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Properties

Label 1-179-179.23-r1-0-0
Degree $1$
Conductor $179$
Sign $-0.115 + 0.993i$
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  + (0.0529 − 0.998i)2-s + (0.844 − 0.535i)3-s + (−0.994 − 0.105i)4-s + (−0.520 − 0.854i)5-s + (−0.489 − 0.871i)6-s + (−0.362 − 0.932i)7-s + (−0.158 + 0.987i)8-s + (0.427 − 0.904i)9-s + (−0.880 + 0.474i)10-s + (−0.713 + 0.700i)11-s + (−0.896 + 0.442i)12-s + (−0.969 + 0.244i)13-s + (−0.949 + 0.312i)14-s + (−0.896 − 0.442i)15-s + (0.977 + 0.210i)16-s + (0.804 − 0.593i)17-s + ⋯
L(s)  = 1  + (0.0529 − 0.998i)2-s + (0.844 − 0.535i)3-s + (−0.994 − 0.105i)4-s + (−0.520 − 0.854i)5-s + (−0.489 − 0.871i)6-s + (−0.362 − 0.932i)7-s + (−0.158 + 0.987i)8-s + (0.427 − 0.904i)9-s + (−0.880 + 0.474i)10-s + (−0.713 + 0.700i)11-s + (−0.896 + 0.442i)12-s + (−0.969 + 0.244i)13-s + (−0.949 + 0.312i)14-s + (−0.896 − 0.442i)15-s + (0.977 + 0.210i)16-s + (0.804 − 0.593i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 179 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.115 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 179 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.115 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(179\)
Sign: $-0.115 + 0.993i$
Analytic conductor: \(19.2362\)
Root analytic conductor: \(19.2362\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{179} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 179,\ (1:\ ),\ -0.115 + 0.993i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.5894014472 - 0.6617664268i\)
\(L(\frac12)\) \(\approx\) \(-0.5894014472 - 0.6617664268i\)
\(L(1)\) \(\approx\) \(0.5021107725 - 0.7888170132i\)
\(L(1)\) \(\approx\) \(0.5021107725 - 0.7888170132i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad179 \( 1 \)
good2 \( 1 + (0.0529 - 0.998i)T \)
3 \( 1 + (0.844 - 0.535i)T \)
5 \( 1 + (-0.520 - 0.854i)T \)
7 \( 1 + (-0.362 - 0.932i)T \)
11 \( 1 + (-0.713 + 0.700i)T \)
13 \( 1 + (-0.969 + 0.244i)T \)
17 \( 1 + (0.804 - 0.593i)T \)
19 \( 1 + (0.960 - 0.278i)T \)
23 \( 1 + (-0.760 + 0.648i)T \)
29 \( 1 + (-0.999 + 0.0352i)T \)
31 \( 1 + (0.990 + 0.140i)T \)
37 \( 1 + (0.999 + 0.0352i)T \)
41 \( 1 + (-0.938 - 0.345i)T \)
43 \( 1 + (-0.458 - 0.888i)T \)
47 \( 1 + (-0.329 + 0.944i)T \)
53 \( 1 + (0.123 - 0.992i)T \)
59 \( 1 + (-0.579 + 0.815i)T \)
61 \( 1 + (-0.783 - 0.621i)T \)
67 \( 1 + (0.158 + 0.987i)T \)
71 \( 1 + (-0.0176 - 0.999i)T \)
73 \( 1 + (0.984 + 0.175i)T \)
79 \( 1 + (-0.550 + 0.835i)T \)
83 \( 1 + (0.295 + 0.955i)T \)
89 \( 1 + (-0.0529 - 0.998i)T \)
97 \( 1 + (0.394 - 0.918i)T \)
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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

−27.39587152399872135003186105798, −26.499596182248678205173729738475, −26.11618791265106890136779216032, −24.99497415565729999329441108441, −24.27769299742407119528756003799, −22.970874244010092837164411438486, −22.047791794909086188574903916789, −21.50234257427106790182358254232, −19.880537972637545924677744942393, −18.78551222174253847601713105601, −18.408756829069210337617608544178, −16.66247171388026439872883625870, −15.81752164761017735937104652308, −15.04008599428150006622234254415, −14.429536405118152339284477565039, −13.30282936250436188319546469771, −12.03679122982268564794505967909, −10.325640928634754377504585646536, −9.53501333653145219916207984631, −8.1713906790877536025280560744, −7.69884838530531339431346581311, −6.19007247716533329519304909685, −5.055461537594487654336235804637, −3.5937266691200294724173324562, −2.724234106375618899213192148371, 0.28184658231221292232254485590, 1.51988792057081742426167363062, 2.94091484575128837794350405449, 4.0354602780581452554350044502, 5.12659709395414715914468744091, 7.348731003712112452191234807196, 7.98447188020278043449183529220, 9.46101336736928919453236965076, 9.94097010934281168169222062299, 11.69066250980795790031754732618, 12.46823846092330118338544972373, 13.3380886154377692377311570991, 14.10872239773975459799835854197, 15.40225782052650708632731370617, 16.793698413124884102272233325888, 17.91068152539464957157783298099, 18.98948467335603290419373207951, 19.91135764471032104407956567645, 20.312262433909488620443584222278, 21.14112664307550407589872049169, 22.625500872001627373726431155840, 23.592803145487481021881392805179, 24.21780218278278302439388370518, 25.624196339564306090848751423723, 26.59776283098449839390024023818

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