Properties

Label 12-2888e6-1.1-c0e6-0-5
Degree $12$
Conductor $5.802\times 10^{20}$
Sign $1$
Analytic cond. $8.96449$
Root an. cond. $1.20054$
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 − 2·27-s − 3·49-s + 3·83-s − 3·88-s − 6·107-s − 6·113-s + 6·121-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 + 2·216-s + ⋯
L(s)  = 1  − 8-s + 3·11-s − 2·27-s − 3·49-s + 3·83-s − 3·88-s − 6·107-s − 6·113-s + 6·121-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 + 2·216-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{18} \cdot 19^{12}\)
Sign: $1$
Analytic conductor: \(8.96449\)
Root analytic conductor: \(1.20054\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2888} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{18} \cdot 19^{12} ,\ ( \ : [0]^{6} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.526905941\)
\(L(\frac12)\) \(\approx\) \(1.526905941\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T^{3} + T^{6} \)
19 \( 1 \)
good3 \( ( 1 + T^{3} + T^{6} )^{2} \)
5 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
7 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
11 \( ( 1 - T )^{6}( 1 + T + T^{2} )^{3} \)
13 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
17 \( ( 1 + T^{3} + T^{6} )^{2} \)
23 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
29 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
31 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
37 \( ( 1 - T )^{6}( 1 + T )^{6} \)
41 \( ( 1 + T^{3} + T^{6} )^{2} \)
43 \( ( 1 + T^{3} + T^{6} )^{2} \)
47 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
53 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
59 \( ( 1 + T^{3} + T^{6} )^{2} \)
61 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
67 \( ( 1 + T^{3} + T^{6} )^{2} \)
71 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
73 \( ( 1 + T^{3} + T^{6} )^{2} \)
79 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
83 \( ( 1 - T )^{6}( 1 + T + T^{2} )^{3} \)
89 \( ( 1 + T^{3} + T^{6} )^{2} \)
97 \( ( 1 + T^{3} + T^{6} )^{2} \)
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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

−4.68709766483578484052480593346, −4.65551833989971582584397563549, −4.58378914598783496287340410022, −4.14757706686331138797609810281, −4.14482564202360871784907291852, −3.93525383357316514256716777302, −3.82228519154998574299724434764, −3.75739115220094840020928697475, −3.56087509032473436438334635633, −3.46728412778949314313551068734, −3.42724340133725705905199335827, −3.16788614294526589651645975774, −2.82071232895791343736159884846, −2.80295473625207870785760040799, −2.58776253955292085717042848389, −2.49061118893622838472850498626, −2.36719589324225337726965757802, −1.95828242719723953100016845398, −1.69878754120843026987059563009, −1.67243791021429230629917055859, −1.46458321187513046661434171785, −1.34219073662482448052155765958, −1.31871239397045894815729063763, −0.60910059503852922791290117459, −0.51616929634515816952066625186, 0.51616929634515816952066625186, 0.60910059503852922791290117459, 1.31871239397045894815729063763, 1.34219073662482448052155765958, 1.46458321187513046661434171785, 1.67243791021429230629917055859, 1.69878754120843026987059563009, 1.95828242719723953100016845398, 2.36719589324225337726965757802, 2.49061118893622838472850498626, 2.58776253955292085717042848389, 2.80295473625207870785760040799, 2.82071232895791343736159884846, 3.16788614294526589651645975774, 3.42724340133725705905199335827, 3.46728412778949314313551068734, 3.56087509032473436438334635633, 3.75739115220094840020928697475, 3.82228519154998574299724434764, 3.93525383357316514256716777302, 4.14482564202360871784907291852, 4.14757706686331138797609810281, 4.58378914598783496287340410022, 4.65551833989971582584397563549, 4.68709766483578484052480593346

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

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