Properties

Label 12-2352e6-1.1-c0e6-0-1
Degree $12$
Conductor $1.693\times 10^{20}$
Sign $1$
Analytic cond. $2.61557$
Root an. cond. $1.08342$
Motivic weight $0$
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  + 3-s − 7-s − 2·19-s − 21-s + 25-s + 2·31-s − 5·37-s − 2·57-s − 7·61-s + 75-s + 2·93-s + 2·103-s − 2·109-s − 5·111-s + 121-s + 127-s + 131-s + 2·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 169-s + 173-s + ⋯
L(s)  = 1  + 3-s − 7-s − 2·19-s − 21-s + 25-s + 2·31-s − 5·37-s − 2·57-s − 7·61-s + 75-s + 2·93-s + 2·103-s − 2·109-s − 5·111-s + 121-s + 127-s + 131-s + 2·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 169-s + 173-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{6} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{6} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{24} \cdot 3^{6} \cdot 7^{12}\)
Sign: $1$
Analytic conductor: \(2.61557\)
Root analytic conductor: \(1.08342\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2352} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{24} \cdot 3^{6} \cdot 7^{12} ,\ ( \ : [0]^{6} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7256461061\)
\(L(\frac12)\) \(\approx\) \(0.7256461061\)
\(L(1)\) 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 \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \)
7 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \)
good5 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} \)
11 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} \)
13 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
17 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} \)
19 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
23 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} \)
29 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
31 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2} \)
37 \( ( 1 + T )^{6}( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} ) \)
41 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} \)
43 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
47 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
53 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
59 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
61 \( ( 1 + T )^{6}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
67 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
71 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} \)
73 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
79 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
83 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
89 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} \)
97 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−4.92354595125333116942223091019, −4.74288479931963217122658685460, −4.41905042605301835567597193727, −4.41137760130188995991913336089, −4.27873992943529624391779349223, −4.14566040655343131505209510875, −4.12989585224044429505395681893, −3.70673848974345101078448718270, −3.61492148778531330229414223324, −3.40419593127907856301500176221, −3.23809845493727868852194853221, −3.17488670903551823327707411186, −3.00572595977367940992522778050, −2.97727333996728002763040828790, −2.84332239119603107924381601805, −2.58905056271359823930045397405, −2.35756799934450588974114719000, −2.06139177548181627442371401479, −2.02800284648896336199436471169, −1.82493380605408697258664078838, −1.48657834125496618821922384319, −1.46523429046331886913914401604, −1.34581639210441885381041289027, −0.71995210668647087997647662698, −0.32137833805072084174409341947, 0.32137833805072084174409341947, 0.71995210668647087997647662698, 1.34581639210441885381041289027, 1.46523429046331886913914401604, 1.48657834125496618821922384319, 1.82493380605408697258664078838, 2.02800284648896336199436471169, 2.06139177548181627442371401479, 2.35756799934450588974114719000, 2.58905056271359823930045397405, 2.84332239119603107924381601805, 2.97727333996728002763040828790, 3.00572595977367940992522778050, 3.17488670903551823327707411186, 3.23809845493727868852194853221, 3.40419593127907856301500176221, 3.61492148778531330229414223324, 3.70673848974345101078448718270, 4.12989585224044429505395681893, 4.14566040655343131505209510875, 4.27873992943529624391779349223, 4.41137760130188995991913336089, 4.41905042605301835567597193727, 4.74288479931963217122658685460, 4.92354595125333116942223091019

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

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