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Properties

Label 48-3549e24-1.1-c0e24-0-1
Degree $48$
Conductor $1.594\times 10^{85}$
Sign $1$
Analytic cond. $908479.$
Root an. cond. $1.33085$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·7-s + 9-s + 16-s + 2·19-s + 2·31-s + 2·37-s + 49-s − 2·63-s + 2·67-s − 2·73-s − 2·97-s − 4·103-s − 2·109-s − 2·112-s + 127-s + 131-s − 4·133-s + 137-s + 139-s + 144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 12·169-s + 2·171-s + ⋯
L(s)  = 1  − 2·7-s + 9-s + 16-s + 2·19-s + 2·31-s + 2·37-s + 49-s − 2·63-s + 2·67-s − 2·73-s − 2·97-s − 4·103-s − 2·109-s − 2·112-s + 127-s + 131-s − 4·133-s + 137-s + 139-s + 144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 12·169-s + 2·171-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 7^{24} \cdot 13^{48}\right)^{s/2} \, \Gamma_{\C}(s)^{24} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 7^{24} \cdot 13^{48}\right)^{s/2} \, \Gamma_{\C}(s)^{24} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(48\)
Conductor: \(3^{24} \cdot 7^{24} \cdot 13^{48}\)
Sign: $1$
Analytic conductor: \(908479.\)
Root analytic conductor: \(1.33085\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((48,\ 3^{24} \cdot 7^{24} \cdot 13^{48} ,\ ( \ : [0]^{24} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3934835876\)
\(L(\frac12)\) \(\approx\) \(0.3934835876\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} - T^{22} + T^{24} \)
7 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} )^{2} \)
13 \( ( 1 + T^{2} )^{12} \)
good2 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} - T^{28} + T^{32} - T^{36} + T^{40} - T^{44} + T^{48} \)
5 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} - T^{28} + T^{32} - T^{36} + T^{40} - T^{44} + T^{48} \)
11 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} - T^{28} + T^{32} - T^{36} + T^{40} - T^{44} + T^{48} \)
17 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} )^{2} \)
19 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} - T^{22} + T^{24} ) \)
23 \( ( 1 + T^{2} )^{24} \)
29 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} )^{2} \)
31 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} - T^{22} + T^{24} ) \)
37 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} - T^{22} + T^{24} ) \)
41 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} - T^{28} + T^{32} - T^{36} + T^{40} - T^{44} + T^{48} \)
43 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} )^{2} \)
47 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} - T^{28} + T^{32} - T^{36} + T^{40} - T^{44} + T^{48} \)
53 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} )^{2} \)
59 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} - T^{28} + T^{32} - T^{36} + T^{40} - T^{44} + T^{48} \)
61 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} - T^{22} + T^{24} )^{2} \)
67 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} - T^{22} + T^{24} ) \)
71 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} - T^{28} + T^{32} - T^{36} + T^{40} - T^{44} + T^{48} \)
73 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} - T^{22} + T^{24} ) \)
79 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} - T^{22} + T^{24} )^{2} \)
83 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} - T^{28} + T^{32} - T^{36} + T^{40} - T^{44} + T^{48} \)
89 \( ( 1 + T^{4} )^{12} \)
97 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} - T^{22} + T^{24} ) \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{48} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−1.81119279118608834767895922480, −1.80081269825433796701421780497, −1.65676176399780600175906242407, −1.53789130973854445947844761135, −1.48140985160578549683391060901, −1.47020766012851595700528227601, −1.41599366021870940561677447192, −1.39185398060350387900262518158, −1.37260495493148986286846834007, −1.27933256768390118874555434952, −1.25708637453697551956134307860, −1.24016210027528494815394632051, −1.21077182900279851243227981519, −1.12166569018203924539436260245, −0.876957218499768739678577921333, −0.867570820381302890298747618614, −0.862059983734764813922707177666, −0.841930588672628310663563241272, −0.817545335473844661265199987845, −0.803195381995767767048060133083, −0.73497368450995999736915385952, −0.62665274310650836540833534781, −0.60329697418487206095159337628, −0.22761969172063096905199092358, −0.090050708139072926104166980771, 0.090050708139072926104166980771, 0.22761969172063096905199092358, 0.60329697418487206095159337628, 0.62665274310650836540833534781, 0.73497368450995999736915385952, 0.803195381995767767048060133083, 0.817545335473844661265199987845, 0.841930588672628310663563241272, 0.862059983734764813922707177666, 0.867570820381302890298747618614, 0.876957218499768739678577921333, 1.12166569018203924539436260245, 1.21077182900279851243227981519, 1.24016210027528494815394632051, 1.25708637453697551956134307860, 1.27933256768390118874555434952, 1.37260495493148986286846834007, 1.39185398060350387900262518158, 1.41599366021870940561677447192, 1.47020766012851595700528227601, 1.48140985160578549683391060901, 1.53789130973854445947844761135, 1.65676176399780600175906242407, 1.80081269825433796701421780497, 1.81119279118608834767895922480

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

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