Properties

Label 12-3744e6-1.1-c0e6-0-0
Degree $12$
Conductor $2.754\times 10^{21}$
Sign $1$
Analytic cond. $42.5557$
Root an. cond. $1.36693$
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  − 3·13-s + 27-s − 3·31-s − 12·107-s + 3·113-s − 3·121-s − 2·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯
L(s)  = 1  − 3·13-s + 27-s − 3·31-s − 12·107-s + 3·113-s − 3·121-s − 2·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 3^{12} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 3^{12} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{30} \cdot 3^{12} \cdot 13^{6}\)
Sign: $1$
Analytic conductor: \(42.5557\)
Root analytic conductor: \(1.36693\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{3744} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{30} \cdot 3^{12} \cdot 13^{6} ,\ ( \ : [0]^{6} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.03498185796\)
\(L(\frac12)\) \(\approx\) \(0.03498185796\)
\(L(1)\) 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 \( 1 - T^{3} + T^{6} \)
13 \( ( 1 + T + T^{2} )^{3} \)
good5 \( ( 1 + T^{3} + T^{6} )^{2} \)
7 \( ( 1 - T^{3} + T^{6} )^{2} \)
11 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
17 \( ( 1 + T^{3} + T^{6} )^{2} \)
19 \( ( 1 - T )^{6}( 1 + T )^{6} \)
23 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
29 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
31 \( ( 1 + T )^{6}( 1 - T + T^{2} )^{3} \)
37 \( ( 1 + T^{3} + T^{6} )^{2} \)
41 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
43 \( ( 1 - T^{3} + T^{6} )^{2} \)
47 \( ( 1 - T^{3} + T^{6} )^{2} \)
53 \( ( 1 - T )^{6}( 1 + T )^{6} \)
59 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
61 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
67 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
71 \( ( 1 - T^{3} + T^{6} )^{2} \)
73 \( ( 1 - T )^{6}( 1 + T )^{6} \)
79 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
83 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
89 \( ( 1 - T )^{6}( 1 + T )^{6} \)
97 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
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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

−4.58032883635711901996472717360, −4.47549417812639687870051519295, −4.16360597452696556417884068996, −4.13290662533100953462873554894, −4.07772831943094740135274776692, −3.89686197769802497935337394068, −3.75896972628953209098223509902, −3.63843085583174294992786969762, −3.52115451161312057333861382525, −3.28881866872003456776600253792, −2.89344185370798417512906918267, −2.80871150141933921927577870020, −2.80068619899772810642715945836, −2.77427477467094565633722405222, −2.55678852059728142736891584574, −2.44357045637019992500120236814, −2.15790323934057622335037041346, −1.92695788602108619953694636416, −1.87183735885200913296928451193, −1.59438240604667677760719850812, −1.49827954681443338889068047044, −1.22952677383235132616985765277, −1.07628345669822305692950463093, −0.63111814297088148890816658048, −0.06019349182816634045108503560, 0.06019349182816634045108503560, 0.63111814297088148890816658048, 1.07628345669822305692950463093, 1.22952677383235132616985765277, 1.49827954681443338889068047044, 1.59438240604667677760719850812, 1.87183735885200913296928451193, 1.92695788602108619953694636416, 2.15790323934057622335037041346, 2.44357045637019992500120236814, 2.55678852059728142736891584574, 2.77427477467094565633722405222, 2.80068619899772810642715945836, 2.80871150141933921927577870020, 2.89344185370798417512906918267, 3.28881866872003456776600253792, 3.52115451161312057333861382525, 3.63843085583174294992786969762, 3.75896972628953209098223509902, 3.89686197769802497935337394068, 4.07772831943094740135274776692, 4.13290662533100953462873554894, 4.16360597452696556417884068996, 4.47549417812639687870051519295, 4.58032883635711901996472717360

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

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