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Properties

Label 1-179-179.177-r0-0-0
Degree $1$
Conductor $179$
Sign $-0.621 + 0.783i$
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  + (−0.999 − 0.0352i)2-s + (−0.783 − 0.621i)3-s + (0.997 + 0.0705i)4-s + (0.158 − 0.987i)5-s + (0.760 + 0.648i)6-s + (−0.969 + 0.244i)7-s + (−0.994 − 0.105i)8-s + (0.227 + 0.973i)9-s + (−0.192 + 0.981i)10-s + (−0.863 − 0.505i)11-s + (−0.737 − 0.675i)12-s + (−0.635 − 0.772i)13-s + (0.977 − 0.210i)14-s + (−0.737 + 0.675i)15-s + (0.990 + 0.140i)16-s + (0.911 − 0.411i)17-s + ⋯
L(s)  = 1  + (−0.999 − 0.0352i)2-s + (−0.783 − 0.621i)3-s + (0.997 + 0.0705i)4-s + (0.158 − 0.987i)5-s + (0.760 + 0.648i)6-s + (−0.969 + 0.244i)7-s + (−0.994 − 0.105i)8-s + (0.227 + 0.973i)9-s + (−0.192 + 0.981i)10-s + (−0.863 − 0.505i)11-s + (−0.737 − 0.675i)12-s + (−0.635 − 0.772i)13-s + (0.977 − 0.210i)14-s + (−0.737 + 0.675i)15-s + (0.990 + 0.140i)16-s + (0.911 − 0.411i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 179 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.621 + 0.783i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 179 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.621 + 0.783i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(179\)
Sign: $-0.621 + 0.783i$
Analytic conductor: \(0.831272\)
Root analytic conductor: \(0.831272\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{179} (177, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 179,\ (0:\ ),\ -0.621 + 0.783i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01535263818 - 0.03176312780i\)
\(L(\frac12)\) \(\approx\) \(0.01535263818 - 0.03176312780i\)
\(L(1)\) \(\approx\) \(0.3443446463 - 0.1318660139i\)
\(L(1)\) \(\approx\) \(0.3443446463 - 0.1318660139i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad179 \( 1 \)
good2 \( 1 + (-0.999 - 0.0352i)T \)
3 \( 1 + (-0.783 - 0.621i)T \)
5 \( 1 + (0.158 - 0.987i)T \)
7 \( 1 + (-0.969 + 0.244i)T \)
11 \( 1 + (-0.863 - 0.505i)T \)
13 \( 1 + (-0.635 - 0.772i)T \)
17 \( 1 + (0.911 - 0.411i)T \)
19 \( 1 + (-0.329 + 0.944i)T \)
23 \( 1 + (-0.0529 + 0.998i)T \)
29 \( 1 + (-0.520 - 0.854i)T \)
31 \( 1 + (-0.579 + 0.815i)T \)
37 \( 1 + (-0.520 + 0.854i)T \)
41 \( 1 + (-0.688 + 0.725i)T \)
43 \( 1 + (-0.949 + 0.312i)T \)
47 \( 1 + (0.295 + 0.955i)T \)
53 \( 1 + (0.427 + 0.904i)T \)
59 \( 1 + (0.804 - 0.593i)T \)
61 \( 1 + (-0.825 + 0.564i)T \)
67 \( 1 + (-0.994 + 0.105i)T \)
71 \( 1 + (0.489 - 0.871i)T \)
73 \( 1 + (-0.394 - 0.918i)T \)
79 \( 1 + (-0.925 - 0.378i)T \)
83 \( 1 + (0.662 + 0.749i)T \)
89 \( 1 + (-0.999 + 0.0352i)T \)
97 \( 1 + (0.713 - 0.700i)T \)
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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

−28.00157186746382513988804311635, −26.87325888642099047197874237863, −26.116892245349897676150216842336, −25.760413476349577539925464192246, −24.03524358327134584196741310518, −23.173313406369814109676249804361, −22.11100778509043022388110354722, −21.29844601378976155513716512424, −20.119886921441258812011061685552, −18.94425983860041445553940739311, −18.28221323378446262276133633047, −17.163322904631129240258726873607, −16.458510134367742896517796273755, −15.44099863924277716519090218757, −14.632891923385976625032573704431, −12.773026527852358160085465434263, −11.667969927109845936445987473689, −10.48341252450730944963147957142, −10.13808232403423491657348964647, −9.08019931814110933860846654789, −7.25215866080601177748342911194, −6.67553352615394301644365685940, −5.46497816857592410804304225554, −3.63594990316644844430127121122, −2.33094224108062217021945884395, 0.0415164839420004850474782050, 1.4927512405230248663304564348, 3.02861648185063922831381772507, 5.35266457621193956117545053406, 6.01582549245460816435034956958, 7.46379517051668483178048759676, 8.25243455649999716185806948670, 9.659103173255179964139718400128, 10.39929771905404848983787877791, 11.84590122305020511162637661923, 12.51828831987153145992411301780, 13.41817669380629222787636756673, 15.43756057143165611323840559520, 16.402242639750779459255944350803, 16.8806614126663698800089236337, 17.9805702936702284996806690348, 18.87526410514739550660597965306, 19.64476523662822361195690659921, 20.7830502274618992071747591962, 21.80299669141098383613464794351, 23.153711975115408999627381600708, 24.016348865349509749188827780384, 25.08187932826511899452064044552, 25.465535986037278995296742889124, 26.999510474339185094647335978291

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