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Properties

Label 1-163-163.119-r0-0-0
Degree $1$
Conductor $163$
Sign $0.300 - 0.953i$
Analytic cond. $0.756968$
Root an. cond. $0.756968$
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  + (0.323 − 0.946i)2-s + (0.952 + 0.305i)3-s + (−0.790 − 0.612i)4-s + (0.973 + 0.230i)5-s + (0.597 − 0.802i)6-s + (−0.875 − 0.483i)7-s + (−0.835 + 0.549i)8-s + (0.813 + 0.581i)9-s + (0.533 − 0.845i)10-s + (−0.211 − 0.977i)11-s + (−0.565 − 0.824i)12-s + (0.893 − 0.448i)13-s + (−0.740 + 0.672i)14-s + (0.856 + 0.516i)15-s + (0.249 + 0.968i)16-s + (−0.0581 − 0.998i)17-s + ⋯
L(s)  = 1  + (0.323 − 0.946i)2-s + (0.952 + 0.305i)3-s + (−0.790 − 0.612i)4-s + (0.973 + 0.230i)5-s + (0.597 − 0.802i)6-s + (−0.875 − 0.483i)7-s + (−0.835 + 0.549i)8-s + (0.813 + 0.581i)9-s + (0.533 − 0.845i)10-s + (−0.211 − 0.977i)11-s + (−0.565 − 0.824i)12-s + (0.893 − 0.448i)13-s + (−0.740 + 0.672i)14-s + (0.856 + 0.516i)15-s + (0.249 + 0.968i)16-s + (−0.0581 − 0.998i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 163 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.300 - 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 163 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.300 - 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(163\)
Sign: $0.300 - 0.953i$
Analytic conductor: \(0.756968\)
Root analytic conductor: \(0.756968\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{163} (119, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 163,\ (0:\ ),\ 0.300 - 0.953i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.402904728 - 1.028661779i\)
\(L(\frac12)\) \(\approx\) \(1.402904728 - 1.028661779i\)
\(L(1)\) \(\approx\) \(1.406160341 - 0.6945687651i\)
\(L(1)\) \(\approx\) \(1.406160341 - 0.6945687651i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad163 \( 1 \)
good2 \( 1 + (0.323 - 0.946i)T \)
3 \( 1 + (0.952 + 0.305i)T \)
5 \( 1 + (0.973 + 0.230i)T \)
7 \( 1 + (-0.875 - 0.483i)T \)
11 \( 1 + (-0.211 - 0.977i)T \)
13 \( 1 + (0.893 - 0.448i)T \)
17 \( 1 + (-0.0581 - 0.998i)T \)
19 \( 1 + (-0.360 + 0.932i)T \)
23 \( 1 + (0.173 + 0.984i)T \)
29 \( 1 + (0.657 - 0.753i)T \)
31 \( 1 + (-0.686 + 0.727i)T \)
37 \( 1 + (-0.993 + 0.116i)T \)
41 \( 1 + (-0.790 + 0.612i)T \)
43 \( 1 + (-0.999 + 0.0387i)T \)
47 \( 1 + (-0.981 + 0.192i)T \)
53 \( 1 + (-0.939 - 0.342i)T \)
59 \( 1 + (-0.5 + 0.866i)T \)
61 \( 1 + (-0.835 - 0.549i)T \)
67 \( 1 + (0.925 + 0.378i)T \)
71 \( 1 + (0.466 - 0.884i)T \)
73 \( 1 + (0.925 - 0.378i)T \)
79 \( 1 + (0.0968 + 0.995i)T \)
83 \( 1 + (0.0193 + 0.999i)T \)
89 \( 1 + (-0.211 + 0.977i)T \)
97 \( 1 + (-0.910 - 0.413i)T \)
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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

−27.90237989478886355546303740109, −26.11231641211468546239740516586, −25.97704034759880117475002964853, −25.217857272220587498064695043409, −24.29057764975948923096616093848, −23.31488010921426354325988416597, −22.0495442411620591437051315413, −21.29390698273399365470921416432, −20.20936887727505289044063396877, −18.82727796071469499251148020062, −18.07156823808097974842809345039, −16.973391356293847706729810172769, −15.77862220667279997374494783728, −14.96150397972130719215903273146, −13.93382527178657563530149998459, −12.98310025748804123893014068911, −12.56468108055427258483023791912, −10.12897573060172900521225337029, −9.09589404732208679210956887728, −8.49489267012213676631888140956, −6.88358720445481941185383746229, −6.283164846479899958268150289039, −4.7803750333796414552545071802, −3.38795546245304504838213392733, −2.01745037351492052770709377913, 1.50810506909526477441174131883, 3.00447767555385842071757421097, 3.54469217201847289025207655221, 5.212668640401591732720411679, 6.45812922549526356012446113230, 8.29952546385494161975549818772, 9.40731211696331732542912486799, 10.12679697672246251902937765416, 11.0095995356876321433289564164, 12.74333862922123374765881759030, 13.745819002580935575963303389366, 13.859534634455586198632998597934, 15.37270419718871145939606274726, 16.55099980201861429795356923577, 18.12081866253866210986719305561, 18.913117478545105370157070481891, 19.84174258759663397209441749816, 20.84070872626182887247402457985, 21.41058303337235618802449757488, 22.42029216193298992473876104851, 23.37984466605465071017697606037, 24.85233797558246549500925440647, 25.68559715306865762375341489009, 26.67578699760218343578753152426, 27.44275724482525809657565628205

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