Properties

Label 40-384e20-1.1-c9e20-0-0
Degree $40$
Conductor $4.860\times 10^{51}$
Sign $1$
Analytic cond. $8.38249\times 10^{45}$
Root an. cond. $14.0632$
Motivic weight $9$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 6.56e4·9-s − 9.05e5·17-s + 1.46e7·25-s + 7.42e7·41-s − 2.77e8·49-s − 8.95e8·73-s + 2.36e9·81-s + 8.82e8·89-s + 4.33e8·97-s + 3.53e9·113-s + 1.11e10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 5.94e10·153-s + 157-s + 163-s + 167-s + 6.58e10·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
L(s)  = 1  − 3.33·9-s − 2.63·17-s + 7.52·25-s + 4.10·41-s − 6.86·49-s − 3.68·73-s + 55/9·81-s + 1.49·89-s + 0.497·97-s + 2.03·113-s + 4.74·121-s + 8.76·153-s + 6.21·169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{140} \cdot 3^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(10-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{140} \cdot 3^{20}\right)^{s/2} \, \Gamma_{\C}(s+9/2)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(40\)
Conductor: \(2^{140} \cdot 3^{20}\)
Sign: $1$
Analytic conductor: \(8.38249\times 10^{45}\)
Root analytic conductor: \(14.0632\)
Motivic weight: \(9\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{384} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((40,\ 2^{140} \cdot 3^{20} ,\ ( \ : [9/2]^{20} ),\ 1 )\)

Particular Values

\(L(5)\) \(\approx\) \(20.62785461\)
\(L(\frac12)\) \(\approx\) \(20.62785461\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( ( 1 + p^{8} T^{2} )^{10} \)
good5 \( ( 1 - 1468994 p T^{2} + 30651186670437 T^{4} - 96038225164508667192 T^{6} + \)\(19\!\cdots\!62\)\( p^{3} T^{8} - \)\(80\!\cdots\!12\)\( p^{4} T^{10} + \)\(19\!\cdots\!62\)\( p^{21} T^{12} - 96038225164508667192 p^{36} T^{14} + 30651186670437 p^{54} T^{16} - 1468994 p^{73} T^{18} + p^{90} T^{20} )^{2} \)
7 \( ( 1 + 138574110 T^{2} + 22746130578923 p^{3} T^{4} + \)\(13\!\cdots\!96\)\( p^{4} T^{6} + \)\(16\!\cdots\!62\)\( p^{6} T^{8} + \)\(16\!\cdots\!72\)\( p^{8} T^{10} + \)\(16\!\cdots\!62\)\( p^{24} T^{12} + \)\(13\!\cdots\!96\)\( p^{40} T^{14} + 22746130578923 p^{57} T^{16} + 138574110 p^{72} T^{18} + p^{90} T^{20} )^{2} \)
11 \( ( 1 - 5594405214 T^{2} + 1320109756127459717 T^{4} + \)\(36\!\cdots\!04\)\( T^{6} + \)\(22\!\cdots\!46\)\( T^{8} - \)\(33\!\cdots\!28\)\( T^{10} + \)\(22\!\cdots\!46\)\( p^{18} T^{12} + \)\(36\!\cdots\!04\)\( p^{36} T^{14} + 1320109756127459717 p^{54} T^{16} - 5594405214 p^{72} T^{18} + p^{90} T^{20} )^{2} \)
13 \( ( 1 - 32949251106 T^{2} + \)\(66\!\cdots\!85\)\( T^{4} - \)\(11\!\cdots\!80\)\( T^{6} + \)\(15\!\cdots\!98\)\( T^{8} - \)\(17\!\cdots\!84\)\( T^{10} + \)\(15\!\cdots\!98\)\( p^{18} T^{12} - \)\(11\!\cdots\!80\)\( p^{36} T^{14} + \)\(66\!\cdots\!85\)\( p^{54} T^{16} - 32949251106 p^{72} T^{18} + p^{90} T^{20} )^{2} \)
17 \( ( 1 + 226442 T + 298338751229 T^{2} + 55181224456447224 T^{3} + \)\(50\!\cdots\!62\)\( T^{4} + \)\(90\!\cdots\!36\)\( T^{5} + \)\(50\!\cdots\!62\)\( p^{9} T^{6} + 55181224456447224 p^{18} T^{7} + 298338751229 p^{27} T^{8} + 226442 p^{36} T^{9} + p^{45} T^{10} )^{4} \)
19 \( ( 1 - 2023428505646 T^{2} + \)\(20\!\cdots\!93\)\( T^{4} - \)\(13\!\cdots\!32\)\( T^{6} + \)\(66\!\cdots\!02\)\( T^{8} - \)\(24\!\cdots\!56\)\( T^{10} + \)\(66\!\cdots\!02\)\( p^{18} T^{12} - \)\(13\!\cdots\!32\)\( p^{36} T^{14} + \)\(20\!\cdots\!93\)\( p^{54} T^{16} - 2023428505646 p^{72} T^{18} + p^{90} T^{20} )^{2} \)
23 \( ( 1 + 6696077631430 T^{2} + \)\(23\!\cdots\!61\)\( T^{4} + \)\(58\!\cdots\!60\)\( T^{6} + \)\(12\!\cdots\!74\)\( T^{8} + \)\(25\!\cdots\!16\)\( T^{10} + \)\(12\!\cdots\!74\)\( p^{18} T^{12} + \)\(58\!\cdots\!60\)\( p^{36} T^{14} + \)\(23\!\cdots\!61\)\( p^{54} T^{16} + 6696077631430 p^{72} T^{18} + p^{90} T^{20} )^{2} \)
29 \( ( 1 - 111359043735994 T^{2} + \)\(59\!\cdots\!89\)\( T^{4} - \)\(19\!\cdots\!36\)\( T^{6} + \)\(46\!\cdots\!82\)\( T^{8} - \)\(78\!\cdots\!44\)\( T^{10} + \)\(46\!\cdots\!82\)\( p^{18} T^{12} - \)\(19\!\cdots\!36\)\( p^{36} T^{14} + \)\(59\!\cdots\!89\)\( p^{54} T^{16} - 111359043735994 p^{72} T^{18} + p^{90} T^{20} )^{2} \)
31 \( ( 1 + 89941603007566 T^{2} + \)\(44\!\cdots\!89\)\( T^{4} + \)\(14\!\cdots\!08\)\( T^{6} + \)\(33\!\cdots\!82\)\( T^{8} + \)\(79\!\cdots\!64\)\( T^{10} + \)\(33\!\cdots\!82\)\( p^{18} T^{12} + \)\(14\!\cdots\!08\)\( p^{36} T^{14} + \)\(44\!\cdots\!89\)\( p^{54} T^{16} + 89941603007566 p^{72} T^{18} + p^{90} T^{20} )^{2} \)
37 \( ( 1 - 950204653074226 T^{2} + \)\(43\!\cdots\!69\)\( T^{4} - \)\(12\!\cdots\!40\)\( T^{6} + \)\(26\!\cdots\!14\)\( T^{8} - \)\(40\!\cdots\!72\)\( T^{10} + \)\(26\!\cdots\!14\)\( p^{18} T^{12} - \)\(12\!\cdots\!40\)\( p^{36} T^{14} + \)\(43\!\cdots\!69\)\( p^{54} T^{16} - 950204653074226 p^{72} T^{18} + p^{90} T^{20} )^{2} \)
41 \( ( 1 - 18566002 T + 707323868353013 T^{2} - \)\(99\!\cdots\!60\)\( T^{3} + \)\(29\!\cdots\!34\)\( T^{4} - \)\(28\!\cdots\!36\)\( T^{5} + \)\(29\!\cdots\!34\)\( p^{9} T^{6} - \)\(99\!\cdots\!60\)\( p^{18} T^{7} + 707323868353013 p^{27} T^{8} - 18566002 p^{36} T^{9} + p^{45} T^{10} )^{4} \)
43 \( ( 1 - 1602548191720990 T^{2} + \)\(12\!\cdots\!29\)\( T^{4} - \)\(67\!\cdots\!48\)\( T^{6} + \)\(39\!\cdots\!02\)\( T^{8} - \)\(21\!\cdots\!44\)\( T^{10} + \)\(39\!\cdots\!02\)\( p^{18} T^{12} - \)\(67\!\cdots\!48\)\( p^{36} T^{14} + \)\(12\!\cdots\!29\)\( p^{54} T^{16} - 1602548191720990 p^{72} T^{18} + p^{90} T^{20} )^{2} \)
47 \( ( 1 + 4650398863290934 T^{2} + \)\(10\!\cdots\!29\)\( T^{4} + \)\(17\!\cdots\!80\)\( T^{6} + \)\(24\!\cdots\!02\)\( T^{8} + \)\(29\!\cdots\!96\)\( T^{10} + \)\(24\!\cdots\!02\)\( p^{18} T^{12} + \)\(17\!\cdots\!80\)\( p^{36} T^{14} + \)\(10\!\cdots\!29\)\( p^{54} T^{16} + 4650398863290934 p^{72} T^{18} + p^{90} T^{20} )^{2} \)
53 \( ( 1 - 15550103826912298 T^{2} + \)\(11\!\cdots\!21\)\( T^{4} - \)\(63\!\cdots\!88\)\( T^{6} + \)\(28\!\cdots\!74\)\( T^{8} - \)\(10\!\cdots\!96\)\( T^{10} + \)\(28\!\cdots\!74\)\( p^{18} T^{12} - \)\(63\!\cdots\!88\)\( p^{36} T^{14} + \)\(11\!\cdots\!21\)\( p^{54} T^{16} - 15550103826912298 p^{72} T^{18} + p^{90} T^{20} )^{2} \)
59 \( ( 1 - 66972384180888126 T^{2} + \)\(21\!\cdots\!29\)\( T^{4} - \)\(43\!\cdots\!24\)\( T^{6} + \)\(60\!\cdots\!46\)\( T^{8} - \)\(60\!\cdots\!72\)\( T^{10} + \)\(60\!\cdots\!46\)\( p^{18} T^{12} - \)\(43\!\cdots\!24\)\( p^{36} T^{14} + \)\(21\!\cdots\!29\)\( p^{54} T^{16} - 66972384180888126 p^{72} T^{18} + p^{90} T^{20} )^{2} \)
61 \( ( 1 - 78984113736497442 T^{2} + \)\(28\!\cdots\!73\)\( T^{4} - \)\(64\!\cdots\!84\)\( T^{6} + \)\(10\!\cdots\!90\)\( T^{8} - \)\(13\!\cdots\!16\)\( T^{10} + \)\(10\!\cdots\!90\)\( p^{18} T^{12} - \)\(64\!\cdots\!84\)\( p^{36} T^{14} + \)\(28\!\cdots\!73\)\( p^{54} T^{16} - 78984113736497442 p^{72} T^{18} + p^{90} T^{20} )^{2} \)
67 \( ( 1 - 107853403180866830 T^{2} + \)\(66\!\cdots\!65\)\( T^{4} - \)\(31\!\cdots\!60\)\( T^{6} + \)\(11\!\cdots\!50\)\( T^{8} - \)\(34\!\cdots\!28\)\( T^{10} + \)\(11\!\cdots\!50\)\( p^{18} T^{12} - \)\(31\!\cdots\!60\)\( p^{36} T^{14} + \)\(66\!\cdots\!65\)\( p^{54} T^{16} - 107853403180866830 p^{72} T^{18} + p^{90} T^{20} )^{2} \)
71 \( ( 1 + 290646683595608230 T^{2} + \)\(41\!\cdots\!05\)\( T^{4} + \)\(39\!\cdots\!20\)\( T^{6} + \)\(26\!\cdots\!10\)\( T^{8} + \)\(14\!\cdots\!80\)\( T^{10} + \)\(26\!\cdots\!10\)\( p^{18} T^{12} + \)\(39\!\cdots\!20\)\( p^{36} T^{14} + \)\(41\!\cdots\!05\)\( p^{54} T^{16} + 290646683595608230 p^{72} T^{18} + p^{90} T^{20} )^{2} \)
73 \( ( 1 + 223798474 T + 88967880198633429 T^{2} + \)\(14\!\cdots\!12\)\( T^{3} + \)\(46\!\cdots\!82\)\( T^{4} + \)\(61\!\cdots\!96\)\( T^{5} + \)\(46\!\cdots\!82\)\( p^{9} T^{6} + \)\(14\!\cdots\!12\)\( p^{18} T^{7} + 88967880198633429 p^{27} T^{8} + 223798474 p^{36} T^{9} + p^{45} T^{10} )^{4} \)
79 \( ( 1 + 499851356938410286 T^{2} + \)\(10\!\cdots\!73\)\( T^{4} + \)\(11\!\cdots\!76\)\( T^{6} + \)\(43\!\cdots\!38\)\( T^{8} - \)\(16\!\cdots\!40\)\( T^{10} + \)\(43\!\cdots\!38\)\( p^{18} T^{12} + \)\(11\!\cdots\!76\)\( p^{36} T^{14} + \)\(10\!\cdots\!73\)\( p^{54} T^{16} + 499851356938410286 p^{72} T^{18} + p^{90} T^{20} )^{2} \)
83 \( ( 1 - 1071966143100584622 T^{2} + \)\(61\!\cdots\!73\)\( T^{4} - \)\(23\!\cdots\!28\)\( T^{6} + \)\(65\!\cdots\!06\)\( T^{8} - \)\(13\!\cdots\!40\)\( T^{10} + \)\(65\!\cdots\!06\)\( p^{18} T^{12} - \)\(23\!\cdots\!28\)\( p^{36} T^{14} + \)\(61\!\cdots\!73\)\( p^{54} T^{16} - 1071966143100584622 p^{72} T^{18} + p^{90} T^{20} )^{2} \)
89 \( ( 1 - 220605534 T + 1090737971548238821 T^{2} - \)\(39\!\cdots\!44\)\( T^{3} + \)\(59\!\cdots\!30\)\( T^{4} - \)\(20\!\cdots\!00\)\( T^{5} + \)\(59\!\cdots\!30\)\( p^{9} T^{6} - \)\(39\!\cdots\!44\)\( p^{18} T^{7} + 1090737971548238821 p^{27} T^{8} - 220605534 p^{36} T^{9} + p^{45} T^{10} )^{4} \)
97 \( ( 1 - 108420934 T + 2168537688075866253 T^{2} - \)\(48\!\cdots\!24\)\( T^{3} + \)\(25\!\cdots\!62\)\( p T^{4} - \)\(62\!\cdots\!68\)\( T^{5} + \)\(25\!\cdots\!62\)\( p^{10} T^{6} - \)\(48\!\cdots\!24\)\( p^{18} T^{7} + 2168537688075866253 p^{27} T^{8} - 108420934 p^{36} T^{9} + p^{45} T^{10} )^{4} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{40} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−1.39794352621832384333055638829, −1.38860422970214596896886575295, −1.31921339155735563815116127243, −1.25403093302822948077901721523, −1.22176980244256355873520393601, −1.14959344896462001012048123787, −1.14201622919820316463024833640, −0.938998669107494061630875628861, −0.936199966262799814892795974270, −0.846814027454214207298436432373, −0.835342266239957891534976904303, −0.76204408883239951899596249726, −0.72201261381317915100526183891, −0.71001789757123716137653131468, −0.70355092951673281586299693642, −0.55159218120886621784210649474, −0.51773319051972491620878935797, −0.40420057026726342481434266750, −0.34357799398662077228015654842, −0.28316582757071225314109587507, −0.26632936684015839501259005994, −0.26348877431804642025320863907, −0.14683636397693051917070970724, −0.14240117643848465691326722185, −0.07420119508230752187033218296, 0.07420119508230752187033218296, 0.14240117643848465691326722185, 0.14683636397693051917070970724, 0.26348877431804642025320863907, 0.26632936684015839501259005994, 0.28316582757071225314109587507, 0.34357799398662077228015654842, 0.40420057026726342481434266750, 0.51773319051972491620878935797, 0.55159218120886621784210649474, 0.70355092951673281586299693642, 0.71001789757123716137653131468, 0.72201261381317915100526183891, 0.76204408883239951899596249726, 0.835342266239957891534976904303, 0.846814027454214207298436432373, 0.936199966262799814892795974270, 0.938998669107494061630875628861, 1.14201622919820316463024833640, 1.14959344896462001012048123787, 1.22176980244256355873520393601, 1.25403093302822948077901721523, 1.31921339155735563815116127243, 1.38860422970214596896886575295, 1.39794352621832384333055638829

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

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