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Properties

Label 1-179-179.65-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: $\chi_{179} (65, \cdot )$
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

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

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