Properties

Label 1-179-179.25-r0-0-0
Degree $1$
Conductor $179$
Sign $0.877 + 0.478i$
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  + (−0.949 − 0.312i)2-s + (−0.969 + 0.244i)3-s + (0.804 + 0.593i)4-s + (0.990 − 0.140i)5-s + (0.997 + 0.0705i)6-s + (0.607 + 0.794i)7-s + (−0.579 − 0.815i)8-s + (0.880 − 0.474i)9-s + (−0.984 − 0.175i)10-s + (−0.0529 + 0.998i)11-s + (−0.925 − 0.378i)12-s + (0.0881 − 0.996i)13-s + (−0.329 − 0.944i)14-s + (−0.925 + 0.378i)15-s + (0.295 + 0.955i)16-s + (−0.783 + 0.621i)17-s + ⋯
L(s)  = 1  + (−0.949 − 0.312i)2-s + (−0.969 + 0.244i)3-s + (0.804 + 0.593i)4-s + (0.990 − 0.140i)5-s + (0.997 + 0.0705i)6-s + (0.607 + 0.794i)7-s + (−0.579 − 0.815i)8-s + (0.880 − 0.474i)9-s + (−0.984 − 0.175i)10-s + (−0.0529 + 0.998i)11-s + (−0.925 − 0.378i)12-s + (0.0881 − 0.996i)13-s + (−0.329 − 0.944i)14-s + (−0.925 + 0.378i)15-s + (0.295 + 0.955i)16-s + (−0.783 + 0.621i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 179 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.877 + 0.478i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 179 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.877 + 0.478i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(179\)
Sign: $0.877 + 0.478i$
Analytic conductor: \(0.831272\)
Root analytic conductor: \(0.831272\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{179} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 179,\ (0:\ ),\ 0.877 + 0.478i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6684557457 + 0.1704226368i\)
\(L(\frac12)\) \(\approx\) \(0.6684557457 + 0.1704226368i\)
\(L(1)\) \(\approx\) \(0.6767092122 + 0.05665887559i\)
\(L(1)\) \(\approx\) \(0.6767092122 + 0.05665887559i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad179 \( 1 \)
good2 \( 1 + (-0.949 - 0.312i)T \)
3 \( 1 + (-0.969 + 0.244i)T \)
5 \( 1 + (0.990 - 0.140i)T \)
7 \( 1 + (0.607 + 0.794i)T \)
11 \( 1 + (-0.0529 + 0.998i)T \)
13 \( 1 + (0.0881 - 0.996i)T \)
17 \( 1 + (-0.783 + 0.621i)T \)
19 \( 1 + (-0.123 - 0.992i)T \)
23 \( 1 + (-0.458 + 0.888i)T \)
29 \( 1 + (0.977 - 0.210i)T \)
31 \( 1 + (0.662 + 0.749i)T \)
37 \( 1 + (0.977 + 0.210i)T \)
41 \( 1 + (-0.520 + 0.854i)T \)
43 \( 1 + (0.960 + 0.278i)T \)
47 \( 1 + (0.427 - 0.904i)T \)
53 \( 1 + (-0.737 - 0.675i)T \)
59 \( 1 + (0.844 + 0.535i)T \)
61 \( 1 + (-0.635 - 0.772i)T \)
67 \( 1 + (-0.579 + 0.815i)T \)
71 \( 1 + (-0.994 + 0.105i)T \)
73 \( 1 + (0.489 + 0.871i)T \)
79 \( 1 + (0.938 + 0.345i)T \)
83 \( 1 + (0.227 + 0.973i)T \)
89 \( 1 + (-0.949 + 0.312i)T \)
97 \( 1 + (0.760 - 0.648i)T \)
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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

−27.02474855046772731233495575923, −26.67568027516299156751473039705, −25.33319817185243729844324977970, −24.318812764778994211173363600559, −23.86666931577817543702466246506, −22.5523283886460402128971731089, −21.36698672508946916811988006972, −20.58670230561928472333201556841, −19.06477504893909501618727214593, −18.32060234334901329029524347535, −17.4892974688213318304255777542, −16.73260605058569327700722145558, −16.06485861017129081470856370295, −14.34366668271720350044662528901, −13.61156797863523637145808103857, −11.94324091676436695849326605395, −10.91977159924133035721256327055, −10.335822516129625285381158977, −9.093617907309365312866286698437, −7.78334274963052469432213801198, −6.59153111943868986343446743098, −5.96950100269299665702032611652, −4.59690614755730089342999629630, −2.18824498851023806412703964363, −0.9864559567603895938373107833, 1.37224815995267025318155020914, 2.55479254913837748842260334619, 4.65174298364390478075190235332, 5.81713212633248080393606437796, 6.830265207951098751919420557084, 8.29882259231721562719138967773, 9.470379144527219713348532233609, 10.26103554958450750020185773926, 11.21587542671858760547609382120, 12.26593385737231878034194530296, 13.11045287298420129038054552138, 15.106277251097566224354219394317, 15.757959617880067660994715316, 17.17870861583800999121861910166, 17.8405085566785106079672653368, 18.03880333890808705960214183986, 19.67002389893450457066871997952, 20.78447924798224403549211796580, 21.61665916880240092671859308123, 22.210692893836833735927527269757, 23.735287877782232238964230941332, 24.848243111801560458902097483166, 25.49225552883099097434044048516, 26.66405413395094709468736604300, 27.7635542387322234820570440262

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