Properties

Label 1-167-167.2-r0-0-0
Degree $1$
Conductor $167$
Sign $-0.307 - 0.951i$
Analytic cond. $0.775544$
Root an. cond. $0.775544$
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  + (−0.644 − 0.764i)2-s + (−0.584 − 0.811i)3-s + (−0.169 + 0.985i)4-s + (0.0567 + 0.998i)5-s + (−0.243 + 0.969i)6-s + (−0.914 + 0.404i)7-s + (0.862 − 0.505i)8-s + (−0.316 + 0.948i)9-s + (0.726 − 0.686i)10-s + (−0.0189 − 0.999i)11-s + (0.898 − 0.438i)12-s + (0.421 − 0.906i)13-s + (0.898 + 0.438i)14-s + (0.776 − 0.629i)15-s + (−0.942 − 0.334i)16-s + (0.132 − 0.991i)17-s + ⋯
L(s)  = 1  + (−0.644 − 0.764i)2-s + (−0.584 − 0.811i)3-s + (−0.169 + 0.985i)4-s + (0.0567 + 0.998i)5-s + (−0.243 + 0.969i)6-s + (−0.914 + 0.404i)7-s + (0.862 − 0.505i)8-s + (−0.316 + 0.948i)9-s + (0.726 − 0.686i)10-s + (−0.0189 − 0.999i)11-s + (0.898 − 0.438i)12-s + (0.421 − 0.906i)13-s + (0.898 + 0.438i)14-s + (0.776 − 0.629i)15-s + (−0.942 − 0.334i)16-s + (0.132 − 0.991i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 167 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.307 - 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 167 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.307 - 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(167\)
Sign: $-0.307 - 0.951i$
Analytic conductor: \(0.775544\)
Root analytic conductor: \(0.775544\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{167} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 167,\ (0:\ ),\ -0.307 - 0.951i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3188120549 - 0.4382452862i\)
\(L(\frac12)\) \(\approx\) \(0.3188120549 - 0.4382452862i\)
\(L(1)\) \(\approx\) \(0.5165292633 - 0.2954482504i\)
\(L(1)\) \(\approx\) \(0.5165292633 - 0.2954482504i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad167 \( 1 \)
good2 \( 1 + (-0.644 - 0.764i)T \)
3 \( 1 + (-0.584 - 0.811i)T \)
5 \( 1 + (0.0567 + 0.998i)T \)
7 \( 1 + (-0.914 + 0.404i)T \)
11 \( 1 + (-0.0189 - 0.999i)T \)
13 \( 1 + (0.421 - 0.906i)T \)
17 \( 1 + (0.132 - 0.991i)T \)
19 \( 1 + (0.988 - 0.150i)T \)
23 \( 1 + (0.614 - 0.788i)T \)
29 \( 1 + (0.614 + 0.788i)T \)
31 \( 1 + (-0.387 - 0.922i)T \)
37 \( 1 + (-0.316 - 0.948i)T \)
41 \( 1 + (-0.700 + 0.713i)T \)
43 \( 1 + (0.974 - 0.225i)T \)
47 \( 1 + (0.351 - 0.936i)T \)
53 \( 1 + (-0.0944 + 0.995i)T \)
59 \( 1 + (0.132 + 0.991i)T \)
61 \( 1 + (0.822 - 0.569i)T \)
67 \( 1 + (0.0567 - 0.998i)T \)
71 \( 1 + (0.553 - 0.832i)T \)
73 \( 1 + (-0.942 + 0.334i)T \)
79 \( 1 + (-0.521 - 0.853i)T \)
83 \( 1 + (-0.644 + 0.764i)T \)
89 \( 1 + (0.776 + 0.629i)T \)
97 \( 1 + (-0.387 + 0.922i)T \)
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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

−27.89114162937091002126425529890, −26.980019812250952101976757163488, −25.98790729307011208456784421017, −25.332065123510517804846595976796, −23.89488417502656464787143675859, −23.34030623220327091513601737063, −22.35216454531452157286923290186, −20.99158489564868658588545872615, −20.08707230334427688630887923441, −19.09184085374981046009312472362, −17.554741124193379781485666222344, −17.064401620563538244132147872210, −16.04529698388146674854402071773, −15.61416348272098327354982099277, −14.20318096471615037676742921730, −12.9104275605649211095181131597, −11.64514633808027338611739954918, −10.2016891497880403680290931175, −9.58580251020324617588886347227, −8.685664212426673230785922506398, −7.13779340442374099664883253862, −6.04896318002091832546881797298, −4.97480080001324987058730203799, −3.92599046705267753893908962066, −1.28291235605041336456920756475, 0.688825372133209256239735653952, 2.58807379571740181615895996314, 3.28608835832414354069802531167, 5.56320115986948741987038702874, 6.73369217052010413096889586116, 7.67689714756059528809192956801, 8.98868044430001536915919413128, 10.32040443405823589603775875346, 11.11158157587789684248755916611, 12.0334354348827240015747088518, 13.10669463585770096633492698239, 13.92714824809725353157246343861, 15.807047612507102542968899825, 16.67653138427286689959310750311, 18.02202516640070781301856819215, 18.488054250578851308524303112425, 19.20066270354592920753458633383, 20.22840149980238223852350600904, 21.74185609073088389680509796705, 22.45666948430605047855331523557, 23.05747233581535018449566502969, 24.746850138810244238651699132635, 25.48127952482998046486567003000, 26.52385496210557511964487320855, 27.42906320136291868386937001554

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