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Properties

Label 1-179-179.10-r1-0-0
Degree $1$
Conductor $179$
Sign $0.425 + 0.905i$
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  + (0.192 − 0.981i)2-s + (−0.520 + 0.854i)3-s + (−0.925 − 0.378i)4-s + (0.0881 − 0.996i)5-s + (0.737 + 0.675i)6-s + (−0.977 − 0.210i)7-s + (−0.550 + 0.835i)8-s + (−0.458 − 0.888i)9-s + (−0.960 − 0.278i)10-s + (−0.227 − 0.973i)11-s + (0.804 − 0.593i)12-s + (0.990 + 0.140i)13-s + (−0.394 + 0.918i)14-s + (0.804 + 0.593i)15-s + (0.713 + 0.700i)16-s + (−0.688 − 0.725i)17-s + ⋯
L(s)  = 1  + (0.192 − 0.981i)2-s + (−0.520 + 0.854i)3-s + (−0.925 − 0.378i)4-s + (0.0881 − 0.996i)5-s + (0.737 + 0.675i)6-s + (−0.977 − 0.210i)7-s + (−0.550 + 0.835i)8-s + (−0.458 − 0.888i)9-s + (−0.960 − 0.278i)10-s + (−0.227 − 0.973i)11-s + (0.804 − 0.593i)12-s + (0.990 + 0.140i)13-s + (−0.394 + 0.918i)14-s + (0.804 + 0.593i)15-s + (0.713 + 0.700i)16-s + (−0.688 − 0.725i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 179 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.425 + 0.905i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 179 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.425 + 0.905i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(179\)
Sign: $0.425 + 0.905i$
Analytic conductor: \(19.2362\)
Root analytic conductor: \(19.2362\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{179} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 179,\ (1:\ ),\ 0.425 + 0.905i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2487089618 + 0.1579767276i\)
\(L(\frac12)\) \(\approx\) \(0.2487089618 + 0.1579767276i\)
\(L(1)\) \(\approx\) \(0.6088588264 - 0.2775442847i\)
\(L(1)\) \(\approx\) \(0.6088588264 - 0.2775442847i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad179 \( 1 \)
good2 \( 1 + (0.192 - 0.981i)T \)
3 \( 1 + (-0.520 + 0.854i)T \)
5 \( 1 + (0.0881 - 0.996i)T \)
7 \( 1 + (-0.977 - 0.210i)T \)
11 \( 1 + (-0.227 - 0.973i)T \)
13 \( 1 + (0.990 + 0.140i)T \)
17 \( 1 + (-0.688 - 0.725i)T \)
19 \( 1 + (0.489 + 0.871i)T \)
23 \( 1 + (-0.880 + 0.474i)T \)
29 \( 1 + (0.607 + 0.794i)T \)
31 \( 1 + (-0.863 + 0.505i)T \)
37 \( 1 + (-0.607 + 0.794i)T \)
41 \( 1 + (0.969 + 0.244i)T \)
43 \( 1 + (-0.984 + 0.175i)T \)
47 \( 1 + (0.760 + 0.648i)T \)
53 \( 1 + (-0.997 + 0.0705i)T \)
59 \( 1 + (0.938 - 0.345i)T \)
61 \( 1 + (0.158 + 0.987i)T \)
67 \( 1 + (0.550 + 0.835i)T \)
71 \( 1 + (0.896 + 0.442i)T \)
73 \( 1 + (0.123 - 0.992i)T \)
79 \( 1 + (-0.844 - 0.535i)T \)
83 \( 1 + (-0.0529 - 0.998i)T \)
89 \( 1 + (-0.192 - 0.981i)T \)
97 \( 1 + (-0.427 + 0.904i)T \)
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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

−26.432890139570529746724653567010, −25.85668628082952275951976847608, −25.13669132212479144274337488539, −23.97467664869999159711534529745, −23.07426381585581657912153842353, −22.542395161376247357980237693824, −21.75892503947250785943960594455, −19.84514452654354478702211153361, −18.747404619185433449702618699495, −18.05703406308849611692268323061, −17.34905505191070413451599944776, −16.00074778473516467274128032684, −15.29470745730215997965526649610, −13.97654580112037123633328198259, −13.17563496834853581047965923776, −12.31228717282110910605300561082, −10.88792239577825218572089314928, −9.661066378108483653043709078783, −8.22896353423694539685713362102, −7.05514993397784630339108424000, −6.479349012638477175278229984876, −5.58584456377927808404027417747, −3.89812131936008671390214004532, −2.37880108017402026793279814586, −0.12238280044443904893125554487, 1.11499384864624405768851849123, 3.20694557487648203070389934837, 4.044006513629047070458746973441, 5.28535612775831945454028769491, 6.13488260383239716816013534636, 8.51916978598955814998675353021, 9.313230318546211345224254761822, 10.22279642693068951679408342534, 11.25885884700158679530966581803, 12.197013477667627326750574290887, 13.24884206007776612337904404649, 14.095795599916144281612341815590, 15.9627729292667587819259980014, 16.236938505076317744596562550097, 17.58122578511788782484958774008, 18.64473781296213149735243358100, 19.94501957370168753912721614194, 20.54871816124381883820637475653, 21.487939507608456555981843887490, 22.25646885766904493127735696453, 23.24075367078410667285367971597, 23.99025812891623074624629867215, 25.5772720920733651920294997403, 26.73381114042318622065271515792, 27.45434068749480555412444730686

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