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Properties

Label 1-179-179.31-r0-0-0
Degree $1$
Conductor $179$
Sign $0.998 - 0.0627i$
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  + (−0.579 + 0.815i)2-s + (−0.737 − 0.675i)3-s + (−0.329 − 0.944i)4-s + (0.911 + 0.411i)5-s + (0.977 − 0.210i)6-s + (−0.925 − 0.378i)7-s + (0.960 + 0.278i)8-s + (0.0881 + 0.996i)9-s + (−0.863 + 0.505i)10-s + (0.158 + 0.987i)11-s + (−0.394 + 0.918i)12-s + (−0.261 − 0.965i)13-s + (0.844 − 0.535i)14-s + (−0.394 − 0.918i)15-s + (−0.783 + 0.621i)16-s + (0.427 − 0.904i)17-s + ⋯
L(s)  = 1  + (−0.579 + 0.815i)2-s + (−0.737 − 0.675i)3-s + (−0.329 − 0.944i)4-s + (0.911 + 0.411i)5-s + (0.977 − 0.210i)6-s + (−0.925 − 0.378i)7-s + (0.960 + 0.278i)8-s + (0.0881 + 0.996i)9-s + (−0.863 + 0.505i)10-s + (0.158 + 0.987i)11-s + (−0.394 + 0.918i)12-s + (−0.261 − 0.965i)13-s + (0.844 − 0.535i)14-s + (−0.394 − 0.918i)15-s + (−0.783 + 0.621i)16-s + (0.427 − 0.904i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 179 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0627i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 179 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0627i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(179\)
Sign: $0.998 - 0.0627i$
Analytic conductor: \(0.831272\)
Root analytic conductor: \(0.831272\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{179} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 179,\ (0:\ ),\ 0.998 - 0.0627i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6854696628 + 0.02151119179i\)
\(L(\frac12)\) \(\approx\) \(0.6854696628 + 0.02151119179i\)
\(L(1)\) \(\approx\) \(0.6892445959 + 0.06019923386i\)
\(L(1)\) \(\approx\) \(0.6892445959 + 0.06019923386i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad179 \( 1 \)
good2 \( 1 + (-0.579 + 0.815i)T \)
3 \( 1 + (-0.737 - 0.675i)T \)
5 \( 1 + (0.911 + 0.411i)T \)
7 \( 1 + (-0.925 - 0.378i)T \)
11 \( 1 + (0.158 + 0.987i)T \)
13 \( 1 + (-0.261 - 0.965i)T \)
17 \( 1 + (0.427 - 0.904i)T \)
19 \( 1 + (0.362 - 0.932i)T \)
23 \( 1 + (0.990 + 0.140i)T \)
29 \( 1 + (0.804 + 0.593i)T \)
31 \( 1 + (-0.825 - 0.564i)T \)
37 \( 1 + (0.804 - 0.593i)T \)
41 \( 1 + (0.997 - 0.0705i)T \)
43 \( 1 + (0.662 - 0.749i)T \)
47 \( 1 + (-0.969 - 0.244i)T \)
53 \( 1 + (0.607 + 0.794i)T \)
59 \( 1 + (-0.123 - 0.992i)T \)
61 \( 1 + (0.880 + 0.474i)T \)
67 \( 1 + (0.960 - 0.278i)T \)
71 \( 1 + (-0.949 - 0.312i)T \)
73 \( 1 + (-0.999 + 0.0352i)T \)
79 \( 1 + (0.489 - 0.871i)T \)
83 \( 1 + (-0.635 + 0.772i)T \)
89 \( 1 + (-0.579 - 0.815i)T \)
97 \( 1 + (-0.520 + 0.854i)T \)
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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

−27.51185735120832720610598065014, −26.57113265923526653870967560585, −25.79252957042090570764682642928, −24.66583641672081478450669957364, −23.20165815545137930394918894507, −22.12891700081553954225714938974, −21.48442484953369152206851379019, −20.94895563764452149584892757197, −19.51525282705304038147066580339, −18.66379956552213643968172379787, −17.56389554123066681677320204476, −16.51615029524837626139553376300, −16.33370640741234409950601482775, −14.42091059490419597431598429431, −13.0837863259149134107354541102, −12.27237766470375844625130643098, −11.21238712874573840256384423529, −10.10128466526891268872494943444, −9.46193902453005547329265352392, −8.57190121149098556920366081304, −6.57635205659634454718753910336, −5.5887678488197216262834992846, −4.14243934957180682439844895270, −2.92616584915091246267999093203, −1.21265527684763347341323669663, 0.90680519509414205430829416921, 2.56074188384225755954630290059, 4.963044476276446831569002804, 5.85775492186319002713751967560, 6.988048201491860871921476640224, 7.37564932370555131720179284619, 9.26713603871675177430935995399, 10.06332238494196698625641383314, 11.00229801869098994360183953761, 12.70686902996700985344805762176, 13.44815095262612305248543326117, 14.561589515078221989896942350560, 15.81356789440787557556625313240, 16.83073874614968468078931711945, 17.64365540485423422240151251461, 18.19079249942986279168405032798, 19.2847620478091528486263746767, 20.24242909332096813627907673748, 22.10846276615915178029445233477, 22.76990077895911140662267261639, 23.43598018338088974063962287282, 24.84914371346412035268111452037, 25.2696239293651687296677297084, 26.13328055911850712315079383909, 27.34738106830853142540872106131

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