Properties

Label 1-179-179.14-r0-0-0
Degree $1$
Conductor $179$
Sign $0.0460 + 0.998i$
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  + (0.977 − 0.210i)2-s + (−0.635 + 0.772i)3-s + (0.911 − 0.411i)4-s + (−0.579 + 0.815i)5-s + (−0.458 + 0.888i)6-s + (0.0881 + 0.996i)7-s + (0.804 − 0.593i)8-s + (−0.192 − 0.981i)9-s + (−0.394 + 0.918i)10-s + (−0.999 + 0.0352i)11-s + (−0.261 + 0.965i)12-s + (0.550 + 0.835i)13-s + (0.295 + 0.955i)14-s + (−0.261 − 0.965i)15-s + (0.662 − 0.749i)16-s + (−0.825 + 0.564i)17-s + ⋯
L(s)  = 1  + (0.977 − 0.210i)2-s + (−0.635 + 0.772i)3-s + (0.911 − 0.411i)4-s + (−0.579 + 0.815i)5-s + (−0.458 + 0.888i)6-s + (0.0881 + 0.996i)7-s + (0.804 − 0.593i)8-s + (−0.192 − 0.981i)9-s + (−0.394 + 0.918i)10-s + (−0.999 + 0.0352i)11-s + (−0.261 + 0.965i)12-s + (0.550 + 0.835i)13-s + (0.295 + 0.955i)14-s + (−0.261 − 0.965i)15-s + (0.662 − 0.749i)16-s + (−0.825 + 0.564i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 179 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0460 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 179 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0460 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(179\)
Sign: $0.0460 + 0.998i$
Analytic conductor: \(0.831272\)
Root analytic conductor: \(0.831272\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{179} (14, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 179,\ (0:\ ),\ 0.0460 + 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.029897262 + 0.9834928506i\)
\(L(\frac12)\) \(\approx\) \(1.029897262 + 0.9834928506i\)
\(L(1)\) \(\approx\) \(1.242811312 + 0.5118906308i\)
\(L(1)\) \(\approx\) \(1.242811312 + 0.5118906308i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad179 \( 1 \)
good2 \( 1 + (0.977 - 0.210i)T \)
3 \( 1 + (-0.635 + 0.772i)T \)
5 \( 1 + (-0.579 + 0.815i)T \)
7 \( 1 + (0.0881 + 0.996i)T \)
11 \( 1 + (-0.999 + 0.0352i)T \)
13 \( 1 + (0.550 + 0.835i)T \)
17 \( 1 + (-0.825 + 0.564i)T \)
19 \( 1 + (0.427 + 0.904i)T \)
23 \( 1 + (-0.949 + 0.312i)T \)
29 \( 1 + (0.990 + 0.140i)T \)
31 \( 1 + (0.844 - 0.535i)T \)
37 \( 1 + (0.990 - 0.140i)T \)
41 \( 1 + (0.158 - 0.987i)T \)
43 \( 1 + (-0.329 + 0.944i)T \)
47 \( 1 + (0.227 - 0.973i)T \)
53 \( 1 + (0.880 - 0.474i)T \)
59 \( 1 + (-0.783 - 0.621i)T \)
61 \( 1 + (-0.896 - 0.442i)T \)
67 \( 1 + (0.804 + 0.593i)T \)
71 \( 1 + (0.997 + 0.0705i)T \)
73 \( 1 + (0.760 - 0.648i)T \)
79 \( 1 + (-0.688 - 0.725i)T \)
83 \( 1 + (0.362 + 0.932i)T \)
89 \( 1 + (0.977 + 0.210i)T \)
97 \( 1 + (-0.0529 - 0.998i)T \)
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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

−27.12417360859292253895295918354, −25.91079983124655823027890644577, −24.70174706183722098795752936841, −24.08588934063894647896658192693, −23.29828228310778025168045958236, −22.79773263386506648766599611928, −21.45755993642660255503158335223, −20.1836739131974704829788850016, −19.87721225700696985127982575796, −18.13956843138670606728907901161, −17.21317581154613348195670919581, −16.13562975918147643272928220594, −15.58171915654629823658805944296, −13.73750335707994796493920050518, −13.30701932752617322975455697029, −12.36560791447423610987202125174, −11.35854946510464950402950905371, −10.509606626851374161370391407914, −8.20740349035345381026874391608, −7.555293889746136863073705110607, −6.42138240845633806577398776118, −5.14918046939268366144918617438, −4.39207536918251017081818624700, −2.74260289886389759193383965809, −0.93046803086656141226476643050, 2.293297899488374863580140743542, 3.5471039055584309210999805760, 4.54717685285489050548320942402, 5.79082643862858578367575855, 6.53555602750254651099490926187, 8.13986756993728115533369150718, 9.85343327138571768883514607608, 10.83326135432077023896983273915, 11.620798343121299943520151598210, 12.36942682509449655441643766631, 13.8724121685009289590071158738, 14.98348990069548942644913516874, 15.63582901717796966438104216794, 16.27148151263053323175275596403, 18.00998170184029483528204751257, 18.855864991613528473167242786270, 20.12812306433144483788960285895, 21.341087116968034768992502804883, 21.74255770446249431819694779946, 22.768394512126155701584745135214, 23.422400780395588562233802805264, 24.33803620705743181856817953375, 25.78937108161218031954933556518, 26.55510983299727516434182828603, 27.859921907532676617454613083098

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