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Properties

Label 1-179-179.135-r0-0-0
Degree $1$
Conductor $179$
Sign $0.949 + 0.314i$
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  + (−0.825 − 0.564i)2-s + (−0.394 + 0.918i)3-s + (0.362 + 0.932i)4-s + (0.427 + 0.904i)5-s + (0.844 − 0.535i)6-s + (0.489 − 0.871i)7-s + (0.227 − 0.973i)8-s + (−0.688 − 0.725i)9-s + (0.158 − 0.987i)10-s + (0.911 − 0.411i)11-s + (−0.999 − 0.0352i)12-s + (0.760 − 0.648i)13-s + (−0.896 + 0.442i)14-s + (−0.999 + 0.0352i)15-s + (−0.737 + 0.675i)16-s + (0.607 − 0.794i)17-s + ⋯
L(s)  = 1  + (−0.825 − 0.564i)2-s + (−0.394 + 0.918i)3-s + (0.362 + 0.932i)4-s + (0.427 + 0.904i)5-s + (0.844 − 0.535i)6-s + (0.489 − 0.871i)7-s + (0.227 − 0.973i)8-s + (−0.688 − 0.725i)9-s + (0.158 − 0.987i)10-s + (0.911 − 0.411i)11-s + (−0.999 − 0.0352i)12-s + (0.760 − 0.648i)13-s + (−0.896 + 0.442i)14-s + (−0.999 + 0.0352i)15-s + (−0.737 + 0.675i)16-s + (0.607 − 0.794i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 179 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.949 + 0.314i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 179 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.949 + 0.314i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(179\)
Sign: $0.949 + 0.314i$
Analytic conductor: \(0.831272\)
Root analytic conductor: \(0.831272\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{179} (135, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 179,\ (0:\ ),\ 0.949 + 0.314i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8086969275 + 0.1306809558i\)
\(L(\frac12)\) \(\approx\) \(0.8086969275 + 0.1306809558i\)
\(L(1)\) \(\approx\) \(0.7808324371 + 0.06643568024i\)
\(L(1)\) \(\approx\) \(0.7808324371 + 0.06643568024i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad179 \( 1 \)
good2 \( 1 + (-0.825 - 0.564i)T \)
3 \( 1 + (-0.394 + 0.918i)T \)
5 \( 1 + (0.427 + 0.904i)T \)
7 \( 1 + (0.489 - 0.871i)T \)
11 \( 1 + (0.911 - 0.411i)T \)
13 \( 1 + (0.760 - 0.648i)T \)
17 \( 1 + (0.607 - 0.794i)T \)
19 \( 1 + (0.550 + 0.835i)T \)
23 \( 1 + (-0.783 + 0.621i)T \)
29 \( 1 + (-0.123 + 0.992i)T \)
31 \( 1 + (0.880 - 0.474i)T \)
37 \( 1 + (-0.123 - 0.992i)T \)
41 \( 1 + (-0.329 + 0.944i)T \)
43 \( 1 + (-0.635 - 0.772i)T \)
47 \( 1 + (-0.925 + 0.378i)T \)
53 \( 1 + (0.938 + 0.345i)T \)
59 \( 1 + (-0.192 + 0.981i)T \)
61 \( 1 + (0.713 - 0.700i)T \)
67 \( 1 + (0.227 + 0.973i)T \)
71 \( 1 + (0.662 + 0.749i)T \)
73 \( 1 + (-0.579 - 0.815i)T \)
79 \( 1 + (-0.949 - 0.312i)T \)
83 \( 1 + (-0.261 + 0.965i)T \)
89 \( 1 + (-0.825 + 0.564i)T \)
97 \( 1 + (0.804 - 0.593i)T \)
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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

−27.66580801731531349027655611569, −26.03098444234471985783944344524, −25.26080837529894264139812875593, −24.438428442452550260376413710300, −24.02613547124352073256889585080, −22.811031580866133526191691287857, −21.45299449394362975933319324172, −20.2413310962983392817070135973, −19.27953583880953140964261757753, −18.35846949944411279877731356915, −17.52262628341222249997037823157, −16.86064933381891417944067394921, −15.80603262155641843612091697480, −14.489948803930551435172741306398, −13.51756924032253584637258811514, −12.10938092369074471779557628998, −11.491086581821183793104152941737, −9.890374749797818557481555594987, −8.71650096115090800588247922466, −8.1614020035527392090930073401, −6.64528605845781803007196154472, −5.90116986573962786620705750768, −4.80678696996159501115412602814, −2.04995252998505174994616245150, −1.237923682800653400606960549047, 1.25375625510470607301424164097, 3.18633167496866408660892816046, 3.9097009412072975428679439721, 5.72876846654935315970142041532, 6.99273694899148855956156160279, 8.239942863190137612543837288557, 9.61411288581566336488848292208, 10.26409537707403957186883749998, 11.1499498945090559351433762128, 11.84110118885341830304555997558, 13.6627321778413865547759874915, 14.57981819904410431844052301833, 15.96907565445442509708255137669, 16.804991275301707655928038179844, 17.722399167439763199036015770029, 18.42339008489977402722752207007, 19.82806322262777553440026981315, 20.65370948498606322605672654753, 21.4709215647525070598498255890, 22.406917483712057046950853029577, 23.16697906715055236241429157321, 24.91612064308213823243633505631, 25.87506586002445121044390614676, 26.73179542206646552381993358779, 27.34400097269404831890786088890

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