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Properties

Label 1-179-179.5-r0-0-0
Degree $1$
Conductor $179$
Sign $-0.895 + 0.444i$
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  + (0.158 − 0.987i)2-s + (−0.123 − 0.992i)3-s + (−0.949 − 0.312i)4-s + (0.997 − 0.0705i)5-s + (−0.999 − 0.0352i)6-s + (−0.896 − 0.442i)7-s + (−0.458 + 0.888i)8-s + (−0.969 + 0.244i)9-s + (0.0881 − 0.996i)10-s + (−0.688 − 0.725i)11-s + (−0.192 + 0.981i)12-s + (−0.737 + 0.675i)13-s + (−0.579 + 0.815i)14-s + (−0.192 − 0.981i)15-s + (0.804 + 0.593i)16-s + (−0.329 − 0.944i)17-s + ⋯
L(s)  = 1  + (0.158 − 0.987i)2-s + (−0.123 − 0.992i)3-s + (−0.949 − 0.312i)4-s + (0.997 − 0.0705i)5-s + (−0.999 − 0.0352i)6-s + (−0.896 − 0.442i)7-s + (−0.458 + 0.888i)8-s + (−0.969 + 0.244i)9-s + (0.0881 − 0.996i)10-s + (−0.688 − 0.725i)11-s + (−0.192 + 0.981i)12-s + (−0.737 + 0.675i)13-s + (−0.579 + 0.815i)14-s + (−0.192 − 0.981i)15-s + (0.804 + 0.593i)16-s + (−0.329 − 0.944i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 179 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.895 + 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 179 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.895 + 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(179\)
Sign: $-0.895 + 0.444i$
Analytic conductor: \(0.831272\)
Root analytic conductor: \(0.831272\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{179} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 179,\ (0:\ ),\ -0.895 + 0.444i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1885731761 - 0.8048286021i\)
\(L(\frac12)\) \(\approx\) \(-0.1885731761 - 0.8048286021i\)
\(L(1)\) \(\approx\) \(0.4622020006 - 0.7599074742i\)
\(L(1)\) \(\approx\) \(0.4622020006 - 0.7599074742i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad179 \( 1 \)
good2 \( 1 + (0.158 - 0.987i)T \)
3 \( 1 + (-0.123 - 0.992i)T \)
5 \( 1 + (0.997 - 0.0705i)T \)
7 \( 1 + (-0.896 - 0.442i)T \)
11 \( 1 + (-0.688 - 0.725i)T \)
13 \( 1 + (-0.737 + 0.675i)T \)
17 \( 1 + (-0.329 - 0.944i)T \)
19 \( 1 + (0.662 - 0.749i)T \)
23 \( 1 + (-0.520 - 0.854i)T \)
29 \( 1 + (-0.994 + 0.105i)T \)
31 \( 1 + (0.911 + 0.411i)T \)
37 \( 1 + (-0.994 - 0.105i)T \)
41 \( 1 + (0.489 + 0.871i)T \)
43 \( 1 + (0.990 + 0.140i)T \)
47 \( 1 + (0.844 - 0.535i)T \)
53 \( 1 + (0.362 - 0.932i)T \)
59 \( 1 + (0.960 + 0.278i)T \)
61 \( 1 + (0.427 - 0.904i)T \)
67 \( 1 + (-0.458 - 0.888i)T \)
71 \( 1 + (-0.0529 - 0.998i)T \)
73 \( 1 + (-0.863 - 0.505i)T \)
79 \( 1 + (-0.984 - 0.175i)T \)
83 \( 1 + (-0.783 - 0.621i)T \)
89 \( 1 + (0.158 + 0.987i)T \)
97 \( 1 + (0.938 - 0.345i)T \)
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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

−27.844579937483103197636032879278, −26.56820359066767814803289501025, −25.912801988237090913475736122098, −25.33145102487263241462545183826, −24.21049025330671599666730592285, −22.74931250068937707464665232986, −22.37060241138807787024738103046, −21.49132304306200387970367313155, −20.46652383069754041626383165982, −18.96755881739416587839582790690, −17.64880639556602718732460324427, −17.179851480616317716198141960, −15.946386428001932825393724042525, −15.321406298774008194853683966938, −14.36401868920536341219575353249, −13.210339759250554875962565377048, −12.288259709958964053466375582696, −10.222446635162123092417012395655, −9.81170352935011654039581339652, −8.77978804120123173544040420688, −7.34792146701697654897388446344, −5.814607212465758207521919180299, −5.51493176083617395487410242134, −4.03591090229535709076325563532, −2.68536746838423641037867775362, 0.617437824081154723836120441561, 2.209740093636191423804560276611, 3.01530112480571582159404374620, 4.91722843857672753817506792939, 6.01673573012457586912118054617, 7.1687891781951407964672355295, 8.78649530701697039077583655193, 9.71504119154829663684555837354, 10.79963018392349602805642506443, 11.93940186021946335760784096226, 12.97538894437258164887082694971, 13.61489861150888728628298983570, 14.24978497988578721181888410021, 16.30295577373673985892643093898, 17.37756939927469218171674823246, 18.2669746267679310563718341454, 19.04790678997795461866937514001, 19.96356889500969116484664500566, 20.9182512502533410886924572719, 22.12763538814699702868553860863, 22.68257190767662041025025832080, 23.94605115980059756365501002495, 24.62567400723720224321954155896, 26.08926649585509317175370893090, 26.6023935920027919596511675429

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