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Properties

Label 1-167-167.33-r0-0-0
Degree $1$
Conductor $167$
Sign $0.845 - 0.533i$
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  + (−0.800 + 0.599i)2-s + (0.862 + 0.505i)3-s + (0.280 − 0.959i)4-s + (−0.0944 − 0.995i)5-s + (−0.993 + 0.113i)6-s + (−0.169 − 0.985i)7-s + (0.351 + 0.936i)8-s + (0.489 + 0.872i)9-s + (0.672 + 0.739i)10-s + (−0.881 − 0.472i)11-s + (0.726 − 0.686i)12-s + (−0.316 − 0.948i)13-s + (0.726 + 0.686i)14-s + (0.421 − 0.906i)15-s + (−0.843 − 0.537i)16-s + (0.954 − 0.298i)17-s + ⋯
L(s)  = 1  + (−0.800 + 0.599i)2-s + (0.862 + 0.505i)3-s + (0.280 − 0.959i)4-s + (−0.0944 − 0.995i)5-s + (−0.993 + 0.113i)6-s + (−0.169 − 0.985i)7-s + (0.351 + 0.936i)8-s + (0.489 + 0.872i)9-s + (0.672 + 0.739i)10-s + (−0.881 − 0.472i)11-s + (0.726 − 0.686i)12-s + (−0.316 − 0.948i)13-s + (0.726 + 0.686i)14-s + (0.421 − 0.906i)15-s + (−0.843 − 0.537i)16-s + (0.954 − 0.298i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 167 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.845 - 0.533i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 167 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.845 - 0.533i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(167\)
Sign: $0.845 - 0.533i$
Analytic conductor: \(0.775544\)
Root analytic conductor: \(0.775544\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{167} (33, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 167,\ (0:\ ),\ 0.845 - 0.533i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8622099999 - 0.2492790195i\)
\(L(\frac12)\) \(\approx\) \(0.8622099999 - 0.2492790195i\)
\(L(1)\) \(\approx\) \(0.8829790533 + 0.01441640776i\)
\(L(1)\) \(\approx\) \(0.8829790533 + 0.01441640776i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad167 \( 1 \)
good2 \( 1 + (-0.800 + 0.599i)T \)
3 \( 1 + (0.862 + 0.505i)T \)
5 \( 1 + (-0.0944 - 0.995i)T \)
7 \( 1 + (-0.169 - 0.985i)T \)
11 \( 1 + (-0.881 - 0.472i)T \)
13 \( 1 + (-0.316 - 0.948i)T \)
17 \( 1 + (0.954 - 0.298i)T \)
19 \( 1 + (-0.700 - 0.713i)T \)
23 \( 1 + (0.0567 - 0.998i)T \)
29 \( 1 + (0.0567 + 0.998i)T \)
31 \( 1 + (0.614 + 0.788i)T \)
37 \( 1 + (0.489 - 0.872i)T \)
41 \( 1 + (-0.243 + 0.969i)T \)
43 \( 1 + (0.929 - 0.369i)T \)
47 \( 1 + (0.997 + 0.0756i)T \)
53 \( 1 + (0.776 + 0.629i)T \)
59 \( 1 + (0.954 + 0.298i)T \)
61 \( 1 + (-0.999 - 0.0378i)T \)
67 \( 1 + (-0.0944 + 0.995i)T \)
71 \( 1 + (0.898 + 0.438i)T \)
73 \( 1 + (-0.843 + 0.537i)T \)
79 \( 1 + (0.132 - 0.991i)T \)
83 \( 1 + (-0.800 - 0.599i)T \)
89 \( 1 + (0.421 + 0.906i)T \)
97 \( 1 + (0.614 - 0.788i)T \)
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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

−27.63458593842081993798572447109, −26.65073481617689875271069730423, −25.68043143527176624688206286002, −25.50144098405551114791741877023, −24.036281750916998295174645380679, −22.753856133432140965949349110914, −21.41622592270511273620082980239, −20.98629204483744458476251783258, −19.49124401959090216493886891020, −18.86504339339693035013784746642, −18.43490833940812537034794811374, −17.24944811894981529449726817490, −15.66401810021420656505927958956, −14.89922939055119726550667919241, −13.613350824913860117156546639379, −12.42237359654565052769535554904, −11.66322196064166863714383142611, −10.17821969075255260893349822852, −9.4438712830585134548537402484, −8.16041775905588110676340259721, −7.43066497896847063164571177068, −6.21078920998010657686441779420, −3.85694329210584536460518306269, −2.66279736107209988815049634247, −1.98118182757264541222679553712, 0.89008109720362775465696223602, 2.766368240582429707859721152434, 4.4652728443619787799594103926, 5.47079405981649065656891433104, 7.28588218788197249519645378158, 8.11478858469464136688494983209, 8.92681135459832761615478330300, 10.11924509157976937523490445622, 10.72849266786511046156732888480, 12.75627332503557185071938846551, 13.750671307162939262360177042870, 14.80119765831229319290962983614, 15.94905823895253759908614158976, 16.46059747748930676339234730786, 17.49728893278519103257857903583, 18.84686332123517661133199582490, 19.87004736747714416814923506684, 20.37528937172035353857873368433, 21.33501755877954321651770146923, 23.07716131695183237378522889296, 23.94982499117806796146789439335, 24.89274249232660270613049126665, 25.694598687249593459164624648930, 26.625815396731675837185289783136, 27.311536241071992476773907615296

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