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Properties

Label 1-179-179.107-r0-0-0
Degree $1$
Conductor $179$
Sign $-0.815 - 0.578i$
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  + (−0.123 − 0.992i)2-s + (0.713 − 0.700i)3-s + (−0.969 + 0.244i)4-s + (0.227 − 0.973i)5-s + (−0.783 − 0.621i)6-s + (0.760 − 0.648i)7-s + (0.362 + 0.932i)8-s + (0.0176 − 0.999i)9-s + (−0.994 − 0.105i)10-s + (0.960 − 0.278i)11-s + (−0.520 + 0.854i)12-s + (−0.0529 + 0.998i)13-s + (−0.737 − 0.675i)14-s + (−0.520 − 0.854i)15-s + (0.880 − 0.474i)16-s + (0.0881 + 0.996i)17-s + ⋯
L(s)  = 1  + (−0.123 − 0.992i)2-s + (0.713 − 0.700i)3-s + (−0.969 + 0.244i)4-s + (0.227 − 0.973i)5-s + (−0.783 − 0.621i)6-s + (0.760 − 0.648i)7-s + (0.362 + 0.932i)8-s + (0.0176 − 0.999i)9-s + (−0.994 − 0.105i)10-s + (0.960 − 0.278i)11-s + (−0.520 + 0.854i)12-s + (−0.0529 + 0.998i)13-s + (−0.737 − 0.675i)14-s + (−0.520 − 0.854i)15-s + (0.880 − 0.474i)16-s + (0.0881 + 0.996i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 179 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.815 - 0.578i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 179 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.815 - 0.578i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(179\)
Sign: $-0.815 - 0.578i$
Analytic conductor: \(0.831272\)
Root analytic conductor: \(0.831272\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{179} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 179,\ (0:\ ),\ -0.815 - 0.578i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4146198794 - 1.301830221i\)
\(L(\frac12)\) \(\approx\) \(0.4146198794 - 1.301830221i\)
\(L(1)\) \(\approx\) \(0.8032557909 - 0.9468710690i\)
\(L(1)\) \(\approx\) \(0.8032557909 - 0.9468710690i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad179 \( 1 \)
good2 \( 1 + (-0.123 - 0.992i)T \)
3 \( 1 + (0.713 - 0.700i)T \)
5 \( 1 + (0.227 - 0.973i)T \)
7 \( 1 + (0.760 - 0.648i)T \)
11 \( 1 + (0.960 - 0.278i)T \)
13 \( 1 + (-0.0529 + 0.998i)T \)
17 \( 1 + (0.0881 + 0.996i)T \)
19 \( 1 + (-0.925 + 0.378i)T \)
23 \( 1 + (-0.825 - 0.564i)T \)
29 \( 1 + (0.427 + 0.904i)T \)
31 \( 1 + (-0.192 + 0.981i)T \)
37 \( 1 + (0.427 - 0.904i)T \)
41 \( 1 + (0.295 + 0.955i)T \)
43 \( 1 + (-0.896 + 0.442i)T \)
47 \( 1 + (-0.261 + 0.965i)T \)
53 \( 1 + (-0.688 + 0.725i)T \)
59 \( 1 + (0.607 - 0.794i)T \)
61 \( 1 + (-0.863 - 0.505i)T \)
67 \( 1 + (0.362 - 0.932i)T \)
71 \( 1 + (0.844 + 0.535i)T \)
73 \( 1 + (0.804 + 0.593i)T \)
79 \( 1 + (0.977 + 0.210i)T \)
83 \( 1 + (-0.984 - 0.175i)T \)
89 \( 1 + (-0.123 + 0.992i)T \)
97 \( 1 + (0.911 - 0.411i)T \)
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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

−27.43320796228600762243774411425, −26.790922582926721842440237242, −25.46018470299540929598040336950, −25.364349437303331754728437994531, −24.20927439362243232308611262308, −22.71606830887563682817062275997, −22.17437206624367619187866804899, −21.244656762072773622698859660549, −19.88245822762521119359622352650, −18.82351157581988282888463168494, −17.89805342160210353212736047596, −16.99958874492103014790653622262, −15.525536128117032035124391302175, −15.072657506290818296632327447207, −14.297365355491225362338810308073, −13.42743257526809659218382040431, −11.62023361797627948303174237299, −10.27132403564333422050160461491, −9.43573196771130594489287289205, −8.34760330961178528877436650005, −7.435754754044882742765122465980, −6.10313379794272594303087029898, −4.942509805925305969178048745556, −3.723260603378928807790449741640, −2.256848412332764023305005172935, 1.28458895757569290341110450350, 1.9026342532179075004692939399, 3.754879335103262895064092723028, 4.55802029892799073561870215289, 6.382546949965690643779521436806, 8.05346407132326876502871473297, 8.6648177538146367469252387577, 9.63599041452468495041868507128, 11.030944722778905095824940189660, 12.20193849587453084989885999452, 12.83367808821507189836174549390, 14.10198883519958071040659540597, 14.40817031287690127155307910560, 16.63268719833554656231363681486, 17.37516621760398490419041382646, 18.38275903918386023363284098181, 19.64705997310521472265015659940, 19.92094543732395579461786069656, 21.111405296736380723846758180535, 21.61037032279552190695404782407, 23.420184102301600353940729320335, 23.95588432982585497816901253114, 25.012452918584096399254500247, 26.15771078644004026316336012366, 27.07872858501752252606414606847

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