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Properties

Label 12-263e6-1.1-c0e6-0-0
Degree $12$
Conductor $3.309\times 10^{14}$
Sign $1$
Analytic cond. $5.11301\times 10^{-6}$
Root an. cond. $0.362290$
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  − 2-s − 3-s + 6-s − 11-s − 13-s − 17-s + 22-s − 23-s + 6·25-s + 26-s − 31-s + 33-s + 34-s − 37-s + 39-s − 43-s + 46-s + 6·49-s − 6·50-s + 51-s − 61-s + 62-s − 66-s + 69-s + 74-s − 6·75-s − 78-s + ⋯
L(s)  = 1  − 2-s − 3-s + 6-s − 11-s − 13-s − 17-s + 22-s − 23-s + 6·25-s + 26-s − 31-s + 33-s + 34-s − 37-s + 39-s − 43-s + 46-s + 6·49-s − 6·50-s + 51-s − 61-s + 62-s − 66-s + 69-s + 74-s − 6·75-s − 78-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(263^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(263^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(263^{6}\)
Sign: $1$
Analytic conductor: \(5.11301\times 10^{-6}\)
Root analytic conductor: \(0.362290\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 263^{6} ,\ ( \ : [0]^{6} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.06185005614\)
\(L(\frac12)\) \(\approx\) \(0.06185005614\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad263 \( ( 1 - T )^{6} \)
good2 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} \)
3 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} \)
5 \( ( 1 - T )^{6}( 1 + T )^{6} \)
7 \( ( 1 - T )^{6}( 1 + T )^{6} \)
11 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} \)
13 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} \)
17 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} \)
19 \( ( 1 - T )^{6}( 1 + T )^{6} \)
23 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} \)
29 \( ( 1 - T )^{6}( 1 + T )^{6} \)
31 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} \)
37 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} \)
41 \( ( 1 - T )^{6}( 1 + T )^{6} \)
43 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} \)
47 \( ( 1 - T )^{6}( 1 + T )^{6} \)
53 \( ( 1 - T )^{6}( 1 + T )^{6} \)
59 \( ( 1 - T )^{6}( 1 + T )^{6} \)
61 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} \)
67 \( ( 1 - T )^{6}( 1 + T )^{6} \)
71 \( ( 1 - T )^{6}( 1 + T )^{6} \)
73 \( ( 1 - T )^{6}( 1 + T )^{6} \)
79 \( ( 1 - T )^{6}( 1 + T )^{6} \)
83 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} \)
89 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} \)
97 \( ( 1 - T )^{6}( 1 + T )^{6} \)
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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

−7.01018730248721100775498988457, −6.79745572810821041853106421168, −6.50549405940831323373473221354, −6.29514499203447553318925663214, −6.21034677642492272626270284947, −5.77037938295177235184125497000, −5.65769121541756419659803929282, −5.61046838821559415491322030925, −5.21676679535671954273197033030, −5.12149260583007580434589540280, −5.03038801102175437299744466725, −4.85149005045640174852370450181, −4.59875790746741040819536959662, −4.25160568616770671434310195972, −4.21229277106725329425056511189, −3.84810543074318642338122195045, −3.68038397151614772052762315342, −3.17899105379353884731962730483, −3.04456167103902164349486647248, −2.72571833867768426736756605605, −2.55240887459485376263238372920, −2.36269671410128617461773958733, −2.07494068410505324404849657877, −1.28005016013897399942236922946, −1.10769750339985271935591557117, 1.10769750339985271935591557117, 1.28005016013897399942236922946, 2.07494068410505324404849657877, 2.36269671410128617461773958733, 2.55240887459485376263238372920, 2.72571833867768426736756605605, 3.04456167103902164349486647248, 3.17899105379353884731962730483, 3.68038397151614772052762315342, 3.84810543074318642338122195045, 4.21229277106725329425056511189, 4.25160568616770671434310195972, 4.59875790746741040819536959662, 4.85149005045640174852370450181, 5.03038801102175437299744466725, 5.12149260583007580434589540280, 5.21676679535671954273197033030, 5.61046838821559415491322030925, 5.65769121541756419659803929282, 5.77037938295177235184125497000, 6.21034677642492272626270284947, 6.29514499203447553318925663214, 6.50549405940831323373473221354, 6.79745572810821041853106421168, 7.01018730248721100775498988457

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

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