Properties

Label 1-167-167.14-r0-0-0
Degree $1$
Conductor $167$
Sign $0.229 + 0.973i$
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  + (0.898 + 0.438i)2-s + (−0.982 + 0.188i)3-s + (0.614 + 0.788i)4-s + (0.954 − 0.298i)5-s + (−0.965 − 0.261i)6-s + (−0.387 + 0.922i)7-s + (0.206 + 0.978i)8-s + (0.929 − 0.369i)9-s + (0.988 + 0.150i)10-s + (−0.584 − 0.811i)11-s + (−0.752 − 0.658i)12-s + (0.974 + 0.225i)13-s + (−0.752 + 0.658i)14-s + (−0.881 + 0.472i)15-s + (−0.243 + 0.969i)16-s + (−0.942 + 0.334i)17-s + ⋯
L(s)  = 1  + (0.898 + 0.438i)2-s + (−0.982 + 0.188i)3-s + (0.614 + 0.788i)4-s + (0.954 − 0.298i)5-s + (−0.965 − 0.261i)6-s + (−0.387 + 0.922i)7-s + (0.206 + 0.978i)8-s + (0.929 − 0.369i)9-s + (0.988 + 0.150i)10-s + (−0.584 − 0.811i)11-s + (−0.752 − 0.658i)12-s + (0.974 + 0.225i)13-s + (−0.752 + 0.658i)14-s + (−0.881 + 0.472i)15-s + (−0.243 + 0.969i)16-s + (−0.942 + 0.334i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 167 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 167 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(167\)
Sign: $0.229 + 0.973i$
Analytic conductor: \(0.775544\)
Root analytic conductor: \(0.775544\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{167} (14, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 167,\ (0:\ ),\ 0.229 + 0.973i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.224008342 + 0.9690380534i\)
\(L(\frac12)\) \(\approx\) \(1.224008342 + 0.9690380534i\)
\(L(1)\) \(\approx\) \(1.301624225 + 0.6001522312i\)
\(L(1)\) \(\approx\) \(1.301624225 + 0.6001522312i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad167 \( 1 \)
good2 \( 1 + (0.898 + 0.438i)T \)
3 \( 1 + (-0.982 + 0.188i)T \)
5 \( 1 + (0.954 - 0.298i)T \)
7 \( 1 + (-0.387 + 0.922i)T \)
11 \( 1 + (-0.584 - 0.811i)T \)
13 \( 1 + (0.974 + 0.225i)T \)
17 \( 1 + (-0.942 + 0.334i)T \)
19 \( 1 + (0.280 + 0.959i)T \)
23 \( 1 + (0.132 + 0.991i)T \)
29 \( 1 + (0.132 - 0.991i)T \)
31 \( 1 + (-0.521 - 0.853i)T \)
37 \( 1 + (0.929 + 0.369i)T \)
41 \( 1 + (-0.455 + 0.890i)T \)
43 \( 1 + (0.351 - 0.936i)T \)
47 \( 1 + (-0.644 - 0.764i)T \)
53 \( 1 + (-0.0189 - 0.999i)T \)
59 \( 1 + (-0.942 - 0.334i)T \)
61 \( 1 + (0.421 - 0.906i)T \)
67 \( 1 + (0.954 + 0.298i)T \)
71 \( 1 + (0.489 - 0.872i)T \)
73 \( 1 + (-0.243 - 0.969i)T \)
79 \( 1 + (0.672 - 0.739i)T \)
83 \( 1 + (0.898 - 0.438i)T \)
89 \( 1 + (-0.881 - 0.472i)T \)
97 \( 1 + (-0.521 + 0.853i)T \)
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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

−27.86490252486800242662298212405, −26.36925100889863968650563746049, −25.31960301205307305638240357993, −24.2003496001085471743707945593, −23.2877513298366242359785977379, −22.63066989878215574900488255696, −21.831795896701527986329871194622, −20.78212516274478729678001492738, −19.91314272917618326603742126927, −18.39499317953589133262499529860, −17.7553596615245539202670209073, −16.44782287768062061542267094687, −15.54889060388671650160032645853, −14.10464966013793739942158496575, −13.17269044997264770667529621483, −12.667984216676226403264464369625, −10.97339020346493298499460926098, −10.6700681043773650098217244252, −9.53182959446019522405234860677, −7.08100217677795450740698665269, −6.51389973744560799764577330085, −5.309867820152200887133888630590, −4.34228937401710973108285202629, −2.68416519013696406600656281618, −1.23874632988394041676585895135, 1.97036861204507463094680324766, 3.59537099355936607182550239535, 5.076538262591199271475590945845, 5.933075929893247195330561300892, 6.37940834509723279672679350140, 8.20820046026456489858431354714, 9.52702555435267204628978932769, 10.93346399527379652346975916042, 11.82963884002761224043582045756, 13.03458074141819911616783940568, 13.5210505598086653671538136238, 15.12355018704920882243021641729, 16.00840891812384415159488514714, 16.71048728368944736170431972042, 17.810500853497472096369920250836, 18.70812278037554275958473862685, 20.61194461408828565408296446413, 21.50305145620817319682079756238, 21.92150818160024884493691330695, 22.96216712140836475018609954812, 23.9365538675068388253140065327, 24.76191102043255630876481333239, 25.65757894298481625456236403350, 26.69092161052377536519613725950, 28.184664474569882730476540135542

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