Properties

Label 1-160-160.53-r1-0-0
Degree $1$
Conductor $160$
Sign $0.349 + 0.936i$
Analytic cond. $17.1943$
Root an. cond. $17.1943$
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  + (0.707 + 0.707i)3-s + 7-s i·9-s + (0.707 + 0.707i)11-s + (−0.707 − 0.707i)13-s i·17-s + (0.707 − 0.707i)19-s + (0.707 + 0.707i)21-s + 23-s + (−0.707 + 0.707i)27-s + (−0.707 + 0.707i)29-s + 31-s + i·33-s + (−0.707 + 0.707i)37-s i·39-s + ⋯
L(s)  = 1  + (0.707 + 0.707i)3-s + 7-s i·9-s + (0.707 + 0.707i)11-s + (−0.707 − 0.707i)13-s i·17-s + (0.707 − 0.707i)19-s + (0.707 + 0.707i)21-s + 23-s + (−0.707 + 0.707i)27-s + (−0.707 + 0.707i)29-s + 31-s + i·33-s + (−0.707 + 0.707i)37-s i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.349 + 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.349 + 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(160\)    =    \(2^{5} \cdot 5\)
Sign: $0.349 + 0.936i$
Analytic conductor: \(17.1943\)
Root analytic conductor: \(17.1943\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{160} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 160,\ (1:\ ),\ 0.349 + 0.936i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.148816815 + 1.491590889i\)
\(L(\frac12)\) \(\approx\) \(2.148816815 + 1.491590889i\)
\(L(1)\) \(\approx\) \(1.470894022 + 0.5114078273i\)
\(L(1)\) \(\approx\) \(1.470894022 + 0.5114078273i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + (0.707 + 0.707i)T \)
7 \( 1 + T \)
11 \( 1 + (0.707 + 0.707i)T \)
13 \( 1 + (-0.707 - 0.707i)T \)
17 \( 1 - iT \)
19 \( 1 + (0.707 - 0.707i)T \)
23 \( 1 + T \)
29 \( 1 + (-0.707 + 0.707i)T \)
31 \( 1 + T \)
37 \( 1 + (-0.707 + 0.707i)T \)
41 \( 1 - iT \)
43 \( 1 + (-0.707 + 0.707i)T \)
47 \( 1 - iT \)
53 \( 1 + (-0.707 + 0.707i)T \)
59 \( 1 + (0.707 + 0.707i)T \)
61 \( 1 + (0.707 - 0.707i)T \)
67 \( 1 + (-0.707 - 0.707i)T \)
71 \( 1 + iT \)
73 \( 1 + T \)
79 \( 1 + T \)
83 \( 1 + (-0.707 - 0.707i)T \)
89 \( 1 - iT \)
97 \( 1 - iT \)
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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

−26.8772700163808447094113696699, −26.79441386050121691584279311183, −25.04211042350368501500767605523, −24.67774271384356197280184087497, −23.78969155210530181804724944831, −22.58086204793721789683841916088, −21.25708095976281673569324687614, −20.546920571730737295606979018442, −19.390518396534112999209709240533, −18.64510022538053182665361473322, −17.609051346490958820414195242303, −16.57022580739370447415280170463, −15.043068016809816790963111439126, −14.22412125623694479494758747028, −13.54343115157594121878716337859, −12.02966568609486084335189172120, −11.43264086814436965706259744700, −9.66170867575470480515421265633, −8.68606831096846448991271998897, −7.65365352567862544168510561530, −6.67735464817865291303514703532, −5.1370933734714318549799395946, −3.65688774890070474774653108579, −2.249765426515142671586050765080, −0.998173365827534948056167925069, 1.59529400061551819660994422585, 3.01249315973223556703298724484, 4.39763996440161694660598197469, 5.251753370040648477675390715848, 7.15140140063318227264410481178, 8.195619160049765779064163836416, 9.22484066410464547862467147139, 10.29145744559799899957040031992, 11.32599803034095714864976945806, 12.64852121647354066699859211606, 13.95839010010582852450955561859, 14.87959452138324659442560489661, 15.41092760486892670648306914596, 16.96139962904269830615239255695, 17.64997010505586319048528831192, 19.14576196546414717264511479361, 20.07276338261894150872073701000, 20.80597866245902179149605118267, 21.84241963281985294026351987476, 22.632807717684904739332550177368, 24.13189901853721682785845833211, 24.90295706876246711393304455564, 25.8702815026829454345377535121, 26.90411302154783525523119278729, 27.603109599624895985227484898586

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