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Properties

Label 12-2808e6-1.1-c0e6-0-1
Degree $12$
Conductor $4.902\times 10^{20}$
Sign $1$
Analytic cond. $7.57400$
Root an. cond. $1.18379$
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  − 3·2-s + 3·4-s + 2·8-s + 3·13-s − 9·16-s − 9·26-s − 3·31-s + 9·32-s + 9·52-s + 9·62-s + 3·64-s + 6·104-s − 12·107-s − 3·113-s − 3·121-s − 9·124-s − 2·125-s + 127-s − 18·128-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯
L(s)  = 1  − 3·2-s + 3·4-s + 2·8-s + 3·13-s − 9·16-s − 9·26-s − 3·31-s + 9·32-s + 9·52-s + 9·62-s + 3·64-s + 6·104-s − 12·107-s − 3·113-s − 3·121-s − 9·124-s − 2·125-s + 127-s − 18·128-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{18} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{18} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{18} \cdot 3^{18} \cdot 13^{6}\)
Sign: $1$
Analytic conductor: \(7.57400\)
Root analytic conductor: \(1.18379\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{18} \cdot 3^{18} \cdot 13^{6} ,\ ( \ : [0]^{6} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.08263573192\)
\(L(\frac12)\) \(\approx\) \(0.08263573192\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 + T + T^{2} )^{3} \)
3 \( 1 \)
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

−4.90389680623335144345109766395, −4.51382251616826448514028692641, −4.24189420886046969216899724626, −4.19823612867088008797091594702, −4.16806957194241548386804165606, −4.05975863225977907507757771955, −3.82096945185121174575224085854, −3.77772152122551983612911582404, −3.74258519202426201591370616436, −3.58164931579764138223197677082, −3.28725047446398793402895011719, −2.92940487187282073148640491457, −2.78969335713640462938453510180, −2.70186698595627849511834931484, −2.64508964392216495612867237549, −2.35897544538286635196880520181, −1.92498346421381928771151318593, −1.77086727421514565803142752803, −1.72655268863680024321275219210, −1.53824782267641602597131885995, −1.38790115458303616844080222582, −1.22929477832101223348740461954, −1.07325261089522034517326325899, −0.74537706121267937539713751819, −0.22018328809072736702437035216, 0.22018328809072736702437035216, 0.74537706121267937539713751819, 1.07325261089522034517326325899, 1.22929477832101223348740461954, 1.38790115458303616844080222582, 1.53824782267641602597131885995, 1.72655268863680024321275219210, 1.77086727421514565803142752803, 1.92498346421381928771151318593, 2.35897544538286635196880520181, 2.64508964392216495612867237549, 2.70186698595627849511834931484, 2.78969335713640462938453510180, 2.92940487187282073148640491457, 3.28725047446398793402895011719, 3.58164931579764138223197677082, 3.74258519202426201591370616436, 3.77772152122551983612911582404, 3.82096945185121174575224085854, 4.05975863225977907507757771955, 4.16806957194241548386804165606, 4.19823612867088008797091594702, 4.24189420886046969216899724626, 4.51382251616826448514028692641, 4.90389680623335144345109766395

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

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