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Properties

Label 1-4143-4143.4142-r1-0-0
Degree $1$
Conductor $4143$
Sign $1$
Analytic cond. $445.227$
Root an. cond. $445.227$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s − 5-s + 7-s + 8-s − 10-s + 11-s + 13-s + 14-s + 16-s − 17-s − 19-s − 20-s + 22-s − 23-s + 25-s + 26-s + 28-s + 29-s − 31-s + 32-s − 34-s − 35-s + 37-s − 38-s − 40-s + 41-s + ⋯
L(s)  = 1  + 2-s + 4-s − 5-s + 7-s + 8-s − 10-s + 11-s + 13-s + 14-s + 16-s − 17-s − 19-s − 20-s + 22-s − 23-s + 25-s + 26-s + 28-s + 29-s − 31-s + 32-s − 34-s − 35-s + 37-s − 38-s − 40-s + 41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4143 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4143 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4143\)    =    \(3 \cdot 1381\)
Sign: $1$
Analytic conductor: \(445.227\)
Root analytic conductor: \(445.227\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{4143} (4142, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 4143,\ (1:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(5.603198276\)
\(L(\frac12)\) \(\approx\) \(5.603198276\)
\(L(1)\) \(\approx\) \(2.147558893\)
\(L(1)\) \(\approx\) \(2.147558893\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
1381 \( 1 \)
good2 \( 1 + T \)
5 \( 1 - T \)
7 \( 1 + T \)
11 \( 1 + T \)
13 \( 1 + T \)
17 \( 1 - T \)
19 \( 1 - T \)
23 \( 1 - T \)
29 \( 1 + T \)
31 \( 1 - T \)
37 \( 1 + T \)
41 \( 1 + T \)
43 \( 1 - T \)
47 \( 1 - T \)
53 \( 1 + T \)
59 \( 1 + T \)
61 \( 1 + T \)
67 \( 1 - T \)
71 \( 1 - T \)
73 \( 1 + T \)
79 \( 1 + T \)
83 \( 1 + T \)
89 \( 1 + T \)
97 \( 1 - T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.14014635173934833770967763559, −17.61905042970137678431750595648, −16.45392103196747543084082233934, −16.27547295335824427102980116781, −15.25210624574367330517267658590, −14.87805564628516409060379706021, −14.28406533615333051793914983608, −13.50220894208075986907480024164, −12.81105452109488036445589587967, −11.97693360101712764677501251260, −11.50287792247604336843827674446, −11.02103256218384077945527222246, −10.37043858452467453699765396622, −9.02469157146044863265399178748, −8.32906472355121996879452922966, −7.82194349656089821562260890143, −6.796979158195894035329374015380, −6.39673440981936344544236610744, −5.44331125435710654013298147386, −4.44318370984326571092570384418, −4.17029037264844347045240908932, −3.519823564611594104079627279214, −2.385616312149320522533942560224, −1.64303265760779659893151102565, −0.72987533941949730715084345688, 0.72987533941949730715084345688, 1.64303265760779659893151102565, 2.385616312149320522533942560224, 3.519823564611594104079627279214, 4.17029037264844347045240908932, 4.44318370984326571092570384418, 5.44331125435710654013298147386, 6.39673440981936344544236610744, 6.796979158195894035329374015380, 7.82194349656089821562260890143, 8.32906472355121996879452922966, 9.02469157146044863265399178748, 10.37043858452467453699765396622, 11.02103256218384077945527222246, 11.50287792247604336843827674446, 11.97693360101712764677501251260, 12.81105452109488036445589587967, 13.50220894208075986907480024164, 14.28406533615333051793914983608, 14.87805564628516409060379706021, 15.25210624574367330517267658590, 16.27547295335824427102980116781, 16.45392103196747543084082233934, 17.61905042970137678431750595648, 18.14014635173934833770967763559

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