Properties

Label 12-1944e6-1.1-c0e6-0-1
Degree $12$
Conductor $5.397\times 10^{19}$
Sign $1$
Analytic cond. $0.833912$
Root an. cond. $0.984978$
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 + 6·43-s − 3·59-s + 6·67-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 + 6·43-s − 3·59-s + 6·67-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

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{30}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{30}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{18} \cdot 3^{30}\)
Sign: $1$
Analytic conductor: \(0.833912\)
Root analytic conductor: \(0.984978\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1944} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{18} \cdot 3^{30} ,\ ( \ : [0]^{6} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7872623435\)
\(L(\frac12)\) \(\approx\) \(0.7872623435\)
\(L(1)\) 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 \( ( 1 - T )^{6}( 1 + T^{3} + T^{6} ) \)
47 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
53 \( ( 1 - T )^{6}( 1 + T )^{6} \)
59 \( ( 1 + T + T^{2} )^{3}( 1 + T^{3} + T^{6} ) \)
61 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
67 \( ( 1 - T )^{6}( 1 + T^{3} + T^{6} ) \)
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

−4.94565169933415971208404761297, −4.93241644630640544412886096179, −4.91111447577254248250570847255, −4.82304306845759910727165732387, −4.20744685159412530731377526403, −4.19328651141861537719199733377, −3.97233515403431741698803097371, −3.94533710061696132756330164745, −3.89748154702166409664975750356, −3.84805081366725728563999595556, −3.25609525346659460459144805620, −3.08093391364812346600425015509, −3.04904879953784496565247383891, −2.96713504216340982369358047273, −2.93715750199233864845552312308, −2.59575344926785343145080441703, −2.39545538881496084653710512343, −2.32175171856978797643088917230, −2.04461484772460385476743235488, −1.98329943019660303178165381658, −1.66181791972155906343863642944, −1.52292273774040810470822398476, −0.880761588254465777176600955243, −0.799154903726255189152722892458, −0.51371093473590003338310190372, 0.51371093473590003338310190372, 0.799154903726255189152722892458, 0.880761588254465777176600955243, 1.52292273774040810470822398476, 1.66181791972155906343863642944, 1.98329943019660303178165381658, 2.04461484772460385476743235488, 2.32175171856978797643088917230, 2.39545538881496084653710512343, 2.59575344926785343145080441703, 2.93715750199233864845552312308, 2.96713504216340982369358047273, 3.04904879953784496565247383891, 3.08093391364812346600425015509, 3.25609525346659460459144805620, 3.84805081366725728563999595556, 3.89748154702166409664975750356, 3.94533710061696132756330164745, 3.97233515403431741698803097371, 4.19328651141861537719199733377, 4.20744685159412530731377526403, 4.82304306845759910727165732387, 4.91111447577254248250570847255, 4.93241644630640544412886096179, 4.94565169933415971208404761297

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

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