5503707398163 p^{9} T^{17} - 23269715269217 p^{10} T^{18} - 8644970107279 p^{14} T^{19} - 14972079928069 p^{14} T^{20} + 30287247349133 p^{17} T^{21} + 463040432289495 p^{19} T^{22} + 755208424061101 p^{22} T^{23} + 1989191215833889 p^{24} T^{24} + 755208424061101 p^{28} T^{25} + 463040432289495 p^{31} T^{26} + 30287247349133 p^{35} T^{27} - 14972079928069 p^{38} T^{28} - 8644970107279 p^{44} T^{29} - 23269715269217 p^{46} T^{30} - 5503707398163 p^{51} T^{31} + 3985654245793 p^{54} T^{32} + 1966701913993 p^{60} T^{33} + 711705439691 p^{65} T^{34} + 72748200675 p^{71} T^{35} + 5161458111 p^{77} T^{36} + 1004037725 p^{82} T^{37} + 233889547 p^{87} T^{38} - 451107 p^{93} T^{39} - 2186621 p^{99} T^{40} - 584213 p^{104} T^{41} + 21325 p^{109} T^{42} + 21063 p^{115} T^{43} + 7723 p^{120} T^{44} + 203 p^{127} T^{45} + 75 p^{132} T^{46} + 3 p^{140} T^{47} + p^{144} T^{48} \)
3 \( 1 + 2 T - 1133 p T^{2} + 29858 T^{3} + 5688526 T^{4} - 38921558 p T^{5} - 198809267 p^{3} T^{6} + 23260719910 p^{2} T^{7} + 224118834076 p^{2} T^{8} - 959902682498 p^{5} T^{9} + 72459550346363 p^{3} T^{10} + 2135569065154682 p^{4} T^{11} - 50359039205031182 p^{4} T^{12} - 325286418273348718 p^{5} T^{13} + 16206904999005156605 p^{5} T^{14} + 1668531950895512194 p^{7} T^{15} - \)\(36\!\cdots\!08\)\( p^{6} T^{16} + \)\(15\!\cdots\!10\)\( p^{7} T^{17} + \)\(54\!\cdots\!35\)\( p^{7} T^{18} - \)\(61\!\cdots\!26\)\( p^{8} T^{19} - \)\(21\!\cdots\!06\)\( p^{8} T^{20} + \)\(13\!\cdots\!74\)\( p^{9} T^{21} - \)\(85\!\cdots\!93\)\( p^{16} T^{22} - \)\(45\!\cdots\!14\)\( p^{11} T^{23} + \)\(69\!\cdots\!14\)\( p^{10} T^{24} - \)\(45\!\cdots\!14\)\( p^{17} T^{25} - \)\(85\!\cdots\!93\)\( p^{28} T^{26} + \)\(13\!\cdots\!74\)\( p^{27} T^{27} - \)\(21\!\cdots\!06\)\( p^{32} T^{28} - \)\(61\!\cdots\!26\)\( p^{38} T^{29} + \)\(54\!\cdots\!35\)\( p^{43} T^{30} + \)\(15\!\cdots\!10\)\( p^{49} T^{31} - \)\(36\!\cdots\!08\)\( p^{54} T^{32} + 1668531950895512194 p^{61} T^{33} + 16206904999005156605 p^{65} T^{34} - 325286418273348718 p^{71} T^{35} - 50359039205031182 p^{76} T^{36} + 2135569065154682 p^{82} T^{37} + 72459550346363 p^{87} T^{38} - 959902682498 p^{95} T^{39} + 224118834076 p^{98} T^{40} + 23260719910 p^{104} T^{41} - 198809267 p^{111} T^{42} - 38921558 p^{115} T^{43} + 5688526 p^{120} T^{44} + 29858 p^{126} T^{45} - 1133 p^{133} T^{46} + 2 p^{138} T^{47} + p^{144} T^{48} \)
5 \( 1 + 114 T + 6498 T^{2} + 4087852 T^{3} + 551045299 T^{4} + 221026272 p^{2} T^{5} + 43235996074 p^{3} T^{6} + 7768415045918 p^{3} T^{7} - 67121912714197 p^{4} T^{8} + 41207654436312 p^{6} T^{9} + 86570161201704668 p^{6} T^{10} - 818779249926950492 p^{7} T^{11} - 16192011382286222784 p^{8} T^{12} + \)\(18\!\cdots\!08\)\( p^{10} T^{13} - \)\(11\!\cdots\!88\)\( p^{11} T^{14} - \)\(54\!\cdots\!08\)\( p^{11} T^{15} + \)\(13\!\cdots\!21\)\( p^{12} T^{16} - \)\(93\!\cdots\!54\)\( p^{13} T^{17} - \)\(24\!\cdots\!58\)\( p^{15} T^{18} + \)\(16\!\cdots\!08\)\( p^{15} T^{19} - \)\(31\!\cdots\!07\)\( p^{16} T^{20} - \)\(45\!\cdots\!12\)\( p^{17} T^{21} - \)\(27\!\cdots\!82\)\( p^{18} T^{22} - \)\(14\!\cdots\!86\)\( p^{19} T^{23} - \)\(10\!\cdots\!14\)\( p^{20} T^{24} - \)\(14\!\cdots\!86\)\( p^{25} T^{25} - \)\(27\!\cdots\!82\)\( p^{30} T^{26} - \)\(45\!\cdots\!12\)\( p^{35} T^{27} - \)\(31\!\cdots\!07\)\( p^{40} T^{28} + \)\(16\!\cdots\!08\)\( p^{45} T^{29} - \)\(24\!\cdots\!58\)\( p^{51} T^{30} - \)\(93\!\cdots\!54\)\( p^{55} T^{31} + \)\(13\!\cdots\!21\)\( p^{60} T^{32} - \)\(54\!\cdots\!08\)\( p^{65} T^{33} - \)\(11\!\cdots\!88\)\( p^{71} T^{34} + \)\(18\!\cdots\!08\)\( p^{76} T^{35} - 16192011382286222784 p^{80} T^{36} - 818779249926950492 p^{85} T^{37} + 86570161201704668 p^{90} T^{38} + 41207654436312 p^{96} T^{39} - 67121912714197 p^{100} T^{40} + 7768415045918 p^{105} T^{41} + 43235996074 p^{111} T^{42} + 221026272 p^{116} T^{43} + 551045299 p^{120} T^{44} + 4087852 p^{126} T^{45} + 6498 p^{132} T^{46} + 114 p^{138} T^{47} + p^{144} T^{48} \)
7 \( 1 - 316T - 3.77e5T^{2} + 1.99e8T^{3} + 5.71e10T^{4} - 6.48e13T^{5} + 2.43e15T^{6} + 1.13e19T^{7} - 2.95e21T^{8} - 1.02e24T^{9} + 6.70e26T^{10} - 2.01e28T^{11} - 7.88e31T^{12} + 2.19e34T^{13} + 4.92e36T^{14} - 3.72e39T^{15} + 9.06e40T^{16} + 3.99e44T^{17} - 6.68e46T^{18} - 3.79e49T^{19} + 1.11e52T^{20} + 2.97e54T^{21} - 1.59e57T^{22} - 1.23e59T^{23}+O(T^{24}) \)
11 \( 1 - 228T - 3.49e6T^{2} + 3.66e8T^{3} + 7.11e12T^{4} + 2.16e15T^{5} - 1.10e19T^{6} - 8.68e19T^{7} + 8.62e24T^{8} - 3.63e27T^{9} - 1.72e31T^{10} + 1.72e34T^{11} + 5.85e37T^{12} + 1.87e39T^{13} - 8.67e43T^{14} - 1.73e47T^{15} + 1.42e50T^{16} + 2.96e53T^{17} + 6.53e55T^{18} - 3.18e59T^{19}+O(T^{20}) \)
17 \( 1 - 2.08e4T + 4.00e8T^{2} - 5.32e12T^{3} + 6.47e16T^{4} - 6.59e20T^{5} + 6.22e24T^{6} - 5.24e28T^{7} + 4.14e32T^{8} - 3.01e36T^{9} + 2.07e40T^{10} - 1.34e44T^{11} + 8.31e47T^{12} - 4.91e51T^{13} + 2.81e55T^{14} - 1.56e59T^{15} + 8.60e62T^{16}+O(T^{17}) \)
19 \( 1 + 1.40e4T + 1.65e8T^{2} + 2.93e11T^{3} - 5.89e15T^{4} - 1.65e20T^{5} - 1.20e24T^{6} - 7.76e27T^{7} + 2.33e31T^{8} + 4.48e35T^{9} + 6.44e39T^{10} + 2.60e43T^{11} + 1.14e47T^{12} - 1.34e51T^{13} - 8.40e54T^{14} - 9.86e58T^{15}+O(T^{16}) \)
23 \( 1 - 2.96e4T + 1.44e9T^{2} - 3.40e13T^{3} + 9.88e17T^{4} - 2.07e22T^{5} + 4.63e26T^{6} - 8.97e30T^{7} + 1.70e35T^{8} - 3.07e39T^{9} + 5.22e43T^{10} - 8.79e47T^{11} + 1.39e52T^{12} - 2.19e56T^{13} + 3.31e60T^{14}+O(T^{15}) \)
29 \( 1 - 1.02e3T - 3.47e9T^{2} - 1.30e13T^{3} + 5.41e18T^{4} + 4.41e22T^{5} - 5.26e27T^{6} - 5.37e31T^{7} + 4.15e36T^{8} + 3.01e40T^{9} - 3.47e45T^{10} - 9.04e48T^{11} + 3.03e54T^{12} + 7.03e57T^{13}+O(T^{14}) \)
31 \( 1 + 1.18e5T + 6.97e9T^{2} + 3.32e14T^{3} + 1.88e19T^{4} + 1.01e24T^{5} + 4.43e28T^{6} + 1.78e33T^{7} + 7.73e37T^{8} + 3.24e42T^{9} + 1.19e47T^{10} + 4.26e51T^{11} + 1.61e56T^{12} + 5.80e60T^{13}+O(T^{14}) \)
37 \( 1 + 3.14e4T + 1.15e10T^{2} + 7.47e14T^{3} + 7.79e19T^{4} + 7.36e24T^{5} + 4.85e29T^{6} + 4.64e34T^{7} + 2.97e39T^{8} + 2.31e44T^{9} + 1.59e49T^{10} + 1.03e54T^{11} + 7.21e58T^{12} + 4.33e63T^{13}+O(T^{14}) \)
41 \( 1 + 3.57e5T + 5.84e10T^{2} + 6.34e15T^{3} + 5.76e20T^{4} + 4.93e25T^{5} + 4.10e30T^{6} + 3.25e35T^{7} + 2.32e40T^{8} + 1.39e45T^{9} + 6.19e49T^{10} + 9.02e53T^{11} - 1.83e59T^{12}+O(T^{13}) \)
43 \( 1 + 4.80e5T + 1.60e11T^{2} + 4.03e16T^{3} + 8.49e21T^{4} + 1.54e27T^{5} + 2.49e32T^{6} + 3.67e37T^{7} + 4.98e42T^{8} + 6.30e47T^{9} + 7.52e52T^{10} + 8.53e57T^{11} + 9.27e62T^{12}+O(T^{13}) \)
47 \( 1 - 1.40e5T + 9.80e9T^{2} + 3.82e15T^{3} + 2.74e19T^{4} - 7.22e25T^{5} + 1.71e31T^{6} + 8.21e35T^{7} - 1.70e41T^{8} + 6.07e45T^{9} + 6.09e51T^{10} - 2.57e56T^{11} - 3.13e61T^{12}+O(T^{13}) \)
53 \( 1 + 1.39e6T + 1.16e12T^{2} + 7.19e17T^{3} + 3.59e23T^{4} + 1.53e29T^{5} + 5.73e34T^{6} + 1.93e40T^{7} + 5.93e45T^{8} + 1.68e51T^{9} + 4.44e56T^{10} + 1.10e62T^{11}+O(T^{12}) \)
59 \( 1 - 1.56e6T + 1.36e12T^{2} - 8.42e17T^{3} + 4.10e23T^{4} - 1.65e29T^{5} + 5.65e34T^{6} - 1.67e40T^{7} + 4.32e45T^{8} - 9.73e50T^{9} + 1.89e56T^{10} - 3.09e61T^{11}+O(T^{12}) \)
61 \( 1 - 7.68e5T - 9.79e10T^{2} + 2.00e17T^{3} - 2.16e21T^{4} - 3.11e28T^{5} + 1.39e33T^{6} + 3.77e39T^{7} - 2.18e44T^{8} - 3.78e50T^{9} + 2.36e55T^{10} + 3.21e61T^{11}+O(T^{12}) \)
67 \( 1 + 1.48e6T + 1.01e12T^{2} + 3.78e17T^{3} + 5.69e22T^{4} - 1.85e28T^{5} - 1.31e34T^{6} - 3.05e39T^{7} + 9.05e43T^{8} + 2.83e50T^{9} + 9.13e55T^{10} + 1.40e61T^{11}+O(T^{12}) \)
71 \( 1 - 2.13e6T + 2.86e12T^{2} - 2.74e18T^{3} + 2.11e24T^{4} - 1.35e30T^{5} + 7.44e35T^{6} - 3.52e41T^{7} + 1.45e47T^{8} - 5.16e52T^{9} + 1.55e58T^{10} - 3.79e63T^{11}+O(T^{12}) \)
73 \( 1 - 3.26e5T + 5.32e10T^{2} - 1.49e17T^{3} + 1.04e23T^{4} - 4.86e26T^{5} + 5.75e33T^{6} - 8.70e39T^{7} - 5.75e43T^{8} + 1.41e51T^{9} + 2.41e56T^{10} + 5.23e61T^{11}+O(T^{12}) \)
79 \( 1 + 1.73e6T + 4.53e12T^{2} + 6.00e18T^{3} + 9.34e24T^{4} + 1.01e31T^{5} + 1.20e37T^{6} + 1.12e43T^{7} + 1.10e49T^{8} + 9.12e54T^{9} + 7.80e60T^{10}+O(T^{11}) \)
83 \( 1 + 3.64e6T + 6.63e12T^{2} + 8.55e18T^{3} + 9.10e24T^{4} + 8.49e30T^{5} + 7.12e36T^{6} + 5.49e42T^{7} + 4.00e48T^{8} + 2.82e54T^{9} + 1.94e60T^{10}+O(T^{11}) \)
89 \( 1 - 6.32e5T - 5.72e10T^{2} + 7.89e17T^{3} - 1.03e24T^{4} + 3.13e29T^{5} + 3.51e35T^{6} - 7.19e41T^{7} + 5.21e47T^{8} - 4.74e51T^{9} - 2.90e59T^{10}+O(T^{11}) \)
97 \( 1 - 2.25e6T + 6.97e12T^{2} - 1.06e19T^{3} + 1.88e25T^{4} - 2.08e31T^{5} + 2.61e37T^{6} - 1.91e43T^{7} + 1.63e49T^{8} + 2.22e53T^{9} - 6.01e60T^{10}+O(T^{11}) \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{48} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−4.16058267109963425699917147670, −4.10537533539872181637112472632, −3.98366223345054869971854439584, −3.79370955995251489686209277756, −3.66209977729891911120260313639, −3.64732805541704382198508374920, −3.43159906759084694609680032827, −3.27650908085141146546162415786, −3.22527062811828659237485880773, −3.07179460636630205802292128163, −2.94658798675343089936114800787, −2.71114265878497823920794529964, −2.24056711708226962061180638885, −2.10514707109587005814953167834, −2.00385189246237898609294475690, −1.82174535677394454174697630032, −1.80247428155898797350088311543, −1.50671643796533895721957085885, −1.43336555171954563279725081331, −1.26103799657995284059154825787, −1.21494263716344480031016721152, −0.971244595941493625544782531415, −0.43183964991025141328242473483, −0.28226696015413080481852561195, −0.20234623542748597957589959277, 0.20234623542748597957589959277, 0.28226696015413080481852561195, 0.43183964991025141328242473483, 0.971244595941493625544782531415, 1.21494263716344480031016721152, 1.26103799657995284059154825787, 1.43336555171954563279725081331, 1.50671643796533895721957085885, 1.80247428155898797350088311543, 1.82174535677394454174697630032, 2.00385189246237898609294475690, 2.10514707109587005814953167834, 2.24056711708226962061180638885, 2.71114265878497823920794529964, 2.94658798675343089936114800787, 3.07179460636630205802292128163, 3.22527062811828659237485880773, 3.27650908085141146546162415786, 3.43159906759084694609680032827, 3.64732805541704382198508374920, 3.66209977729891911120260313639, 3.79370955995251489686209277756, 3.98366223345054869971854439584, 4.10537533539872181637112472632, 4.16058267109963425699917147670

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

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