Properties

Label 1-161-161.160-r0-0-0
Degree $1$
Conductor $161$
Sign $1$
Analytic cond. $0.747680$
Root an. cond. $0.747680$
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 − 3-s + 4-s + 5-s − 6-s + 8-s + 9-s + 10-s − 11-s − 12-s − 13-s − 15-s + 16-s + 17-s + 18-s + 19-s + 20-s − 22-s − 24-s + 25-s − 26-s − 27-s + 29-s − 30-s − 31-s + 32-s + 33-s + ⋯
L(s)  = 1  + 2-s − 3-s + 4-s + 5-s − 6-s + 8-s + 9-s + 10-s − 11-s − 12-s − 13-s − 15-s + 16-s + 17-s + 18-s + 19-s + 20-s − 22-s − 24-s + 25-s − 26-s − 27-s + 29-s − 30-s − 31-s + 32-s + 33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(161\)    =    \(7 \cdot 23\)
Sign: $1$
Analytic conductor: \(0.747680\)
Root analytic conductor: \(0.747680\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{161} (160, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 161,\ (0:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.750625444\)
\(L(\frac12)\) \(\approx\) \(1.750625444\)
\(L(1)\) \(\approx\) \(1.586762750\)
\(L(1)\) \(\approx\) \(1.586762750\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
23 \( 1 \)
good2 \( 1 + T \)
3 \( 1 - T \)
5 \( 1 + T \)
11 \( 1 - T \)
13 \( 1 - T \)
17 \( 1 + T \)
19 \( 1 + T \)
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

−28.226252687712368459295611300423, −26.77274857872284051945166964378, −25.56632477095717528263813972541, −24.6117817142226311236206426553, −23.79090883058220017624650656748, −22.85859667003473489391548494036, −21.93560577389607203213354983439, −21.3477563724487271676221702084, −20.36655620582330525564999594214, −18.80742558305517790053516488457, −17.717278743183782620940570904444, −16.741384089260543997842076207, −15.902790106330856173014191101230, −14.65074325152942269815807234764, −13.574027095976763595476127239559, −12.647676531253781502250337935563, −11.81400642901997425883153943088, −10.50030792309514014368953394528, −9.87029862675439222505852118235, −7.621130913731959828121210104193, −6.57503960231461631301215816788, −5.36486926249605990713385616200, −4.999082081045909074381129347452, −3.10218030541642144456326707603, −1.65723590715700437225096705131, 1.65723590715700437225096705131, 3.10218030541642144456326707603, 4.999082081045909074381129347452, 5.36486926249605990713385616200, 6.57503960231461631301215816788, 7.621130913731959828121210104193, 9.87029862675439222505852118235, 10.50030792309514014368953394528, 11.81400642901997425883153943088, 12.647676531253781502250337935563, 13.574027095976763595476127239559, 14.65074325152942269815807234764, 15.902790106330856173014191101230, 16.741384089260543997842076207, 17.717278743183782620940570904444, 18.80742558305517790053516488457, 20.36655620582330525564999594214, 21.3477563724487271676221702084, 21.93560577389607203213354983439, 22.85859667003473489391548494036, 23.79090883058220017624650656748, 24.6117817142226311236206426553, 25.56632477095717528263813972541, 26.77274857872284051945166964378, 28.226252687712368459295611300423

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