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Properties

Label 12-2736e6-1.1-c0e6-0-0
Degree $12$
Conductor $4.195\times 10^{20}$
Sign $1$
Analytic cond. $6.48095$
Root an. cond. $1.16852$
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  + 3·13-s − 3·19-s − 3·43-s + 6·61-s + 3·67-s + 3·73-s − 6·79-s − 3·121-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 + ⋯
L(s)  = 1  + 3·13-s − 3·19-s − 3·43-s + 6·61-s + 3·67-s + 3·73-s − 6·79-s − 3·121-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{24} \cdot 3^{12} \cdot 19^{6}\)
Sign: $1$
Analytic conductor: \(6.48095\)
Root analytic conductor: \(1.16852\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{24} \cdot 3^{12} \cdot 19^{6} ,\ ( \ : [0]^{6} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.008617870\)
\(L(\frac12)\) \(\approx\) \(1.008617870\)
\(L(1)\) 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 \( 1 \)
19 \( ( 1 + T + T^{2} )^{3} \)
good5 \( 1 - T^{6} + T^{12} \)
7 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
11 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
13 \( ( 1 - T + T^{2} )^{3}( 1 - T^{3} + T^{6} ) \)
17 \( 1 - T^{6} + T^{12} \)
23 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
29 \( 1 - T^{6} + T^{12} \)
31 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
37 \( ( 1 + T^{3} + T^{6} )^{2} \)
41 \( 1 - T^{6} + T^{12} \)
43 \( ( 1 + T + T^{2} )^{3}( 1 - T^{3} + T^{6} ) \)
47 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
53 \( 1 - T^{6} + T^{12} \)
59 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
61 \( ( 1 - T )^{6}( 1 + T^{3} + T^{6} ) \)
67 \( ( 1 - T + T^{2} )^{3}( 1 + T^{3} + T^{6} ) \)
71 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
73 \( ( 1 - T + T^{2} )^{3}( 1 - T^{3} + T^{6} ) \)
79 \( ( 1 + T )^{6}( 1 + T^{3} + T^{6} ) \)
83 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
89 \( 1 - T^{6} + T^{12} \)
97 \( ( 1 - T^{3} + T^{6} )^{2} \)
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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

−4.86403843603672319696373064768, −4.42394146450408279512184821282, −4.41526488262755357749970646050, −4.27442059749124439939269772986, −4.22458089905394565061156896591, −4.03944920814586975055848328400, −3.79994411513703291413319344133, −3.77109988308146855970313278162, −3.67540167275001715205225433856, −3.45609794551037068429060020729, −3.38042380681135728129232329399, −3.28744066663394950134247637470, −2.96589084186042026135142432250, −2.79863134501186826309233371785, −2.38762544390805577242169113432, −2.35739860201720531077296559279, −2.29168992537748781087707492552, −2.18229922937424077199966067396, −2.08728553481229556377015702274, −1.56404227040650274525021717039, −1.48433857041866618193387588187, −1.27312086160543450059328495415, −1.05638252967925549042839901036, −1.02642732426606904064979820242, −0.32742209667497723651042750906, 0.32742209667497723651042750906, 1.02642732426606904064979820242, 1.05638252967925549042839901036, 1.27312086160543450059328495415, 1.48433857041866618193387588187, 1.56404227040650274525021717039, 2.08728553481229556377015702274, 2.18229922937424077199966067396, 2.29168992537748781087707492552, 2.35739860201720531077296559279, 2.38762544390805577242169113432, 2.79863134501186826309233371785, 2.96589084186042026135142432250, 3.28744066663394950134247637470, 3.38042380681135728129232329399, 3.45609794551037068429060020729, 3.67540167275001715205225433856, 3.77109988308146855970313278162, 3.79994411513703291413319344133, 4.03944920814586975055848328400, 4.22458089905394565061156896591, 4.27442059749124439939269772986, 4.41526488262755357749970646050, 4.42394146450408279512184821282, 4.86403843603672319696373064768

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

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