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Properties

Label 1-179-179.168-r0-0-0
Degree $1$
Conductor $179$
Sign $0.587 - 0.809i$
Analytic cond. $0.831272$
Root an. cond. $0.831272$
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  + (−0.863 − 0.505i)2-s + (0.804 + 0.593i)3-s + (0.489 + 0.871i)4-s + (−0.688 − 0.725i)5-s + (−0.394 − 0.918i)6-s + (0.844 − 0.535i)7-s + (0.0176 − 0.999i)8-s + (0.295 + 0.955i)9-s + (0.227 + 0.973i)10-s + (0.0881 − 0.996i)11-s + (−0.123 + 0.992i)12-s + (−0.783 − 0.621i)13-s + (−0.999 + 0.0352i)14-s + (−0.123 − 0.992i)15-s + (−0.520 + 0.854i)16-s + (0.997 − 0.0705i)17-s + ⋯
L(s)  = 1  + (−0.863 − 0.505i)2-s + (0.804 + 0.593i)3-s + (0.489 + 0.871i)4-s + (−0.688 − 0.725i)5-s + (−0.394 − 0.918i)6-s + (0.844 − 0.535i)7-s + (0.0176 − 0.999i)8-s + (0.295 + 0.955i)9-s + (0.227 + 0.973i)10-s + (0.0881 − 0.996i)11-s + (−0.123 + 0.992i)12-s + (−0.783 − 0.621i)13-s + (−0.999 + 0.0352i)14-s + (−0.123 − 0.992i)15-s + (−0.520 + 0.854i)16-s + (0.997 − 0.0705i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 179 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.587 - 0.809i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 179 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.587 - 0.809i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(179\)
Sign: $0.587 - 0.809i$
Analytic conductor: \(0.831272\)
Root analytic conductor: \(0.831272\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{179} (168, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 179,\ (0:\ ),\ 0.587 - 0.809i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8442668280 - 0.4301709315i\)
\(L(\frac12)\) \(\approx\) \(0.8442668280 - 0.4301709315i\)
\(L(1)\) \(\approx\) \(0.8708952746 - 0.2305518590i\)
\(L(1)\) \(\approx\) \(0.8708952746 - 0.2305518590i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad179 \( 1 \)
good2 \( 1 + (-0.863 - 0.505i)T \)
3 \( 1 + (0.804 + 0.593i)T \)
5 \( 1 + (-0.688 - 0.725i)T \)
7 \( 1 + (0.844 - 0.535i)T \)
11 \( 1 + (0.0881 - 0.996i)T \)
13 \( 1 + (-0.783 - 0.621i)T \)
17 \( 1 + (0.997 - 0.0705i)T \)
19 \( 1 + (-0.949 - 0.312i)T \)
23 \( 1 + (0.713 - 0.700i)T \)
29 \( 1 + (0.938 - 0.345i)T \)
31 \( 1 + (0.158 + 0.987i)T \)
37 \( 1 + (0.938 + 0.345i)T \)
41 \( 1 + (-0.925 - 0.378i)T \)
43 \( 1 + (-0.0529 - 0.998i)T \)
47 \( 1 + (0.977 + 0.210i)T \)
53 \( 1 + (-0.329 - 0.944i)T \)
59 \( 1 + (-0.994 + 0.105i)T \)
61 \( 1 + (0.911 + 0.411i)T \)
67 \( 1 + (0.0176 + 0.999i)T \)
71 \( 1 + (-0.984 + 0.175i)T \)
73 \( 1 + (-0.192 + 0.981i)T \)
79 \( 1 + (-0.896 + 0.442i)T \)
83 \( 1 + (0.990 + 0.140i)T \)
89 \( 1 + (-0.863 + 0.505i)T \)
97 \( 1 + (0.607 + 0.794i)T \)
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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

−27.29013312497338444664054336912, −26.530675924140884293021148309853, −25.48698088270049457985944665186, −24.97905001901148051888230172460, −23.7399589969892726333178447259, −23.28519080605471465074335529079, −21.502743348254948503707436142763, −20.38233973738567573751087807340, −19.3987909548496970170792559172, −18.78860737138144694407607883561, −17.9732537324699211197105805648, −16.96666206569230905526644417393, −15.34989522689588244894290037771, −14.85121960962383424945493286969, −14.210038502464044759184381398339, −12.34461745850891914277786742125, −11.50168752079586417452743367386, −10.11820425352321213155075062035, −9.06193172428932486229146448360, −7.897321032374692636140256330700, −7.41213321549757766251074060902, −6.30069313973756550673642118169, −4.55462966802464316652665711656, −2.70523643748430565273178897103, −1.64668711816849017857728596523, 1.03017923871515373849929935723, 2.72505311647733575179430348047, 3.87808451071217849982268131963, 4.96126145895742506627192965215, 7.285092542134516066094673518332, 8.28331445328515311181552584991, 8.679425486163306813062719068782, 10.10069410283146180610855628837, 10.903144347778354621291821165318, 12.04173098681788359683411450741, 13.19548705879711802033629230742, 14.503909403915106084532672668592, 15.588257689983851449948556532945, 16.62052978785899537082267152878, 17.24284113125079662454868446093, 18.8567437800433604327893439368, 19.5321180869549083413144917293, 20.40476050988645831063555609831, 21.04678967704992456996363768296, 21.907993729746358755493894220510, 23.5603397843801666811247821610, 24.67746675780118473524517568939, 25.36084467497024375143325984087, 26.82987380077572979897333986553, 27.09498113671943285959238925866

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