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Properties

Label 1-167-167.57-r0-0-0
Degree $1$
Conductor $167$
Sign $0.948 - 0.315i$
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  + (−0.700 − 0.713i)2-s + (0.898 + 0.438i)3-s + (−0.0189 + 0.999i)4-s + (0.862 − 0.505i)5-s + (−0.316 − 0.948i)6-s + (0.954 + 0.298i)7-s + (0.726 − 0.686i)8-s + (0.614 + 0.788i)9-s + (−0.965 − 0.261i)10-s + (−0.644 − 0.764i)11-s + (−0.455 + 0.890i)12-s + (−0.387 + 0.922i)13-s + (−0.455 − 0.890i)14-s + (0.997 − 0.0756i)15-s + (−0.999 − 0.0378i)16-s + (−0.982 − 0.188i)17-s + ⋯
L(s)  = 1  + (−0.700 − 0.713i)2-s + (0.898 + 0.438i)3-s + (−0.0189 + 0.999i)4-s + (0.862 − 0.505i)5-s + (−0.316 − 0.948i)6-s + (0.954 + 0.298i)7-s + (0.726 − 0.686i)8-s + (0.614 + 0.788i)9-s + (−0.965 − 0.261i)10-s + (−0.644 − 0.764i)11-s + (−0.455 + 0.890i)12-s + (−0.387 + 0.922i)13-s + (−0.455 − 0.890i)14-s + (0.997 − 0.0756i)15-s + (−0.999 − 0.0378i)16-s + (−0.982 − 0.188i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 167 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.948 - 0.315i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 167 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.948 - 0.315i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(167\)
Sign: $0.948 - 0.315i$
Analytic conductor: \(0.775544\)
Root analytic conductor: \(0.775544\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{167} (57, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 167,\ (0:\ ),\ 0.948 - 0.315i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.252186044 - 0.2029850499i\)
\(L(\frac12)\) \(\approx\) \(1.252186044 - 0.2029850499i\)
\(L(1)\) \(\approx\) \(1.144163002 - 0.1756068435i\)
\(L(1)\) \(\approx\) \(1.144163002 - 0.1756068435i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad167 \( 1 \)
good2 \( 1 + (-0.700 - 0.713i)T \)
3 \( 1 + (0.898 + 0.438i)T \)
5 \( 1 + (0.862 - 0.505i)T \)
7 \( 1 + (0.954 + 0.298i)T \)
11 \( 1 + (-0.644 - 0.764i)T \)
13 \( 1 + (-0.387 + 0.922i)T \)
17 \( 1 + (-0.982 - 0.188i)T \)
19 \( 1 + (0.776 + 0.629i)T \)
23 \( 1 + (-0.584 - 0.811i)T \)
29 \( 1 + (-0.584 + 0.811i)T \)
31 \( 1 + (-0.843 + 0.537i)T \)
37 \( 1 + (0.614 - 0.788i)T \)
41 \( 1 + (0.421 - 0.906i)T \)
43 \( 1 + (-0.521 - 0.853i)T \)
47 \( 1 + (0.672 - 0.739i)T \)
53 \( 1 + (0.351 - 0.936i)T \)
59 \( 1 + (-0.982 + 0.188i)T \)
61 \( 1 + (-0.914 + 0.404i)T \)
67 \( 1 + (0.862 + 0.505i)T \)
71 \( 1 + (0.280 + 0.959i)T \)
73 \( 1 + (-0.999 + 0.0378i)T \)
79 \( 1 + (-0.993 - 0.113i)T \)
83 \( 1 + (-0.700 + 0.713i)T \)
89 \( 1 + (0.997 + 0.0756i)T \)
97 \( 1 + (-0.843 - 0.537i)T \)
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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

−27.36998275113770243063893211587, −26.3909408886987411555032138721, −25.89090291962582292377286462650, −24.84120121380992857789281444392, −24.29691912641784803854390963161, −23.18562210925903862745115407799, −21.81104107648288545274478767469, −20.43308205448867370344480994648, −19.94781555527048029047688436460, −18.44313745043042734818037455590, −17.92952980793117694646458331138, −17.26718598431132364255423581326, −15.41590759253487467572664263765, −14.9349332495241258970532675080, −13.86867350279908304391595181178, −13.12519864421458480088051712045, −11.18036234103260304483250444874, −10.02461303757233956211638269383, −9.253466824657812121645756647180, −7.83173711878849218436353359771, −7.3997338554839482295678237665, −6.03242236855028558488250157306, −4.7189677930783657055536658190, −2.574976905929628596380734741463, −1.5643309183174977689808051125, 1.723833859097812884028029192839, 2.501914688457218608929283646676, 4.07966060846525411524660490030, 5.27571890087754701132962030261, 7.33446240083310502500784939739, 8.582889447626001099170833644121, 9.01988594680712704386286956940, 10.17919808117879518819124949403, 11.12858089738991146980140396847, 12.47708178310759845696248462393, 13.62856131414081089744817483893, 14.359612455920170613361240718095, 15.98362805756189383636587502604, 16.74309902316039253130777930857, 18.07549283089721550896693727415, 18.673992726992570688944559810455, 20.071975474406195388428873108987, 20.6414267125854964736498327484, 21.57747853487713717397875371626, 21.952691633665441360028173270917, 24.24236082652778243023902690575, 24.79755789297022648835836655803, 25.96732370487574800164628856105, 26.67600333803525456789619446552, 27.471139558693262460522649108956

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