Properties

Label 12-407e6-1.1-c0e6-0-0
Degree $12$
Conductor $4.545\times 10^{15}$
Sign $1$
Analytic cond. $7.02278\times 10^{-5}$
Root an. cond. $0.450687$
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  − 3·3-s + 3·9-s − 3·11-s + 27-s + 12·31-s + 9·33-s − 3·53-s + 6·59-s − 64-s − 3·67-s − 3·71-s − 6·81-s − 3·89-s − 36·93-s − 9·99-s + 3·121-s − 2·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 9·159-s + 163-s + 167-s + ⋯
L(s)  = 1  − 3·3-s + 3·9-s − 3·11-s + 27-s + 12·31-s + 9·33-s − 3·53-s + 6·59-s − 64-s − 3·67-s − 3·71-s − 6·81-s − 3·89-s − 36·93-s − 9·99-s + 3·121-s − 2·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 9·159-s + 163-s + 167-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(11^{6} \cdot 37^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(11^{6} \cdot 37^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(11^{6} \cdot 37^{6}\)
Sign: $1$
Analytic conductor: \(7.02278\times 10^{-5}\)
Root analytic conductor: \(0.450687\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 11^{6} \cdot 37^{6} ,\ ( \ : [0]^{6} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.09112991219\)
\(L(\frac12)\) \(\approx\) \(0.09112991219\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( ( 1 + T + T^{2} )^{3} \)
37 \( 1 + T^{3} + T^{6} \)
good2 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
3 \( ( 1 + T + T^{2} )^{3}( 1 + T^{3} + T^{6} ) \)
5 \( ( 1 + T^{3} + T^{6} )^{2} \)
7 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
13 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
17 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
19 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
23 \( ( 1 + T^{3} + T^{6} )^{2} \)
29 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
31 \( ( 1 - T )^{12} \)
41 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
43 \( ( 1 - T )^{6}( 1 + T )^{6} \)
47 \( ( 1 + T^{3} + T^{6} )^{2} \)
53 \( ( 1 + T + T^{2} )^{3}( 1 + T^{3} + T^{6} ) \)
59 \( ( 1 - T )^{6}( 1 + T^{3} + T^{6} ) \)
61 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
67 \( ( 1 + T + T^{2} )^{3}( 1 + T^{3} + T^{6} ) \)
71 \( ( 1 + T + T^{2} )^{3}( 1 + T^{3} + T^{6} ) \)
73 \( ( 1 - T )^{6}( 1 + T )^{6} \)
79 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
83 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
89 \( ( 1 + T + T^{2} )^{3}( 1 + T^{3} + T^{6} ) \)
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

−6.20274850566523525593278106534, −6.13407929120898718749817132677, −6.00479517200805152061105620017, −5.81170389665132519448374669403, −5.76290871876270903178548918995, −5.49154630204879132175238337142, −5.36090117218041470415267888230, −4.88949111719141926285001843134, −4.88938404430479415485044603182, −4.85585393060742151947985136386, −4.75483690409214865644332432709, −4.60217740992373578203418581281, −4.15563747961215323007051919158, −4.14419623490745225779043186665, −4.13250490568667006801659723813, −3.22744442883835758129111703026, −2.98438436443007968456624537123, −2.93919599092711986645224029126, −2.82900007417402049918220310946, −2.62814320490132022408455502918, −2.58909232800666601550057405467, −2.20327293651571592584385886639, −1.34472141062828023445034705506, −1.06933456498190192970040794797, −0.866913797025874151951299178537, 0.866913797025874151951299178537, 1.06933456498190192970040794797, 1.34472141062828023445034705506, 2.20327293651571592584385886639, 2.58909232800666601550057405467, 2.62814320490132022408455502918, 2.82900007417402049918220310946, 2.93919599092711986645224029126, 2.98438436443007968456624537123, 3.22744442883835758129111703026, 4.13250490568667006801659723813, 4.14419623490745225779043186665, 4.15563747961215323007051919158, 4.60217740992373578203418581281, 4.75483690409214865644332432709, 4.85585393060742151947985136386, 4.88938404430479415485044603182, 4.88949111719141926285001843134, 5.36090117218041470415267888230, 5.49154630204879132175238337142, 5.76290871876270903178548918995, 5.81170389665132519448374669403, 6.00479517200805152061105620017, 6.13407929120898718749817132677, 6.20274850566523525593278106534

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

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