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Properties

Label 1-179-179.46-r0-0-0
Degree $1$
Conductor $179$
Sign $0.984 + 0.173i$
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  + (0.0881 − 0.996i)2-s + (−0.994 − 0.105i)3-s + (−0.984 − 0.175i)4-s + (−0.925 + 0.378i)5-s + (−0.192 + 0.981i)6-s + (−0.579 − 0.815i)7-s + (−0.261 + 0.965i)8-s + (0.977 + 0.210i)9-s + (0.295 + 0.955i)10-s + (−0.969 + 0.244i)11-s + (0.960 + 0.278i)12-s + (0.804 + 0.593i)13-s + (−0.863 + 0.505i)14-s + (0.960 − 0.278i)15-s + (0.938 + 0.345i)16-s + (0.489 − 0.871i)17-s + ⋯
L(s)  = 1  + (0.0881 − 0.996i)2-s + (−0.994 − 0.105i)3-s + (−0.984 − 0.175i)4-s + (−0.925 + 0.378i)5-s + (−0.192 + 0.981i)6-s + (−0.579 − 0.815i)7-s + (−0.261 + 0.965i)8-s + (0.977 + 0.210i)9-s + (0.295 + 0.955i)10-s + (−0.969 + 0.244i)11-s + (0.960 + 0.278i)12-s + (0.804 + 0.593i)13-s + (−0.863 + 0.505i)14-s + (0.960 − 0.278i)15-s + (0.938 + 0.345i)16-s + (0.489 − 0.871i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 179 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.984 + 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 179 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.984 + 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(179\)
Sign: $0.984 + 0.173i$
Analytic conductor: \(0.831272\)
Root analytic conductor: \(0.831272\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{179} (46, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 179,\ (0:\ ),\ 0.984 + 0.173i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4141993754 + 0.03630751995i\)
\(L(\frac12)\) \(\approx\) \(0.4141993754 + 0.03630751995i\)
\(L(1)\) \(\approx\) \(0.5174477786 - 0.1843361228i\)
\(L(1)\) \(\approx\) \(0.5174477786 - 0.1843361228i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad179 \( 1 \)
good2 \( 1 + (0.0881 - 0.996i)T \)
3 \( 1 + (-0.994 - 0.105i)T \)
5 \( 1 + (-0.925 + 0.378i)T \)
7 \( 1 + (-0.579 - 0.815i)T \)
11 \( 1 + (-0.969 + 0.244i)T \)
13 \( 1 + (0.804 + 0.593i)T \)
17 \( 1 + (0.489 - 0.871i)T \)
19 \( 1 + (-0.0529 + 0.998i)T \)
23 \( 1 + (0.607 + 0.794i)T \)
29 \( 1 + (0.550 + 0.835i)T \)
31 \( 1 + (-0.688 + 0.725i)T \)
37 \( 1 + (0.550 - 0.835i)T \)
41 \( 1 + (-0.896 + 0.442i)T \)
43 \( 1 + (0.713 + 0.700i)T \)
47 \( 1 + (-0.999 - 0.0352i)T \)
53 \( 1 + (-0.949 + 0.312i)T \)
59 \( 1 + (0.0176 + 0.999i)T \)
61 \( 1 + (0.997 + 0.0705i)T \)
67 \( 1 + (-0.261 - 0.965i)T \)
71 \( 1 + (0.880 + 0.474i)T \)
73 \( 1 + (0.227 + 0.973i)T \)
79 \( 1 + (-0.825 + 0.564i)T \)
83 \( 1 + (-0.520 + 0.854i)T \)
89 \( 1 + (0.0881 + 0.996i)T \)
97 \( 1 + (0.362 + 0.932i)T \)
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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

−27.36522286664784729497566691094, −26.35257715888556767901265431568, −25.35335111575902160272719077737, −24.14964186458741796001284855402, −23.59795320635458444179631395778, −22.775187639207066217601449623070, −21.950233979895255176085495253618, −20.84484397466259718631526484034, −19.05896968514447655371846890516, −18.50356085751379393949974452033, −17.34377374207719427478943645628, −16.339791555080289131650241194652, −15.64160018964979408041290800391, −15.12390222061351738513264698595, −13.08399800921640890960919566624, −12.70531452122254279887421172823, −11.41333852125678827819124409789, −10.168698073694453851307532587271, −8.77426311932245496193055073570, −7.87024708935900353230442109877, −6.55944669865358508207846404473, −5.629620447060330758152007676384, −4.70710276807480701090694008759, −3.38689596936359247897930604113, −0.44456946909003111602851471663, 1.192035103191524959046719762172, 3.17974235650117001242418660759, 4.173066705779856741948445985346, 5.332806686719139770505628007535, 6.84609000960295296013478121465, 7.92028736465067660377929476601, 9.639726266843214431960639301329, 10.644588716887641042635202686381, 11.2417707051674135986633563472, 12.30609317996725730135745265005, 13.117898277260729368151626426140, 14.28754281970129213988351294966, 15.86168653435527842247284484355, 16.584753048100436562276071152752, 18.03333137279792034008686147450, 18.63607787100432228913007506976, 19.56662110586998308353186759158, 20.66772761344652039787011819036, 21.59224019536116937156163872977, 22.98656857952626620872805507119, 23.09074612831476792934017469954, 23.81968231464013001557412415119, 25.7483034719536194937247472535, 26.9321182437496810807028675253, 27.391833409861899397957236653774

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