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Properties

Label 1-163-163.152-r0-0-0
Degree $1$
Conductor $163$
Sign $-0.995 - 0.0980i$
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  + (0.249 − 0.968i)2-s + (0.323 − 0.946i)3-s + (−0.875 − 0.483i)4-s + (0.597 − 0.802i)5-s + (−0.835 − 0.549i)6-s + (−0.431 − 0.902i)7-s + (−0.686 + 0.727i)8-s + (−0.790 − 0.612i)9-s + (−0.627 − 0.778i)10-s + (0.657 + 0.753i)11-s + (−0.740 + 0.672i)12-s + (−0.286 + 0.957i)13-s + (−0.981 + 0.192i)14-s + (−0.565 − 0.824i)15-s + (0.533 + 0.845i)16-s + (0.973 + 0.230i)17-s + ⋯
L(s)  = 1  + (0.249 − 0.968i)2-s + (0.323 − 0.946i)3-s + (−0.875 − 0.483i)4-s + (0.597 − 0.802i)5-s + (−0.835 − 0.549i)6-s + (−0.431 − 0.902i)7-s + (−0.686 + 0.727i)8-s + (−0.790 − 0.612i)9-s + (−0.627 − 0.778i)10-s + (0.657 + 0.753i)11-s + (−0.740 + 0.672i)12-s + (−0.286 + 0.957i)13-s + (−0.981 + 0.192i)14-s + (−0.565 − 0.824i)15-s + (0.533 + 0.845i)16-s + (0.973 + 0.230i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 163 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 163 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(163\)
Sign: $-0.995 - 0.0980i$
Analytic conductor: \(0.756968\)
Root analytic conductor: \(0.756968\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{163} (152, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 163,\ (0:\ ),\ -0.995 - 0.0980i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.06343283000 - 1.290405114i\)
\(L(\frac12)\) \(\approx\) \(0.06343283000 - 1.290405114i\)
\(L(1)\) \(\approx\) \(0.6507130354 - 1.032717246i\)
\(L(1)\) \(\approx\) \(0.6507130354 - 1.032717246i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad163 \( 1 \)
good2 \( 1 + (0.249 - 0.968i)T \)
3 \( 1 + (0.323 - 0.946i)T \)
5 \( 1 + (0.597 - 0.802i)T \)
7 \( 1 + (-0.431 - 0.902i)T \)
11 \( 1 + (0.657 + 0.753i)T \)
13 \( 1 + (-0.286 + 0.957i)T \)
17 \( 1 + (0.973 + 0.230i)T \)
19 \( 1 + (0.0968 - 0.995i)T \)
23 \( 1 + (0.766 + 0.642i)T \)
29 \( 1 + (-0.963 - 0.268i)T \)
31 \( 1 + (-0.993 - 0.116i)T \)
37 \( 1 + (0.893 + 0.448i)T \)
41 \( 1 + (-0.875 + 0.483i)T \)
43 \( 1 + (0.987 + 0.154i)T \)
47 \( 1 + (0.713 + 0.700i)T \)
53 \( 1 + (0.173 - 0.984i)T \)
59 \( 1 + (-0.5 - 0.866i)T \)
61 \( 1 + (-0.686 - 0.727i)T \)
67 \( 1 + (0.0193 - 0.999i)T \)
71 \( 1 + (-0.360 - 0.932i)T \)
73 \( 1 + (0.0193 + 0.999i)T \)
79 \( 1 + (0.925 + 0.378i)T \)
83 \( 1 + (0.996 + 0.0774i)T \)
89 \( 1 + (0.657 - 0.753i)T \)
97 \( 1 + (-0.135 - 0.990i)T \)
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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

−27.68045359993896677763477581989, −27.12198821826011114282951395332, −26.12520216726635737772395709765, −25.20623546419863991616457385795, −24.89891506762916231396218427338, −23.06908315787776507878375221030, −22.2605408062956976576476286031, −21.82433424154456337048475538918, −20.74945109409942566353554353940, −19.085686690857910401567258075871, −18.317681513836683179642019420676, −16.95350126776386557164193905757, −16.27782285688542346497928151888, −14.97723225079366005692254873413, −14.66772387376948398490752081504, −13.56776755312832330749803145720, −12.23992300200076741726736678426, −10.64800898998943223732733182212, −9.560089239944521550513621838098, −8.7815258760177688953564574727, −7.46853710065318122904868065007, −5.89468830390962866453390554225, −5.49114217129366491409599006060, −3.67342536704785035437701032559, −2.85776529546752351703986634857, 1.063648422056685529301517312654, 2.02914748001754183783177795334, 3.57020377652786154882141625826, 4.84864378128093376565047507122, 6.29183659558171777502975235739, 7.559787660398761028290205375394, 9.18592676544856721436500603790, 9.603171947056247057540664665, 11.243648467312253327940299453087, 12.32940408648546124240436192888, 13.08155419839687036761679782225, 13.83316923743695789941479257505, 14.74103641376884996616954337076, 16.87065581270969913984785742736, 17.41601211497818677933785961682, 18.683875466573449545998959995047, 19.66196880085250989219100657773, 20.22688558961701822152974703454, 21.17804658232261403805332001909, 22.37302602615161318830435449987, 23.530273477951981933073867401084, 24.037268522815122196693796846120, 25.36030906590857736278827136999, 26.19537337921933199590463200120, 27.564860666022446814105370050432

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