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Properties

Label 12-2960e6-1.1-c0e6-0-1
Degree $12$
Conductor $6.726\times 10^{20}$
Sign $1$
Analytic cond. $10.3918$
Root an. cond. $1.21541$
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  + 3·5-s + 3·17-s + 3·25-s + 3·41-s − 6·61-s + 9·85-s + 6·89-s + 3·109-s − 3·121-s − 2·125-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 + 9·205-s + ⋯
L(s)  = 1  + 3·5-s + 3·17-s + 3·25-s + 3·41-s − 6·61-s + 9·85-s + 6·89-s + 3·109-s − 3·121-s − 2·125-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 + 9·205-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{6} \cdot 37^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{6} \cdot 37^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{24} \cdot 5^{6} \cdot 37^{6}\)
Sign: $1$
Analytic conductor: \(10.3918\)
Root analytic conductor: \(1.21541\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{24} \cdot 5^{6} \cdot 37^{6} ,\ ( \ : [0]^{6} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.799660815\)
\(L(\frac12)\) \(\approx\) \(4.799660815\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( ( 1 - T + T^{2} )^{3} \)
37 \( 1 + T^{3} + T^{6} \)
good3 \( 1 - T^{6} + T^{12} \)
7 \( 1 - T^{6} + T^{12} \)
11 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
13 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
17 \( ( 1 - T + T^{2} )^{3}( 1 + T^{3} + T^{6} ) \)
19 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
23 \( ( 1 - T^{2} + T^{4} )^{3} \)
29 \( ( 1 + T^{3} + T^{6} )^{2} \)
31 \( ( 1 - T )^{6}( 1 + T )^{6} \)
41 \( ( 1 - T + T^{2} )^{3}( 1 - T^{3} + T^{6} ) \)
43 \( ( 1 + T^{2} )^{6} \)
47 \( ( 1 - T^{2} + T^{4} )^{3} \)
53 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
59 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
61 \( ( 1 + T )^{6}( 1 - T^{3} + T^{6} ) \)
67 \( 1 - T^{6} + T^{12} \)
71 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
73 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
79 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
83 \( 1 - T^{6} + T^{12} \)
89 \( ( 1 - T )^{6}( 1 + T^{3} + T^{6} ) \)
97 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
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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

−4.86490024805418520897403517120, −4.71392421655945255805320476196, −4.44381553768921732595813673864, −4.14452673835279198071711663017, −4.12408902638415485054756385436, −4.10326813641609086752666829977, −4.06177365460030625128401030788, −3.52770570051339012888827305077, −3.36006806512681040598811595529, −3.31963559672060161439984066105, −3.26071587161163226157379346029, −3.13365276779110987950459235299, −2.98023392957289680448694892082, −2.58389485632583811498182655909, −2.57679909886225275027461410748, −2.47251635465140625155527843455, −2.11467765686802138023919121503, −1.99479666487297060881955332557, −1.92479720447447357976641160795, −1.78280011909008805058367522047, −1.49358667221553987625071396665, −1.30847329791483175148100414287, −1.06162985192457472635288034548, −1.01686930626138233168466744919, −0.63792047485703077912084748771, 0.63792047485703077912084748771, 1.01686930626138233168466744919, 1.06162985192457472635288034548, 1.30847329791483175148100414287, 1.49358667221553987625071396665, 1.78280011909008805058367522047, 1.92479720447447357976641160795, 1.99479666487297060881955332557, 2.11467765686802138023919121503, 2.47251635465140625155527843455, 2.57679909886225275027461410748, 2.58389485632583811498182655909, 2.98023392957289680448694892082, 3.13365276779110987950459235299, 3.26071587161163226157379346029, 3.31963559672060161439984066105, 3.36006806512681040598811595529, 3.52770570051339012888827305077, 4.06177365460030625128401030788, 4.10326813641609086752666829977, 4.12408902638415485054756385436, 4.14452673835279198071711663017, 4.44381553768921732595813673864, 4.71392421655945255805320476196, 4.86490024805418520897403517120

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

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