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Properties

Label 2-2151-239.238-c0-0-0
Degree $2$
Conductor $2151$
Sign $-1$
Analytic cond. $1.07348$
Root an. cond. $1.03609$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 5-s + 1.41i·7-s − 8-s − 10-s − 11-s − 1.41i·13-s + 1.41i·14-s − 16-s − 17-s − 1.41i·19-s − 22-s + 1.41i·23-s − 1.41i·26-s − 29-s + ⋯
L(s)  = 1  + 2-s − 5-s + 1.41i·7-s − 8-s − 10-s − 11-s − 1.41i·13-s + 1.41i·14-s − 16-s − 17-s − 1.41i·19-s − 22-s + 1.41i·23-s − 1.41i·26-s − 29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2151\)    =    \(3^{2} \cdot 239\)
Sign: $-1$
Analytic conductor: \(1.07348\)
Root analytic conductor: \(1.03609\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2151} (955, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2151,\ (\ :0),\ -1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1584004200\)
\(L(\frac12)\) \(\approx\) \(0.1584004200\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
239 \( 1 + T \)
good2 \( 1 - T + T^{2} \)
5 \( 1 + T + T^{2} \)
7 \( 1 - 1.41iT - T^{2} \)
11 \( 1 + T + T^{2} \)
13 \( 1 + 1.41iT - T^{2} \)
17 \( 1 + T + T^{2} \)
19 \( 1 + 1.41iT - T^{2} \)
23 \( 1 - 1.41iT - T^{2} \)
29 \( 1 + T + T^{2} \)
31 \( 1 + T + T^{2} \)
37 \( 1 - 1.41iT - T^{2} \)
41 \( 1 + 1.41iT - T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 - 1.41iT - T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T + T^{2} \)
67 \( 1 + T + T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - T + T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 - T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.364558653309133186863229632688, −8.913712851769911105734396598342, −8.053816892174974553752283109470, −7.40066449448733838107564404479, −6.23028335320851513442158649447, −5.28174142241706029063108391798, −5.16361768773396945947607737351, −3.94927823860764189670122396399, −3.10134008867550264656629972698, −2.42536525069669998329821145827, 0.07416276483991733702633029023, 2.10659290656490315807921617376, 3.50387419107720342803070054452, 4.12307580894715968002608103038, 4.45908829379912200882644955193, 5.52756195735526834969096377398, 6.55295933975131773851341150436, 7.22532037688207414665326407006, 7.981588645673202158222035717812, 8.768854186393531276627995672271

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