Properties

Label 12-648e6-1.1-c0e6-0-0
Degree $12$
Conductor $7.404\times 10^{16}$
Sign $1$
Analytic cond. $0.00114391$
Root an. cond. $0.568677$
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  + 8-s + 3·11-s + 3·41-s − 3·43-s − 6·59-s − 3·67-s + 3·88-s − 3·89-s − 3·97-s + 3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯
L(s)  = 1  + 8-s + 3·11-s + 3·41-s − 3·43-s − 6·59-s − 3·67-s + 3·88-s − 3·89-s − 3·97-s + 3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{18} \cdot 3^{24}\)
Sign: $1$
Analytic conductor: \(0.00114391\)
Root analytic conductor: \(0.568677\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{648} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{18} \cdot 3^{24} ,\ ( \ : [0]^{6} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7489681680\)
\(L(\frac12)\) \(\approx\) \(0.7489681680\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T^{3} + T^{6} \)
3 \( 1 \)
good5 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
7 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
11 \( ( 1 - T + T^{2} )^{3}( 1 - T^{3} + T^{6} ) \)
13 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
17 \( ( 1 - T^{3} + T^{6} )^{2} \)
19 \( ( 1 + T^{3} + T^{6} )^{2} \)
23 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
29 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
31 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
37 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
41 \( ( 1 - T + T^{2} )^{3}( 1 - T^{3} + T^{6} ) \)
43 \( ( 1 + T + T^{2} )^{3}( 1 + T^{3} + T^{6} ) \)
47 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
53 \( ( 1 - T )^{6}( 1 + T )^{6} \)
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 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
73 \( ( 1 + T^{3} + T^{6} )^{2} \)
79 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
83 \( ( 1 - T^{3} + T^{6} )^{2} \)
89 \( ( 1 + T )^{6}( 1 - T + T^{2} )^{3} \)
97 \( ( 1 + T + T^{2} )^{3}( 1 + T^{3} + T^{6} ) \)
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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

−5.98450871464762345955013208741, −5.80841108431118560691542198890, −5.60850636494450442091549474917, −5.57865849778524076113325276868, −5.08936252215798086106877877919, −4.90411437313865413709001679114, −4.76728833577939548002084748431, −4.64058123968197968418501812715, −4.51387278780308344635790454175, −4.36269865603094444550033539475, −4.27009057692490596875312490315, −3.84692162109391219141052490840, −3.81137319305538519004421568370, −3.76314719529268763888071934575, −3.45490098698584044943874717969, −3.04572850693123334834849917021, −2.94297834986848735156757820177, −2.84081477783645816140093298989, −2.78836931095467073363034603674, −2.15089482639313799589237408829, −1.89428405712027679101302479928, −1.70823222980527210448508480564, −1.37093533327596151907169413100, −1.34644698177982694049245314355, −1.20756043082390257455401919427, 1.20756043082390257455401919427, 1.34644698177982694049245314355, 1.37093533327596151907169413100, 1.70823222980527210448508480564, 1.89428405712027679101302479928, 2.15089482639313799589237408829, 2.78836931095467073363034603674, 2.84081477783645816140093298989, 2.94297834986848735156757820177, 3.04572850693123334834849917021, 3.45490098698584044943874717969, 3.76314719529268763888071934575, 3.81137319305538519004421568370, 3.84692162109391219141052490840, 4.27009057692490596875312490315, 4.36269865603094444550033539475, 4.51387278780308344635790454175, 4.64058123968197968418501812715, 4.76728833577939548002084748431, 4.90411437313865413709001679114, 5.08936252215798086106877877919, 5.57865849778524076113325276868, 5.60850636494450442091549474917, 5.80841108431118560691542198890, 5.98450871464762345955013208741

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

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