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Properties

Label 1-179-179.138-r0-0-0
Degree $1$
Conductor $179$
Sign $-0.525 + 0.851i$
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  + (−0.688 + 0.725i)2-s + (0.960 + 0.278i)3-s + (−0.0529 − 0.998i)4-s + (0.489 + 0.871i)5-s + (−0.863 + 0.505i)6-s + (−0.825 + 0.564i)7-s + (0.760 + 0.648i)8-s + (0.844 + 0.535i)9-s + (−0.969 − 0.244i)10-s + (−0.925 − 0.378i)11-s + (0.227 − 0.973i)12-s + (−0.123 + 0.992i)13-s + (0.158 − 0.987i)14-s + (0.227 + 0.973i)15-s + (−0.994 + 0.105i)16-s + (−0.949 − 0.312i)17-s + ⋯
L(s)  = 1  + (−0.688 + 0.725i)2-s + (0.960 + 0.278i)3-s + (−0.0529 − 0.998i)4-s + (0.489 + 0.871i)5-s + (−0.863 + 0.505i)6-s + (−0.825 + 0.564i)7-s + (0.760 + 0.648i)8-s + (0.844 + 0.535i)9-s + (−0.969 − 0.244i)10-s + (−0.925 − 0.378i)11-s + (0.227 − 0.973i)12-s + (−0.123 + 0.992i)13-s + (0.158 − 0.987i)14-s + (0.227 + 0.973i)15-s + (−0.994 + 0.105i)16-s + (−0.949 − 0.312i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 179 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.525 + 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 179 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.525 + 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(179\)
Sign: $-0.525 + 0.851i$
Analytic conductor: \(0.831272\)
Root analytic conductor: \(0.831272\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{179} (138, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 179,\ (0:\ ),\ -0.525 + 0.851i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4994902485 + 0.8949873590i\)
\(L(\frac12)\) \(\approx\) \(0.4994902485 + 0.8949873590i\)
\(L(1)\) \(\approx\) \(0.7793883938 + 0.5894812078i\)
\(L(1)\) \(\approx\) \(0.7793883938 + 0.5894812078i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad179 \( 1 \)
good2 \( 1 + (-0.688 + 0.725i)T \)
3 \( 1 + (0.960 + 0.278i)T \)
5 \( 1 + (0.489 + 0.871i)T \)
7 \( 1 + (-0.825 + 0.564i)T \)
11 \( 1 + (-0.925 - 0.378i)T \)
13 \( 1 + (-0.123 + 0.992i)T \)
17 \( 1 + (-0.949 - 0.312i)T \)
19 \( 1 + (0.990 + 0.140i)T \)
23 \( 1 + (0.938 + 0.345i)T \)
29 \( 1 + (0.0176 - 0.999i)T \)
31 \( 1 + (0.997 - 0.0705i)T \)
37 \( 1 + (0.0176 + 0.999i)T \)
41 \( 1 + (-0.984 + 0.175i)T \)
43 \( 1 + (-0.520 - 0.854i)T \)
47 \( 1 + (-0.579 + 0.815i)T \)
53 \( 1 + (0.662 - 0.749i)T \)
59 \( 1 + (-0.458 + 0.888i)T \)
61 \( 1 + (-0.329 - 0.944i)T \)
67 \( 1 + (0.760 - 0.648i)T \)
71 \( 1 + (0.713 - 0.700i)T \)
73 \( 1 + (0.0881 + 0.996i)T \)
79 \( 1 + (0.880 + 0.474i)T \)
83 \( 1 + (0.804 - 0.593i)T \)
89 \( 1 + (-0.688 - 0.725i)T \)
97 \( 1 + (0.550 - 0.835i)T \)
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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

−26.80074357762127867888920221093, −26.22926500759915704443451791149, −25.275240243865961657813590018817, −24.586071457977470656882310171127, −23.14524795672080715526246314557, −21.857937024192652051659575687751, −20.78691760402684500426200400221, −20.16133009879705709649536345416, −19.60702387390376137431898546190, −18.31353586242284326047969603328, −17.56675933229516038959741846282, −16.34961364810920192690589353977, −15.42236192024016351529447447624, −13.59748721206824027557166685152, −13.068500559970425199576772506044, −12.39227342914616424836248206400, −10.55392877885773234199666010573, −9.77863512586274147254529320576, −8.87785283340148690513998012258, −7.91981962784991733987958294205, −6.85789877687561530266903786043, −4.84231915709585926774994877006, −3.39851410252511034335628579025, −2.39593023337909153873160542962, −0.948997079844697178750254514522, 2.10146235787050060041016904328, 3.11069301689298949240111016339, 4.98265040473495067566589357860, 6.36697585024747277667794186720, 7.222489221676485379314344038365, 8.45940534832625873184308470341, 9.49577147929426140151874973880, 10.04895372184632329214003589179, 11.33209406691177749454822364221, 13.43391585518552997588647843715, 13.87860070170609058641422352789, 15.25260767326827399251789124385, 15.61781795372880229654825243093, 16.80200789415964471829253717870, 18.25524128278445064745009253173, 18.814312616986428608072332239653, 19.55037726687128810584318258608, 20.87404245091970411477028551499, 21.92203801920290912451322102999, 22.89190967428733258052912558792, 24.2807310395112427652510754287, 25.06906871682408026765697684395, 25.97580692103312309905630321099, 26.44838064475722550241916447043, 27.14337865233369003551323368397

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