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Properties

Label 1-381-381.380-r0-0-0
Degree $1$
Conductor $381$
Sign $1$
Analytic cond. $1.76935$
Root an. cond. $1.76935$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(381\)    =    \(3 \cdot 127\)
Sign: $1$
Analytic conductor: \(1.76935\)
Root analytic conductor: \(1.76935\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{381} (380, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 381,\ (0:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8943492599\)
\(L(\frac12)\) \(\approx\) \(0.8943492599\)
\(L(1)\) \(\approx\) \(0.7803375671\)
\(L(1)\) \(\approx\) \(0.7803375671\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
127 \( 1 \)
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

−25.00838232172142336370200351237, −23.83507987693949191626892460496, −22.79861990417864498033240088315, −21.698953186549048195902362376465, −20.901454979325997176034772560121, −20.14476442633885400379801642691, −19.11887961327711160020380270558, −18.236987357801661269835314354654, −17.748099764059375885436938951013, −16.586104507051073190632546478469, −15.95459493960004047161312700510, −15.09903729084220708575940202249, −13.52287995634325006554401921380, −13.09021323039172953264574302077, −11.72319468806766554493591274178, −10.61268941531571974408527304379, −9.984619317369109937372823227589, −9.112943285857401662805664573546, −8.25980328153644861598775199251, −6.874006161944720562074328233938, −6.27916480359486153705408477897, −5.176820857680449693761216473450, −3.218468865743940311565412297516, −2.41456870050531142193684063387, −0.99979569085501172249760102723, 0.99979569085501172249760102723, 2.41456870050531142193684063387, 3.218468865743940311565412297516, 5.176820857680449693761216473450, 6.27916480359486153705408477897, 6.874006161944720562074328233938, 8.25980328153644861598775199251, 9.112943285857401662805664573546, 9.984619317369109937372823227589, 10.61268941531571974408527304379, 11.72319468806766554493591274178, 13.09021323039172953264574302077, 13.52287995634325006554401921380, 15.09903729084220708575940202249, 15.95459493960004047161312700510, 16.586104507051073190632546478469, 17.748099764059375885436938951013, 18.236987357801661269835314354654, 19.11887961327711160020380270558, 20.14476442633885400379801642691, 20.901454979325997176034772560121, 21.698953186549048195902362376465, 22.79861990417864498033240088315, 23.83507987693949191626892460496, 25.00838232172142336370200351237

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