Properties

Label 1-2e3-8.3-r1-0-0
Degree $1$
Conductor $8$
Sign $1$
Analytic cond. $0.859719$
Root an. cond. $0.859719$
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  + 3-s − 5-s − 7-s + 9-s + 11-s − 13-s − 15-s + 17-s + 19-s − 21-s − 23-s + 25-s + 27-s − 29-s − 31-s + 33-s + 35-s − 37-s − 39-s + 41-s + 43-s − 45-s − 47-s + 49-s + 51-s − 53-s − 55-s + ⋯
L(s)  = 1  + 3-s − 5-s − 7-s + 9-s + 11-s − 13-s − 15-s + 17-s + 19-s − 21-s − 23-s + 25-s + 27-s − 29-s − 31-s + 33-s + 35-s − 37-s − 39-s + 41-s + 43-s − 45-s − 47-s + 49-s + 51-s − 53-s − 55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(8\)    =    \(2^{3}\)
Sign: $1$
Analytic conductor: \(0.859719\)
Root analytic conductor: \(0.859719\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{8} (3, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 8,\ (1:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.100421409\)
\(L(\frac12)\) \(\approx\) \(1.100421409\)
\(L(1)\) \(\approx\) \(1.110720734\)
\(L(1)\) \(\approx\) \(1.110720734\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
good3 \( 1 + T \)
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

−48.74068793215534022752522930410, −47.6209798877567549178563249871, −45.9510335880281704577596028993, −43.9346152766520649240665150339, −42.752615477159649672801727607675, −41.53981595511799883077360485981, −39.16750681241490296713715528102, −38.0617400808002403731726223796, −36.23116033995471923810665296105, −34.98256753712872018308348058124, −32.53233400657462120871686846132, −31.45201345414038079725886531059, −29.857791921026421623346922599858, −27.5181259144140394812087660182, −26.089470593190540959426835382352, −24.41964822006060936671535842043, −22.37466756829608885834718488552, −20.07363842306806925540295209504, −19.08644166006480394769355338242, −16.16469366756340984309856870967, −14.49097192816054725085740936823, −12.3405011590722115530614093056, −9.50320196197290897854598873093, −7.43447295737022101391739084615, −3.57615483678758907557757872995, 3.57615483678758907557757872995, 7.43447295737022101391739084615, 9.50320196197290897854598873093, 12.3405011590722115530614093056, 14.49097192816054725085740936823, 16.16469366756340984309856870967, 19.08644166006480394769355338242, 20.07363842306806925540295209504, 22.37466756829608885834718488552, 24.41964822006060936671535842043, 26.089470593190540959426835382352, 27.5181259144140394812087660182, 29.857791921026421623346922599858, 31.45201345414038079725886531059, 32.53233400657462120871686846132, 34.98256753712872018308348058124, 36.23116033995471923810665296105, 38.0617400808002403731726223796, 39.16750681241490296713715528102, 41.53981595511799883077360485981, 42.752615477159649672801727607675, 43.9346152766520649240665150339, 45.9510335880281704577596028993, 47.6209798877567549178563249871, 48.74068793215534022752522930410

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