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Properties

Label 1-167-167.11-r0-0-0
Degree $1$
Conductor $167$
Sign $-0.436 - 0.899i$
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  + (−0.0189 − 0.999i)2-s + (0.614 − 0.788i)3-s + (−0.999 + 0.0378i)4-s + (0.489 + 0.872i)5-s + (−0.800 − 0.599i)6-s + (0.822 − 0.569i)7-s + (0.0567 + 0.998i)8-s + (−0.243 − 0.969i)9-s + (0.862 − 0.505i)10-s + (−0.169 − 0.985i)11-s + (−0.584 + 0.811i)12-s + (−0.700 + 0.713i)13-s + (−0.584 − 0.811i)14-s + (0.988 + 0.150i)15-s + (0.997 − 0.0756i)16-s + (0.929 − 0.369i)17-s + ⋯
L(s)  = 1  + (−0.0189 − 0.999i)2-s + (0.614 − 0.788i)3-s + (−0.999 + 0.0378i)4-s + (0.489 + 0.872i)5-s + (−0.800 − 0.599i)6-s + (0.822 − 0.569i)7-s + (0.0567 + 0.998i)8-s + (−0.243 − 0.969i)9-s + (0.862 − 0.505i)10-s + (−0.169 − 0.985i)11-s + (−0.584 + 0.811i)12-s + (−0.700 + 0.713i)13-s + (−0.584 − 0.811i)14-s + (0.988 + 0.150i)15-s + (0.997 − 0.0756i)16-s + (0.929 − 0.369i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 167 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.436 - 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 167 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.436 - 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(167\)
Sign: $-0.436 - 0.899i$
Analytic conductor: \(0.775544\)
Root analytic conductor: \(0.775544\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{167} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 167,\ (0:\ ),\ -0.436 - 0.899i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7227971402 - 1.153966583i\)
\(L(\frac12)\) \(\approx\) \(0.7227971402 - 1.153966583i\)
\(L(1)\) \(\approx\) \(0.9542221488 - 0.8147057709i\)
\(L(1)\) \(\approx\) \(0.9542221488 - 0.8147057709i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad167 \( 1 \)
good2 \( 1 + (-0.0189 - 0.999i)T \)
3 \( 1 + (0.614 - 0.788i)T \)
5 \( 1 + (0.489 + 0.872i)T \)
7 \( 1 + (0.822 - 0.569i)T \)
11 \( 1 + (-0.169 - 0.985i)T \)
13 \( 1 + (-0.700 + 0.713i)T \)
17 \( 1 + (0.929 - 0.369i)T \)
19 \( 1 + (0.206 - 0.978i)T \)
23 \( 1 + (-0.316 - 0.948i)T \)
29 \( 1 + (-0.316 + 0.948i)T \)
31 \( 1 + (0.421 + 0.906i)T \)
37 \( 1 + (-0.243 + 0.969i)T \)
41 \( 1 + (-0.644 + 0.764i)T \)
43 \( 1 + (-0.455 - 0.890i)T \)
47 \( 1 + (-0.0944 + 0.995i)T \)
53 \( 1 + (-0.752 + 0.658i)T \)
59 \( 1 + (0.929 + 0.369i)T \)
61 \( 1 + (0.672 + 0.739i)T \)
67 \( 1 + (0.489 - 0.872i)T \)
71 \( 1 + (-0.843 - 0.537i)T \)
73 \( 1 + (0.997 + 0.0756i)T \)
79 \( 1 + (0.974 - 0.225i)T \)
83 \( 1 + (-0.0189 + 0.999i)T \)
89 \( 1 + (0.988 - 0.150i)T \)
97 \( 1 + (0.421 - 0.906i)T \)
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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

−27.86508334933725435489607273847, −26.95880665878735741473014729450, −25.765490855773695953657604880571, −25.0798759140213809680431330566, −24.51391863821319306577693758140, −23.212331288302971342709491863132, −22.08723043546799759776390894498, −21.167592475891877331423929417213, −20.40921744183196719482252718196, −19.06169083461801643641923518695, −17.7022819670389885835173127613, −17.059904690795436065866596156144, −15.93844724159305380298070758465, −15.03003791117764184094546347364, −14.380730492259021224754230490942, −13.19792446088931554230409990619, −12.10396792334976531615679423505, −10.027885404602993877103728380550, −9.56454806730938108864114576386, −8.24122999394970210690405340574, −7.74795408560940507802976478590, −5.60255092387472711255929435830, −5.09789965999212205290195100045, −3.916182236770061775581088644723, −1.95039071889152134277175390818, 1.25448630153381475672911769466, 2.4975489259555461444807193404, 3.411464602855676976357961303, 5.04648513002018323005439979161, 6.69536480931315302050122868689, 7.8495895343287154148878625010, 8.947606312963450495803734072249, 10.14764030723770470993410461445, 11.1870880884101066684631718126, 12.10075675836110253515210311595, 13.494416779401340536280512408041, 14.07746300534504068851435486845, 14.71148131378860310282279894853, 16.900577466251089173829194130456, 17.93292199735016329220857089078, 18.63346269172999160863065814753, 19.40780786151129815407221567528, 20.47985098015390945401127696649, 21.347286692225278541384407513349, 22.22637527378777236757331631419, 23.55862039290240638083628981179, 24.1921085211610704005257104832, 25.61217013119361892702930280094, 26.63411411167611626516808827522, 27.02777755638776453209200401487

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