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Properties

Label 12-6e18-1.1-c0e6-0-0
Degree $12$
Conductor $1.016\times 10^{14}$
Sign $1$
Analytic cond. $1.56915\times 10^{-6}$
Root an. cond. $0.328326$
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 − 27-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 + ⋯
L(s)  = 1  − 8-s − 3·11-s − 27-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 + ⋯

Functional equation

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

Invariants

Degree: \(12\)
Conductor: \(2^{18} \cdot 3^{18}\)
Sign: $1$
Analytic conductor: \(1.56915\times 10^{-6}\)
Root analytic conductor: \(0.328326\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{18} \cdot 3^{18} ,\ ( \ : [0]^{6} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1095100918\)
\(L(\frac12)\) \(\approx\) \(0.1095100918\)
\(L(1)\) 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 \( 1 + T^{3} + T^{6} \)
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

−6.93695020489685553561509634112, −6.89665279786183129393175293035, −6.87212787298639068654634342695, −6.65256265235872878193443521054, −6.28765311599362488319783499160, −5.94026093401960317177308463747, −5.84167590691965716889083289510, −5.82766409258450834246922328554, −5.48389329429040903075294870649, −5.21108410862649946687208184609, −5.19134054502797220968515953638, −5.00456087207015534385987498116, −4.72744834922822389289818147415, −4.70295673344309367410168992789, −4.22934562509257204306838507095, −3.83801659828596263560765615994, −3.69748497958035591559088146679, −3.47532539152169408503793629324, −3.26958848221300922261022230232, −3.07553742771147273664702880928, −2.56436732427807162727599322871, −2.41271315573078278806371546827, −2.40222102885017417011513209495, −1.78266404823769550531317073504, −1.50292278577570919072792984455, 1.50292278577570919072792984455, 1.78266404823769550531317073504, 2.40222102885017417011513209495, 2.41271315573078278806371546827, 2.56436732427807162727599322871, 3.07553742771147273664702880928, 3.26958848221300922261022230232, 3.47532539152169408503793629324, 3.69748497958035591559088146679, 3.83801659828596263560765615994, 4.22934562509257204306838507095, 4.70295673344309367410168992789, 4.72744834922822389289818147415, 5.00456087207015534385987498116, 5.19134054502797220968515953638, 5.21108410862649946687208184609, 5.48389329429040903075294870649, 5.82766409258450834246922328554, 5.84167590691965716889083289510, 5.94026093401960317177308463747, 6.28765311599362488319783499160, 6.65256265235872878193443521054, 6.87212787298639068654634342695, 6.89665279786183129393175293035, 6.93695020489685553561509634112

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

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