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Properties

Label 1-179-179.51-r0-0-0
Degree $1$
Conductor $179$
Sign $-0.991 + 0.129i$
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  + (−0.969 − 0.244i)2-s + (0.0176 + 0.999i)3-s + (0.880 + 0.474i)4-s + (−0.896 + 0.442i)5-s + (0.227 − 0.973i)6-s + (0.158 + 0.987i)7-s + (−0.737 − 0.675i)8-s + (−0.999 + 0.0352i)9-s + (0.977 − 0.210i)10-s + (0.844 + 0.535i)11-s + (−0.458 + 0.888i)12-s + (−0.994 + 0.105i)13-s + (0.0881 − 0.996i)14-s + (−0.458 − 0.888i)15-s + (0.550 + 0.835i)16-s + (−0.984 − 0.175i)17-s + ⋯
L(s)  = 1  + (−0.969 − 0.244i)2-s + (0.0176 + 0.999i)3-s + (0.880 + 0.474i)4-s + (−0.896 + 0.442i)5-s + (0.227 − 0.973i)6-s + (0.158 + 0.987i)7-s + (−0.737 − 0.675i)8-s + (−0.999 + 0.0352i)9-s + (0.977 − 0.210i)10-s + (0.844 + 0.535i)11-s + (−0.458 + 0.888i)12-s + (−0.994 + 0.105i)13-s + (0.0881 − 0.996i)14-s + (−0.458 − 0.888i)15-s + (0.550 + 0.835i)16-s + (−0.984 − 0.175i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 179 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.129i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 179 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.129i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(179\)
Sign: $-0.991 + 0.129i$
Analytic conductor: \(0.831272\)
Root analytic conductor: \(0.831272\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{179} (51, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 179,\ (0:\ ),\ -0.991 + 0.129i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02547388910 + 0.3913815698i\)
\(L(\frac12)\) \(\approx\) \(0.02547388910 + 0.3913815698i\)
\(L(1)\) \(\approx\) \(0.4375138549 + 0.2843202698i\)
\(L(1)\) \(\approx\) \(0.4375138549 + 0.2843202698i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad179 \( 1 \)
good2 \( 1 + (-0.969 - 0.244i)T \)
3 \( 1 + (0.0176 + 0.999i)T \)
5 \( 1 + (-0.896 + 0.442i)T \)
7 \( 1 + (0.158 + 0.987i)T \)
11 \( 1 + (0.844 + 0.535i)T \)
13 \( 1 + (-0.994 + 0.105i)T \)
17 \( 1 + (-0.984 - 0.175i)T \)
19 \( 1 + (0.713 + 0.700i)T \)
23 \( 1 + (0.362 - 0.932i)T \)
29 \( 1 + (-0.635 - 0.772i)T \)
31 \( 1 + (-0.925 + 0.378i)T \)
37 \( 1 + (-0.635 + 0.772i)T \)
41 \( 1 + (-0.825 - 0.564i)T \)
43 \( 1 + (0.607 + 0.794i)T \)
47 \( 1 + (-0.863 + 0.505i)T \)
53 \( 1 + (-0.0529 + 0.998i)T \)
59 \( 1 + (-0.261 + 0.965i)T \)
61 \( 1 + (0.489 - 0.871i)T \)
67 \( 1 + (-0.737 + 0.675i)T \)
71 \( 1 + (0.427 - 0.904i)T \)
73 \( 1 + (0.295 - 0.955i)T \)
79 \( 1 + (0.911 - 0.411i)T \)
83 \( 1 + (0.938 - 0.345i)T \)
89 \( 1 + (-0.969 + 0.244i)T \)
97 \( 1 + (0.662 + 0.749i)T \)
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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

−26.94094356802788519229054178747, −26.06935398063028201534137761995, −24.76744439753017082799430290640, −24.17690531282978211284805129718, −23.61002300228042668417778380593, −22.29701179627832180564497139915, −20.44845879305395841991838355734, −19.67828718596969936141715295246, −19.42048456618090003507410649022, −18.03338234542103821396220850997, −17.16150745972557424749743017004, −16.509675129543125448419939939739, −15.17856174393820190753477456933, −14.11129171753220122923356134254, −12.84712624738694618325990665714, −11.58523058906238549317722154448, −11.0829873296793054351846368602, −9.365638114014464770443921804619, −8.42832555779249668215400302985, −7.314511400543057586189621952946, −6.9354028484446932680150414209, −5.25404386517899581923048613245, −3.444849833986934324637030145803, −1.68064285627647157507411354848, −0.42016564860287461734267942142, 2.26932598633645440076910621964, 3.42213327724330764488566153949, 4.721922063622076820835834191637, 6.40275533691594991377196854926, 7.65580632219844769532239274006, 8.814528646683739157420200975046, 9.543486214570262699902600625402, 10.69336213181054564429787919389, 11.6831617101380927155963244650, 12.238399947495127690456312070887, 14.61605947367183346078255758533, 15.1802061064191699614443443761, 16.08066502685319544738536449710, 17.04410456357554777902745291611, 18.134651731687879619264662667, 19.189524012892690842892581018971, 19.99323788328569614845787236831, 20.83616533202035261902685055261, 22.17544714052740951812338394165, 22.46080978182912725991615511271, 24.32247037892096970485849496376, 25.14541462203111782340580163987, 26.2933469785103769701540349854, 27.01479068929313967749685293881, 27.61566071138599838195043740626

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