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Properties

Label 12-1215e6-1.1-c0e6-0-1
Degree $12$
Conductor $3.217\times 10^{18}$
Sign $1$
Analytic cond. $0.0497050$
Root an. cond. $0.778693$
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  + 3·5-s + 2·8-s + 3·25-s + 6·40-s − 3·47-s − 3·49-s + 64-s + 6·107-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 + ⋯
L(s)  = 1  + 3·5-s + 2·8-s + 3·25-s + 6·40-s − 3·47-s − 3·49-s + 64-s + 6·107-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{30} \cdot 5^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{30} \cdot 5^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(3^{30} \cdot 5^{6}\)
Sign: $1$
Analytic conductor: \(0.0497050\)
Root analytic conductor: \(0.778693\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 3^{30} \cdot 5^{6} ,\ ( \ : [0]^{6} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.233175251\)
\(L(\frac12)\) \(\approx\) \(2.233175251\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( ( 1 - T + T^{2} )^{3} \)
good2 \( ( 1 - T^{3} + T^{6} )^{2} \)
7 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
11 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
13 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
17 \( ( 1 - T^{3} + T^{6} )^{2} \)
19 \( ( 1 + T^{3} + T^{6} )^{2} \)
23 \( ( 1 - T^{3} + T^{6} )^{2} \)
29 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
31 \( ( 1 + T^{3} + T^{6} )^{2} \)
37 \( ( 1 - T )^{6}( 1 + T )^{6} \)
41 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
43 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
47 \( ( 1 + T )^{6}( 1 - T + T^{2} )^{3} \)
53 \( ( 1 - T^{3} + T^{6} )^{2} \)
59 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
61 \( ( 1 + T^{3} + T^{6} )^{2} \)
67 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
71 \( ( 1 - T )^{6}( 1 + T )^{6} \)
73 \( ( 1 - T )^{6}( 1 + T )^{6} \)
79 \( ( 1 + T^{3} + T^{6} )^{2} \)
83 \( ( 1 - T^{3} + T^{6} )^{2} \)
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

−5.42403670455020529932128691637, −5.14503441570692434650679970820, −5.03965394631363721020175065647, −4.87192026932947252345106588910, −4.82455115323385194506942301479, −4.73530692255601190862159569826, −4.47767367291054647539826962080, −4.24986825529568919630353379582, −4.18792924045637888091612635364, −3.90574347852655801628328596824, −3.74066853534088770495302106891, −3.42730116950810653028055622756, −3.39654279676769974666416316783, −3.22863440300120722587320186993, −2.92282612118883291678072368072, −2.76029924715524281446289401417, −2.57063535871271363650588059072, −2.31390761245212291527448806302, −2.02760841365534393785105269913, −2.01684037435772616157819798221, −1.66709214923435557140922791929, −1.60691308174696087125175910576, −1.53750201066889805159593740776, −1.30717069961066755328970863679, −0.836481394621267295993117142369, 0.836481394621267295993117142369, 1.30717069961066755328970863679, 1.53750201066889805159593740776, 1.60691308174696087125175910576, 1.66709214923435557140922791929, 2.01684037435772616157819798221, 2.02760841365534393785105269913, 2.31390761245212291527448806302, 2.57063535871271363650588059072, 2.76029924715524281446289401417, 2.92282612118883291678072368072, 3.22863440300120722587320186993, 3.39654279676769974666416316783, 3.42730116950810653028055622756, 3.74066853534088770495302106891, 3.90574347852655801628328596824, 4.18792924045637888091612635364, 4.24986825529568919630353379582, 4.47767367291054647539826962080, 4.73530692255601190862159569826, 4.82455115323385194506942301479, 4.87192026932947252345106588910, 5.03965394631363721020175065647, 5.14503441570692434650679970820, 5.42403670455020529932128691637

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

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