Properties

Degree $60$
Conductor $2.475\times 10^{47}$
Sign $1$
Motivic weight $5$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 15·3-s − 84·7-s − 320·8-s − 60·9-s − 126·11-s − 2.61e3·13-s + 1.90e3·17-s − 2.40e3·19-s − 1.26e3·21-s − 3.30e3·23-s − 4.80e3·24-s − 9.03e3·25-s + 2.31e3·27-s + 5.98e3·29-s − 6.84e3·31-s − 1.89e3·33-s − 8.55e3·37-s − 3.91e4·39-s − 4.05e4·41-s + 3.67e3·43-s − 1.33e5·47-s + 9.76e4·49-s + 2.85e4·51-s − 6.46e4·53-s + 2.68e4·56-s − 3.60e4·57-s + 1.69e5·59-s + ⋯
L(s)  = 1  + 0.962·3-s − 0.647·7-s − 1.76·8-s − 0.246·9-s − 0.313·11-s − 4.28·13-s + 1.59·17-s − 1.52·19-s − 0.623·21-s − 1.30·23-s − 1.70·24-s − 2.88·25-s + 0.610·27-s + 1.32·29-s − 1.27·31-s − 0.302·33-s − 1.02·37-s − 4.12·39-s − 3.76·41-s + 0.303·43-s − 8.83·47-s + 5.81·49-s + 1.53·51-s − 3.16·53-s + 1.14·56-s − 1.46·57-s + 6.32·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 19^{30}\right)^{s/2} \, \Gamma_{\C}(s)^{30} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 19^{30}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{30} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(60\)
Conductor: \(2^{30} \cdot 19^{30}\)
Sign: $1$
Motivic weight: \(5\)
Character: induced by $\chi_{38} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((60,\ 2^{30} \cdot 19^{30} ,\ ( \ : [5/2]^{30} ),\ 1 )\)

Particular Values

\(L(3)\) \(\approx\) \(8.83048\)
\(L(\frac12)\) \(\approx\) \(8.83048\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 + p^{6} T^{3} + p^{12} T^{6} )^{5} \)
19 \( 1 + 2.40e3T + 1.57e5T^{2} + 6.87e9T^{3} + 2.76e13T^{4} + 1.85e16T^{5} + 3.56e19T^{6} + 1.37e23T^{7} + 1.41e26T^{8} + 2.47e29T^{9} + 6.40e32T^{10} + 6.59e35T^{11} + 1.10e39T^{12} + 2.94e42T^{13} + 3.78e45T^{14} + 3.90e48T^{15} + 9.38e51T^{16} + 1.80e55T^{17} + 1.67e58T^{18}+O(T^{19}) \)
good3 \( 1 - 5 p T + 95 p T^{2} - 7489 T^{3} + 46567 p T^{4} - 291415 p T^{5} - 10025737 T^{6} + 194944333 p T^{7} - 4303046311 p T^{8} + 242761751018 T^{9} - 1728336669752 p T^{10} + 18378826901444 p T^{11} + 281299798935047 T^{12} - 5138042284154465 p T^{13} + 101554420090294358 p T^{14} - 6571736536896770734 T^{15} + 35897474429260242925 p T^{16} - \)\(36\!\cdots\!81\)\( p T^{17} + \)\(29\!\cdots\!95\)\( T^{18} + \)\(32\!\cdots\!87\)\( p^{2} T^{19} - \)\(21\!\cdots\!42\)\( p^{3} T^{20} + \)\(30\!\cdots\!55\)\( p^{5} T^{21} - \)\(50\!\cdots\!08\)\( p^{5} T^{22} + \)\(21\!\cdots\!41\)\( p^{6} T^{23} + \)\(17\!\cdots\!21\)\( p^{6} T^{24} - \)\(37\!\cdots\!58\)\( p^{8} T^{25} + \)\(26\!\cdots\!10\)\( p^{9} T^{26} - \)\(12\!\cdots\!24\)\( p^{12} T^{27} + \)\(61\!\cdots\!26\)\( p^{11} T^{28} - \)\(33\!\cdots\!83\)\( p^{12} T^{29} + \)\(25\!\cdots\!54\)\( p^{12} T^{30} - \)\(33\!\cdots\!83\)\( p^{17} T^{31} + \)\(61\!\cdots\!26\)\( p^{21} T^{32} - \)\(12\!\cdots\!24\)\( p^{27} T^{33} + \)\(26\!\cdots\!10\)\( p^{29} T^{34} - \)\(37\!\cdots\!58\)\( p^{33} T^{35} + \)\(17\!\cdots\!21\)\( p^{36} T^{36} + \)\(21\!\cdots\!41\)\( p^{41} T^{37} - \)\(50\!\cdots\!08\)\( p^{45} T^{38} + \)\(30\!\cdots\!55\)\( p^{50} T^{39} - \)\(21\!\cdots\!42\)\( p^{53} T^{40} + \)\(32\!\cdots\!87\)\( p^{57} T^{41} + \)\(29\!\cdots\!95\)\( p^{60} T^{42} - \)\(36\!\cdots\!81\)\( p^{66} T^{43} + 35897474429260242925 p^{71} T^{44} - 6571736536896770734 p^{75} T^{45} + 101554420090294358 p^{81} T^{46} - 5138042284154465 p^{86} T^{47} + 281299798935047 p^{90} T^{48} + 18378826901444 p^{96} T^{49} - 1728336669752 p^{101} T^{50} + 242761751018 p^{105} T^{51} - 4303046311 p^{111} T^{52} + 194944333 p^{116} T^{53} - 10025737 p^{120} T^{54} - 291415 p^{126} T^{55} + 46567 p^{131} T^{56} - 7489 p^{135} T^{57} + 95 p^{141} T^{58} - 5 p^{146} T^{59} + p^{150} T^{60} \)
5 \( 1 + 1806 p T^{2} + 299406 T^{3} + 1266813 p^{2} T^{4} + 572892906 p T^{5} + 59726871473 T^{6} + 427733342046 p^{2} T^{7} + 16256149582512 p T^{8} + 9673651419197372 T^{9} - 35519485513593396 p T^{10} - 21075407039516034984 p T^{11} - \)\(62\!\cdots\!22\)\( p T^{12} - \)\(13\!\cdots\!26\)\( p T^{13} - \)\(46\!\cdots\!03\)\( p T^{14} - \)\(17\!\cdots\!52\)\( T^{15} - \)\(19\!\cdots\!92\)\( p T^{16} - \)\(31\!\cdots\!44\)\( p T^{17} - \)\(35\!\cdots\!91\)\( p T^{18} + \)\(16\!\cdots\!44\)\( p T^{19} + \)\(70\!\cdots\!11\)\( p T^{20} + \)\(48\!\cdots\!42\)\( T^{21} + \)\(73\!\cdots\!33\)\( p T^{22} + \)\(49\!\cdots\!78\)\( p^{2} T^{23} + \)\(12\!\cdots\!23\)\( T^{24} + \)\(27\!\cdots\!34\)\( p T^{25} + \)\(10\!\cdots\!79\)\( p^{3} T^{26} - \)\(25\!\cdots\!24\)\( T^{27} - \)\(13\!\cdots\!26\)\( p T^{28} - \)\(90\!\cdots\!18\)\( p^{2} T^{29} - \)\(36\!\cdots\!94\)\( T^{30} - \)\(90\!\cdots\!18\)\( p^{7} T^{31} - \)\(13\!\cdots\!26\)\( p^{11} T^{32} - \)\(25\!\cdots\!24\)\( p^{15} T^{33} + \)\(10\!\cdots\!79\)\( p^{23} T^{34} + \)\(27\!\cdots\!34\)\( p^{26} T^{35} + \)\(12\!\cdots\!23\)\( p^{30} T^{36} + \)\(49\!\cdots\!78\)\( p^{37} T^{37} + \)\(73\!\cdots\!33\)\( p^{41} T^{38} + \)\(48\!\cdots\!42\)\( p^{45} T^{39} + \)\(70\!\cdots\!11\)\( p^{51} T^{40} + \)\(16\!\cdots\!44\)\( p^{56} T^{41} - \)\(35\!\cdots\!91\)\( p^{61} T^{42} - \)\(31\!\cdots\!44\)\( p^{66} T^{43} - \)\(19\!\cdots\!92\)\( p^{71} T^{44} - \)\(17\!\cdots\!52\)\( p^{75} T^{45} - \)\(46\!\cdots\!03\)\( p^{81} T^{46} - \)\(13\!\cdots\!26\)\( p^{86} T^{47} - \)\(62\!\cdots\!22\)\( p^{91} T^{48} - 21075407039516034984 p^{96} T^{49} - 35519485513593396 p^{101} T^{50} + 9673651419197372 p^{105} T^{51} + 16256149582512 p^{111} T^{52} + 427733342046 p^{117} T^{53} + 59726871473 p^{120} T^{54} + 572892906 p^{126} T^{55} + 1266813 p^{132} T^{56} + 299406 p^{135} T^{57} + 1806 p^{141} T^{58} + p^{150} T^{60} \)
7 \( 1 + 84T - 9.06e4T^{2} - 5.73e6T^{3} + 4.21e9T^{4} + 2.15e11T^{5} - 1.14e14T^{6} - 6.30e15T^{7} + 1.50e18T^{8} + 1.73e20T^{9} + 1.25e22T^{10} - 4.13e24T^{11} - 1.06e27T^{12} + 5.17e28T^{13} + 2.45e31T^{14} + 8.13e32T^{15} - 2.84e35T^{16} - 5.50e37T^{17} - 4.66e38T^{18} + 1.19e42T^{19} + 1.10e44T^{20} - 9.65e45T^{21} - 3.07e48T^{22} - 1.60e50T^{23} + 4.70e52T^{24} + 6.34e54T^{25} - 1.34e56T^{26} - 9.86e58T^{27}+O(T^{28}) \)
11 \( 1 + 126T - 9.43e5T^{2} - 1.32e8T^{3} + 3.93e11T^{4} + 4.41e13T^{5} - 1.01e17T^{6} + 6.16e17T^{7} + 2.21e22T^{8} - 5.11e24T^{9} - 5.15e27T^{10} + 2.06e30T^{11} + 1.13e33T^{12} - 5.50e35T^{13} - 1.85e38T^{14} + 1.32e41T^{15} + 1.79e43T^{16} - 3.04e46T^{17} + 1.26e47T^{18} + 6.24e51T^{19} - 6.08e53T^{20} - 1.08e57T^{21} + 2.31e59T^{22}+O(T^{23}) \)
13 \( 1 + 2.61e3T + 3.95e6T^{2} + 4.20e9T^{3} + 3.25e12T^{4} + 2.14e15T^{5} + 1.47e18T^{6} + 1.26e21T^{7} + 1.16e24T^{8} + 8.98e26T^{9} + 5.74e29T^{10} + 3.37e32T^{11} + 2.22e35T^{12} + 1.75e38T^{13} + 1.28e41T^{14} + 7.70e43T^{15} + 3.86e46T^{16} + 1.99e49T^{17} + 1.41e52T^{18} + 1.03e55T^{19} + 5.78e57T^{20} + 2.17e60T^{21}+O(T^{22}) \)
17 \( 1 - 1.90e3T + 2.69e6T^{2} - 8.79e8T^{3} - 2.26e12T^{4} + 2.62e15T^{5} - 1.16e17T^{6} - 4.68e21T^{7} + 8.30e24T^{8} - 2.24e27T^{9} - 4.54e30T^{10} + 9.65e33T^{11} - 3.04e36T^{12} - 3.68e40T^{13} + 5.27e43T^{14} - 5.34e46T^{15} + 9.93e48T^{16} + 4.29e52T^{17} - 7.57e54T^{18} - 5.55e58T^{19}+O(T^{20}) \)
23 \( 1 + 3.30e3T - 4.65e6T^{2} - 5.51e10T^{3} - 2.28e13T^{4} + 5.50e17T^{5} + 8.32e20T^{6} - 3.82e24T^{7} - 1.10e28T^{8} + 2.08e31T^{9} + 1.16e35T^{10} + 6.29e35T^{11} - 7.73e41T^{12} - 1.13e45T^{13} + 3.99e48T^{14} + 1.39e52T^{15} - 6.53e54T^{16} - 1.05e59T^{17}+O(T^{18}) \)
29 \( 1 - 5.98e3T + 5.70e7T^{2} - 2.23e11T^{3} + 1.30e15T^{4} - 8.45e18T^{5} + 4.46e22T^{6} - 2.81e26T^{7} + 1.03e30T^{8} - 5.06e33T^{9} + 2.17e37T^{10} - 9.68e40T^{11} + 4.95e44T^{12} - 1.62e48T^{13} + 8.03e51T^{14} - 2.58e55T^{15} + 1.09e59T^{16}+O(T^{17}) \)
31 \( 1 + 6.84e3T - 1.35e8T^{2} - 8.20e11T^{3} + 8.30e15T^{4} + 3.79e19T^{5} - 2.98e23T^{6} - 7.66e26T^{7} + 6.61e30T^{8} + 1.56e34T^{9} - 9.88e37T^{10} - 1.57e42T^{11} + 3.43e45T^{12} + 8.91e49T^{13} - 2.33e53T^{14} - 2.23e57T^{15} + 7.98e60T^{16}+O(T^{17}) \)
37 \( 1 + 8.55e3T + 1.24e9T^{2} + 8.64e12T^{3} + 7.50e17T^{4} + 4.10e21T^{5} + 2.93e26T^{6} + 1.20e30T^{7} + 8.43e34T^{8} + 2.38e38T^{9} + 1.91e43T^{10} + 3.15e46T^{11} + 3.61e51T^{12} + 2.13e54T^{13} + 5.84e59T^{14} - 1.84e62T^{15}+O(T^{16}) \)
41 \( 1 + 4.05e4T + 1.23e9T^{2} + 2.61e13T^{3} + 4.50e17T^{4} + 6.23e21T^{5} + 7.11e25T^{6} + 6.34e29T^{7} + 3.58e33T^{8} - 7.96e36T^{9} - 5.80e41T^{10} - 9.42e45T^{11} - 9.88e49T^{12} - 5.57e53T^{13} + 3.25e57T^{14}+O(T^{15}) \)
43 \( 1 - 3.67e3T + 3.14e8T^{2} - 9.04e12T^{3} + 1.16e17T^{4} - 2.42e21T^{5} + 5.63e25T^{6} - 7.84e29T^{7} + 1.23e34T^{8} - 2.36e38T^{9} + 3.40e42T^{10} - 4.71e46T^{11} + 7.84e50T^{12} - 1.13e55T^{13} + 1.47e59T^{14}+O(T^{15}) \)
47 \( 1 + 1.33e5T + 8.43e9T^{2} + 3.32e14T^{3} + 9.20e18T^{4} + 1.89e23T^{5} + 2.93e27T^{6} + 2.98e31T^{7} - 2.76e34T^{8} - 1.05e40T^{9} - 3.15e44T^{10} - 6.31e48T^{11} - 9.58e52T^{12} - 1.02e57T^{13} - 3.37e60T^{14}+O(T^{15}) \)
53 \( 1 + 6.46e4T + 2.89e8T^{2} - 8.58e13T^{3} - 2.44e18T^{4} + 1.55e22T^{5} + 1.68e27T^{6} + 1.48e31T^{7} - 3.10e35T^{8} + 3.82e39T^{9} + 2.61e44T^{10} - 6.81e48T^{11} - 3.66e53T^{12} - 1.02e57T^{13} + 1.65e62T^{14}+O(T^{15}) \)
59 \( 1 - 1.69e5T + 1.59e10T^{2} - 1.06e15T^{3} + 5.57e19T^{4} - 2.43e24T^{5} + 9.10e28T^{6} - 2.96e33T^{7} + 8.30e37T^{8} - 1.91e42T^{9} + 3.11e46T^{10} - 9.12e49T^{11} - 1.72e55T^{12} + 7.89e59T^{13}+O(T^{14}) \)
61 \( 1 - 6.16e4T + 2.33e9T^{2} - 1.15e14T^{3} + 5.74e18T^{4} - 2.02e23T^{5} + 6.40e27T^{6} - 2.29e32T^{7} + 7.73e36T^{8} - 2.10e41T^{9} + 5.91e45T^{10} - 1.87e50T^{11} + 5.24e54T^{12} - 1.33e59T^{13}+O(T^{14}) \)
67 \( 1 + 7.21e4T + 1.37e9T^{2} - 4.77e13T^{3} - 3.72e18T^{4} - 9.35e22T^{5} - 2.67e27T^{6} - 5.16e31T^{7} + 3.76e36T^{8} + 4.04e41T^{9} + 2.25e46T^{10} + 8.78e50T^{11} + 2.10e55T^{12} - 2.63e59T^{13}+O(T^{14}) \)
71 \( 1 - 4.17e5T + 8.65e10T^{2} - 1.19e16T^{3} + 1.23e21T^{4} - 1.06e26T^{5} + 7.98e30T^{6} - 5.49e35T^{7} + 3.54e40T^{8} - 2.16e45T^{9} + 1.25e50T^{10} - 7.01e54T^{11} + 3.74e59T^{12} - 1.92e64T^{13}+O(T^{14}) \)
73 \( 1 - 2.39e5T + 2.60e10T^{2} - 1.70e15T^{3} + 7.46e19T^{4} - 2.25e24T^{5} + 3.81e28T^{6} + 1.50e32T^{7} - 1.59e37T^{8} + 1.66e42T^{9} - 3.47e47T^{10} + 3.27e52T^{11} - 1.78e57T^{12} + 5.44e61T^{13}+O(T^{14}) \)
79 \( 1 + 1.45e5T + 1.53e10T^{2} + 1.01e15T^{3} + 8.77e19T^{4} + 6.32e24T^{5} + 4.60e29T^{6} + 2.19e34T^{7} + 1.39e39T^{8} + 7.31e43T^{9} + 4.58e48T^{10} + 1.51e53T^{11} + 1.00e58T^{12}+O(T^{13}) \)
83 \( 1 + 2.01e5T - 7.48e9T^{2} - 2.97e15T^{3} + 1.25e20T^{4} + 2.94e25T^{5} - 1.97e30T^{6} - 1.96e35T^{7} + 2.22e40T^{8} + 7.48e44T^{9} - 1.74e50T^{10} + 1.35e54T^{11} + 1.00e60T^{12}+O(T^{13}) \)
89 \( 1 - 4.16e5T + 8.83e10T^{2} - 1.22e16T^{3} + 1.27e21T^{4} - 1.09e26T^{5} + 8.22e30T^{6} - 5.27e35T^{7} + 2.61e40T^{8} - 5.76e44T^{9} - 6.14e49T^{10} + 9.80e54T^{11} - 7.75e59T^{12}+O(T^{13}) \)
97 \( 1 - 8.58e5T + 3.49e11T^{2} - 8.94e16T^{3} + 1.58e22T^{4} - 1.95e27T^{5} + 1.41e32T^{6} + 1.65e36T^{7} - 2.03e42T^{8} + 2.91e47T^{9} - 2.04e52T^{10} + 1.38e55T^{11} + 1.52e62T^{12}+O(T^{13}) \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{60} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−2.44934761378763437776912287314, −2.30746764115570162124907178238, −2.28176336041672583499196088323, −2.23723577448634154013816981948, −2.13646882255354498416696226385, −2.11731036099219031010878801531, −1.93193693961976909973044143900, −1.91130644582780361871557749589, −1.87415412496480074407177498487, −1.81802609134381555725091027169, −1.75233951222987119927881956637, −1.68571817940766294943066243175, −1.65348924668444002075700189589, −1.06857757292724509442689158003, −1.04371725816851312523941175600, −1.00519814906895539603939734219, −0.899697631745169269183301659300, −0.827859708001106926218566088348, −0.73522371756780933252567367920, −0.44145639808686111498609112641, −0.41985586360217012111305605205, −0.36574411998479145561017828968, −0.26294002770134863496618480389, −0.22627110144835400123007686983, −0.19851323887680771422767506815, 0.19851323887680771422767506815, 0.22627110144835400123007686983, 0.26294002770134863496618480389, 0.36574411998479145561017828968, 0.41985586360217012111305605205, 0.44145639808686111498609112641, 0.73522371756780933252567367920, 0.827859708001106926218566088348, 0.899697631745169269183301659300, 1.00519814906895539603939734219, 1.04371725816851312523941175600, 1.06857757292724509442689158003, 1.65348924668444002075700189589, 1.68571817940766294943066243175, 1.75233951222987119927881956637, 1.81802609134381555725091027169, 1.87415412496480074407177498487, 1.91130644582780361871557749589, 1.93193693961976909973044143900, 2.11731036099219031010878801531, 2.13646882255354498416696226385, 2.23723577448634154013816981948, 2.28176336041672583499196088323, 2.30746764115570162124907178238, 2.44934761378763437776912287314

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

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