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Properties

Label 1-179-179.21-r1-0-0
Degree $1$
Conductor $179$
Sign $0.00939 - 0.999i$
Analytic cond. $19.2362$
Root an. cond. $19.2362$
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  + (−0.911 − 0.411i)2-s + (−0.192 + 0.981i)3-s + (0.662 + 0.749i)4-s + (−0.329 + 0.944i)5-s + (0.579 − 0.815i)6-s + (0.984 + 0.175i)7-s + (−0.295 − 0.955i)8-s + (−0.925 − 0.378i)9-s + (0.688 − 0.725i)10-s + (−0.997 − 0.0705i)11-s + (−0.863 + 0.505i)12-s + (−0.394 − 0.918i)13-s + (−0.825 − 0.564i)14-s + (−0.863 − 0.505i)15-s + (−0.123 + 0.992i)16-s + (0.362 + 0.932i)17-s + ⋯
L(s)  = 1  + (−0.911 − 0.411i)2-s + (−0.192 + 0.981i)3-s + (0.662 + 0.749i)4-s + (−0.329 + 0.944i)5-s + (0.579 − 0.815i)6-s + (0.984 + 0.175i)7-s + (−0.295 − 0.955i)8-s + (−0.925 − 0.378i)9-s + (0.688 − 0.725i)10-s + (−0.997 − 0.0705i)11-s + (−0.863 + 0.505i)12-s + (−0.394 − 0.918i)13-s + (−0.825 − 0.564i)14-s + (−0.863 − 0.505i)15-s + (−0.123 + 0.992i)16-s + (0.362 + 0.932i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 179 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.00939 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 179 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.00939 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(179\)
Sign: $0.00939 - 0.999i$
Analytic conductor: \(19.2362\)
Root analytic conductor: \(19.2362\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{179} (21, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 179,\ (1:\ ),\ 0.00939 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1827565042 - 0.1810467317i\)
\(L(\frac12)\) \(\approx\) \(0.1827565042 - 0.1810467317i\)
\(L(1)\) \(\approx\) \(0.5224140009 + 0.1170451367i\)
\(L(1)\) \(\approx\) \(0.5224140009 + 0.1170451367i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad179 \( 1 \)
good2 \( 1 + (-0.911 - 0.411i)T \)
3 \( 1 + (-0.192 + 0.981i)T \)
5 \( 1 + (-0.329 + 0.944i)T \)
7 \( 1 + (0.984 + 0.175i)T \)
11 \( 1 + (-0.997 - 0.0705i)T \)
13 \( 1 + (-0.394 - 0.918i)T \)
17 \( 1 + (0.362 + 0.932i)T \)
19 \( 1 + (-0.635 - 0.772i)T \)
23 \( 1 + (-0.804 - 0.593i)T \)
29 \( 1 + (0.960 - 0.278i)T \)
31 \( 1 + (0.427 + 0.904i)T \)
37 \( 1 + (-0.960 - 0.278i)T \)
41 \( 1 + (0.949 - 0.312i)T \)
43 \( 1 + (-0.783 + 0.621i)T \)
47 \( 1 + (-0.896 + 0.442i)T \)
53 \( 1 + (-0.550 - 0.835i)T \)
59 \( 1 + (0.227 - 0.973i)T \)
61 \( 1 + (0.607 - 0.794i)T \)
67 \( 1 + (0.295 - 0.955i)T \)
71 \( 1 + (-0.990 + 0.140i)T \)
73 \( 1 + (-0.158 - 0.987i)T \)
79 \( 1 + (0.0529 + 0.998i)T \)
83 \( 1 + (-0.737 - 0.675i)T \)
89 \( 1 + (0.911 - 0.411i)T \)
97 \( 1 + (0.994 + 0.105i)T \)
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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

−27.49572737228215637455908795353, −26.387183935917058552787775686244, −25.25961151697799781411945250109, −24.47558924411526068644450368791, −23.78558201239184909021552322705, −23.24321570449223337821008511629, −21.14200521476027442079602117746, −20.39297848348622415361329766591, −19.36011837552289637015280425552, −18.49492322721026109551062845194, −17.62247856773707716373715217321, −16.79693033466440324672022995785, −15.91991784969084508397784223560, −14.53710546973014419330667008292, −13.56099741039044832585431788061, −12.08197623647901804939873665098, −11.51380974563817555950609127726, −10.12675377178220498428319054127, −8.692830266813562731223711891511, −7.957610088000190028750057260297, −7.22150328318716215041424883402, −5.7453282473268394520446464625, −4.770076823345020906572767459559, −2.23188140830026094388408463020, −1.19990856421190197294309250541, 0.13460466971207510959741201947, 2.34775377458021534076025799948, 3.36758143147742055238118022779, 4.80133728516437886175475753596, 6.309695551847387790099113088908, 7.8713107904381310599304192445, 8.47023100391841741644221892364, 10.13087097702525644193917800847, 10.57085244496359897840182179203, 11.38815618653524337092933660151, 12.52062164555311295851772395492, 14.3993943390427880141271223041, 15.31512612920114245610127113755, 15.990872623647017648882803633366, 17.527844407159748129159422884243, 17.82423692791041841655365390060, 19.133198153970054592357278641414, 20.08260685804837365557102950468, 21.202308419060035272248543011557, 21.66027837745018192677357875916, 22.823520985873833365956844578726, 24.00069656690379637197600432960, 25.432271413064162903265903836964, 26.32818933976943653046572667885, 26.86968405312793473943649682833

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