Properties

Label 12-3120e6-1.1-c1e6-0-3
Degree $12$
Conductor $9.224\times 10^{20}$
Sign $1$
Analytic cond. $2.39105\times 10^{8}$
Root an. cond. $4.99132$
Motivic weight $1$
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  − 3·9-s + 12·11-s − 4·19-s − 5·25-s + 20·29-s − 24·31-s − 20·41-s + 42·49-s − 12·59-s + 4·61-s − 16·71-s + 40·79-s + 6·81-s + 44·89-s − 36·99-s − 60·101-s + 24·109-s + 38·121-s − 8·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + ⋯
L(s)  = 1  − 9-s + 3.61·11-s − 0.917·19-s − 25-s + 3.71·29-s − 4.31·31-s − 3.12·41-s + 6·49-s − 1.56·59-s + 0.512·61-s − 1.89·71-s + 4.50·79-s + 2/3·81-s + 4.66·89-s − 3.61·99-s − 5.97·101-s + 2.29·109-s + 3.45·121-s − 0.715·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{6} \cdot 5^{6} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{6} \cdot 5^{6} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{24} \cdot 3^{6} \cdot 5^{6} \cdot 13^{6}\)
Sign: $1$
Analytic conductor: \(2.39105\times 10^{8}\)
Root analytic conductor: \(4.99132\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{24} \cdot 3^{6} \cdot 5^{6} \cdot 13^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(13.08524835\)
\(L(\frac12)\) \(\approx\) \(13.08524835\)
\(L(\frac{3}{2})\) 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 \( ( 1 + T^{2} )^{3} \)
5 \( 1 + p T^{2} + 8 T^{3} + p^{2} T^{4} + p^{3} T^{6} \)
13 \( ( 1 + T^{2} )^{3} \)
good7 \( ( 1 - p T^{2} )^{6} \)
11 \( ( 1 - 6 T + 35 T^{2} - 112 T^{3} + 35 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
17 \( 1 - 46 T^{2} + 1439 T^{4} - 28068 T^{6} + 1439 p^{2} T^{8} - 46 p^{4} T^{10} + p^{6} T^{12} \)
19 \( ( 1 + 2 T + 33 T^{2} + 92 T^{3} + 33 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
23 \( 1 - 54 T^{2} + 1871 T^{4} - 46868 T^{6} + 1871 p^{2} T^{8} - 54 p^{4} T^{10} + p^{6} T^{12} \)
29 \( ( 1 - 10 T + 43 T^{2} - 108 T^{3} + 43 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
31 \( ( 1 + 12 T + 101 T^{2} + 584 T^{3} + 101 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
37 \( 1 - 130 T^{2} + 9335 T^{4} - 417660 T^{6} + 9335 p^{2} T^{8} - 130 p^{4} T^{10} + p^{6} T^{12} \)
41 \( ( 1 + 10 T + 89 T^{2} + 488 T^{3} + 89 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
43 \( 1 - 202 T^{2} + 19015 T^{4} - 1043212 T^{6} + 19015 p^{2} T^{8} - 202 p^{4} T^{10} + p^{6} T^{12} \)
47 \( 1 - 2 p T^{2} + 7139 T^{4} - 397884 T^{6} + 7139 p^{2} T^{8} - 2 p^{5} T^{10} + p^{6} T^{12} \)
53 \( 1 - 130 T^{2} + 13655 T^{4} - 792060 T^{6} + 13655 p^{2} T^{8} - 130 p^{4} T^{10} + p^{6} T^{12} \)
59 \( ( 1 + 6 T + 99 T^{2} + 752 T^{3} + 99 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
61 \( ( 1 - 2 T + 151 T^{2} - 212 T^{3} + 151 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
67 \( 1 - 130 T^{2} + 12871 T^{4} - 1093564 T^{6} + 12871 p^{2} T^{8} - 130 p^{4} T^{10} + p^{6} T^{12} \)
71 \( ( 1 + 8 T + 215 T^{2} + 1072 T^{3} + 215 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
73 \( 1 - 106 T^{2} + 14783 T^{4} - 1127628 T^{6} + 14783 p^{2} T^{8} - 106 p^{4} T^{10} + p^{6} T^{12} \)
79 \( ( 1 - 20 T + 345 T^{2} - 3320 T^{3} + 345 p T^{4} - 20 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
83 \( 1 - 358 T^{2} + 62555 T^{4} - 6541356 T^{6} + 62555 p^{2} T^{8} - 358 p^{4} T^{10} + p^{6} T^{12} \)
89 \( ( 1 - 22 T + 361 T^{2} - 3672 T^{3} + 361 p T^{4} - 22 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
97 \( ( 1 - 158 T^{2} + p^{2} T^{4} )^{3} \)
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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

−4.58876805687438348338886408060, −4.22658777644963277224711219020, −4.07280584953882041810949398383, −3.88912970204085948270739178323, −3.84284258198109916668165331117, −3.77789327831133070368561813568, −3.77085828579888510112810546576, −3.56479881321858100176329149335, −3.55415593881964060690529633781, −3.02697237492869481377911624690, −2.99047286123964886112421061733, −2.90570967842718281498014123252, −2.61763409108892445073203585862, −2.45703590356028158849151447077, −2.43635022999084056155742735816, −2.14746807883972408889166258360, −1.79708542088910533601733190947, −1.65211422586707982345360402373, −1.64811885978877486401605361282, −1.47437469455835327633122598897, −1.45748957006103370277085726975, −0.71317089395745856639903613525, −0.69706277352882374359123859339, −0.57406262764946145722890282107, −0.43442245076966229069987308226, 0.43442245076966229069987308226, 0.57406262764946145722890282107, 0.69706277352882374359123859339, 0.71317089395745856639903613525, 1.45748957006103370277085726975, 1.47437469455835327633122598897, 1.64811885978877486401605361282, 1.65211422586707982345360402373, 1.79708542088910533601733190947, 2.14746807883972408889166258360, 2.43635022999084056155742735816, 2.45703590356028158849151447077, 2.61763409108892445073203585862, 2.90570967842718281498014123252, 2.99047286123964886112421061733, 3.02697237492869481377911624690, 3.55415593881964060690529633781, 3.56479881321858100176329149335, 3.77085828579888510112810546576, 3.77789327831133070368561813568, 3.84284258198109916668165331117, 3.88912970204085948270739178323, 4.07280584953882041810949398383, 4.22658777644963277224711219020, 4.58876805687438348338886408060

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

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