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Properties

Label 1-179-179.155-r0-0-0
Degree $1$
Conductor $179$
Sign $-0.644 + 0.764i$
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  + (0.713 + 0.700i)2-s + (−0.579 + 0.815i)3-s + (0.0176 + 0.999i)4-s + (0.938 + 0.345i)5-s + (−0.984 + 0.175i)6-s + (0.662 + 0.749i)7-s + (−0.688 + 0.725i)8-s + (−0.329 − 0.944i)9-s + (0.427 + 0.904i)10-s + (0.607 − 0.794i)11-s + (−0.825 − 0.564i)12-s + (0.844 + 0.535i)13-s + (−0.0529 + 0.998i)14-s + (−0.825 + 0.564i)15-s + (−0.999 + 0.0352i)16-s + (−0.994 − 0.105i)17-s + ⋯
L(s)  = 1  + (0.713 + 0.700i)2-s + (−0.579 + 0.815i)3-s + (0.0176 + 0.999i)4-s + (0.938 + 0.345i)5-s + (−0.984 + 0.175i)6-s + (0.662 + 0.749i)7-s + (−0.688 + 0.725i)8-s + (−0.329 − 0.944i)9-s + (0.427 + 0.904i)10-s + (0.607 − 0.794i)11-s + (−0.825 − 0.564i)12-s + (0.844 + 0.535i)13-s + (−0.0529 + 0.998i)14-s + (−0.825 + 0.564i)15-s + (−0.999 + 0.0352i)16-s + (−0.994 − 0.105i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 179 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.644 + 0.764i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 179 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.644 + 0.764i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(179\)
Sign: $-0.644 + 0.764i$
Analytic conductor: \(0.831272\)
Root analytic conductor: \(0.831272\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{179} (155, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 179,\ (0:\ ),\ -0.644 + 0.764i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6897958780 + 1.484663989i\)
\(L(\frac12)\) \(\approx\) \(0.6897958780 + 1.484663989i\)
\(L(1)\) \(\approx\) \(1.048252932 + 1.030921661i\)
\(L(1)\) \(\approx\) \(1.048252932 + 1.030921661i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad179 \( 1 \)
good2 \( 1 + (0.713 + 0.700i)T \)
3 \( 1 + (-0.579 + 0.815i)T \)
5 \( 1 + (0.938 + 0.345i)T \)
7 \( 1 + (0.662 + 0.749i)T \)
11 \( 1 + (0.607 - 0.794i)T \)
13 \( 1 + (0.844 + 0.535i)T \)
17 \( 1 + (-0.994 - 0.105i)T \)
19 \( 1 + (-0.458 - 0.888i)T \)
23 \( 1 + (-0.394 - 0.918i)T \)
29 \( 1 + (-0.863 - 0.505i)T \)
31 \( 1 + (-0.520 - 0.854i)T \)
37 \( 1 + (-0.863 + 0.505i)T \)
41 \( 1 + (0.550 + 0.835i)T \)
43 \( 1 + (0.760 - 0.648i)T \)
47 \( 1 + (-0.949 + 0.312i)T \)
53 \( 1 + (0.960 - 0.278i)T \)
59 \( 1 + (0.158 - 0.987i)T \)
61 \( 1 + (0.804 - 0.593i)T \)
67 \( 1 + (-0.688 - 0.725i)T \)
71 \( 1 + (-0.261 + 0.965i)T \)
73 \( 1 + (0.880 + 0.474i)T \)
79 \( 1 + (-0.635 + 0.772i)T \)
83 \( 1 + (0.977 - 0.210i)T \)
89 \( 1 + (0.713 - 0.700i)T \)
97 \( 1 + (-0.192 + 0.981i)T \)
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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

−27.60566343268854331248915331724, −25.71015213551879412626823787031, −24.76322750030409621536582057892, −24.0202588292262602317874290949, −23.07346042702097549132859126563, −22.34359797656642195489301403181, −21.19013418141320069574116380368, −20.342702076212015332316074591473, −19.484826108305214003902841476369, −17.95899896955548779560765842864, −17.705019387669376653060664194843, −16.36398296310231853503714485454, −14.7251717502649127412103847698, −13.783170838936353204589638782178, −13.08909210321717132472767123214, −12.2013151815358164055687844128, −11.02074369392533434052434152553, −10.31248898278755763648309800826, −8.85711149556998943537896768355, −7.20188061245233301506486598929, −6.09261047426464572832128342443, −5.190063786727532387217046856694, −3.96608873092435057070370519842, −1.975493666840729609040183552125, −1.3474055639856155823560795465, 2.343063421488176213429363018587, 3.85825057881838790567596863859, 4.97147544547844116080967228233, 6.016408819833905079935378753883, 6.57755477372232603972366001181, 8.59164647368414587087771470290, 9.24775126025297203580594123258, 11.02597534103530987976671609543, 11.5320805497261588352830723088, 13.02168412086440490755408534902, 14.1053092498643013808776377668, 14.909215373178544937968381187514, 15.84553937447447878141865275659, 16.85988440798493018577840458416, 17.638580133433845916295950861989, 18.536001980846042073924682715, 20.5938119930943902208564691249, 21.353667267713906784349448778130, 22.01822300638370130119189737839, 22.59123209887648197200802988932, 24.02360841055457790665402214644, 24.5972441879545871324515792049, 25.90298867578550981322222818917, 26.43450590804501692865337876755, 27.625254581826438021614336893302

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