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 of factors

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.