Properties

Label 1-161-161.11-r1-0-0
Degree $1$
Conductor $161$
Sign $-0.369 - 0.929i$
Analytic cond. $17.3018$
Root an. cond. $17.3018$
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  + (0.580 + 0.814i)2-s + (0.235 + 0.971i)3-s + (−0.327 + 0.945i)4-s + (−0.0475 − 0.998i)5-s + (−0.654 + 0.755i)6-s + (−0.959 + 0.281i)8-s + (−0.888 + 0.458i)9-s + (0.786 − 0.618i)10-s + (−0.580 + 0.814i)11-s + (−0.995 − 0.0950i)12-s + (−0.142 − 0.989i)13-s + (0.959 − 0.281i)15-s + (−0.786 − 0.618i)16-s + (−0.981 − 0.189i)17-s + (−0.888 − 0.458i)18-s + (−0.981 + 0.189i)19-s + ⋯
L(s)  = 1  + (0.580 + 0.814i)2-s + (0.235 + 0.971i)3-s + (−0.327 + 0.945i)4-s + (−0.0475 − 0.998i)5-s + (−0.654 + 0.755i)6-s + (−0.959 + 0.281i)8-s + (−0.888 + 0.458i)9-s + (0.786 − 0.618i)10-s + (−0.580 + 0.814i)11-s + (−0.995 − 0.0950i)12-s + (−0.142 − 0.989i)13-s + (0.959 − 0.281i)15-s + (−0.786 − 0.618i)16-s + (−0.981 − 0.189i)17-s + (−0.888 − 0.458i)18-s + (−0.981 + 0.189i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.369 - 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.369 - 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(161\)    =    \(7 \cdot 23\)
Sign: $-0.369 - 0.929i$
Analytic conductor: \(17.3018\)
Root analytic conductor: \(17.3018\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{161} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 161,\ (1:\ ),\ -0.369 - 0.929i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.3487621028 + 0.5140939128i\)
\(L(\frac12)\) \(\approx\) \(-0.3487621028 + 0.5140939128i\)
\(L(1)\) \(\approx\) \(0.7352706273 + 0.7001623044i\)
\(L(1)\) \(\approx\) \(0.7352706273 + 0.7001623044i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (0.580 + 0.814i)T \)
3 \( 1 + (0.235 + 0.971i)T \)
5 \( 1 + (-0.0475 - 0.998i)T \)
11 \( 1 + (-0.580 + 0.814i)T \)
13 \( 1 + (-0.142 - 0.989i)T \)
17 \( 1 + (-0.981 - 0.189i)T \)
19 \( 1 + (-0.981 + 0.189i)T \)
29 \( 1 + (-0.654 + 0.755i)T \)
31 \( 1 + (0.723 + 0.690i)T \)
37 \( 1 + (0.888 - 0.458i)T \)
41 \( 1 + (0.841 - 0.540i)T \)
43 \( 1 + (0.959 + 0.281i)T \)
47 \( 1 + (-0.5 + 0.866i)T \)
53 \( 1 + (-0.928 + 0.371i)T \)
59 \( 1 + (-0.786 + 0.618i)T \)
61 \( 1 + (-0.235 + 0.971i)T \)
67 \( 1 + (0.995 - 0.0950i)T \)
71 \( 1 + (0.415 - 0.909i)T \)
73 \( 1 + (-0.327 + 0.945i)T \)
79 \( 1 + (-0.928 - 0.371i)T \)
83 \( 1 + (-0.841 - 0.540i)T \)
89 \( 1 + (-0.723 + 0.690i)T \)
97 \( 1 + (-0.841 + 0.540i)T \)
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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

−26.73734844833553837796774858400, −26.06763572021268370615945490693, −24.587679772447523693908868119944, −23.8176417864788880530479941774, −23.02380512337407414749092131760, −21.97369874663593291448260918294, −21.10582104622031704826895668421, −19.79910394936571503647591149909, −18.97293195855197599174673528338, −18.52347517281722582536495283945, −17.31159545693537825782218680221, −15.45575996593617507826002107273, −14.46585195481076893289332461178, −13.64711240088171473888151747981, −12.85927584379009640904849970619, −11.47440361367286017154448146753, −11.01422052840258371209996324038, −9.52079238937702789842041982643, −8.19395245743010051588302606069, −6.70280541304940321254779339431, −5.984998676380960591210374727401, −4.17461811688863519365028736229, −2.812167269016762669862730711020, −2.01598718694963179676158027921, −0.16769247617180531571146399579, 2.64342636419317748374270402007, 4.18379840247570893787694685982, 4.862072926579797821580994920334, 5.88313410613558091592282789082, 7.614396425591890794513910946626, 8.56683407998283963050955204718, 9.49301882799202186307106572081, 10.885942531974675262365851382274, 12.45215109444938498148954085742, 13.140891956312396340327613266941, 14.45005131436571981814132147498, 15.441973926414169578026966719108, 15.97923589754381673427704917385, 17.08859070096915409007065038098, 17.81888644454051045422619405591, 19.81604550187924924308006982390, 20.650067997552832378920861147746, 21.38009541126192744450059160054, 22.50078683661870688278021315948, 23.27752376173952008893334287553, 24.43799081900750847685233957600, 25.32325089669796105361155133845, 26.06501879414753889396121337055, 27.2247654267525997823765107502, 27.890112169989094648667018584487

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