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Properties

Label 1-167-167.29-r0-0-0
Degree $1$
Conductor $167$
Sign $0.818 + 0.574i$
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  + (0.614 + 0.788i)2-s + (0.929 − 0.369i)3-s + (−0.243 + 0.969i)4-s + (0.822 − 0.569i)5-s + (0.862 + 0.505i)6-s + (−0.700 − 0.713i)7-s + (−0.914 + 0.404i)8-s + (0.726 − 0.686i)9-s + (0.954 + 0.298i)10-s + (−0.316 + 0.948i)11-s + (0.132 + 0.991i)12-s + (0.898 + 0.438i)13-s + (0.132 − 0.991i)14-s + (0.553 − 0.832i)15-s + (−0.881 − 0.472i)16-s + (0.776 − 0.629i)17-s + ⋯
L(s)  = 1  + (0.614 + 0.788i)2-s + (0.929 − 0.369i)3-s + (−0.243 + 0.969i)4-s + (0.822 − 0.569i)5-s + (0.862 + 0.505i)6-s + (−0.700 − 0.713i)7-s + (−0.914 + 0.404i)8-s + (0.726 − 0.686i)9-s + (0.954 + 0.298i)10-s + (−0.316 + 0.948i)11-s + (0.132 + 0.991i)12-s + (0.898 + 0.438i)13-s + (0.132 − 0.991i)14-s + (0.553 − 0.832i)15-s + (−0.881 − 0.472i)16-s + (0.776 − 0.629i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 167 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.818 + 0.574i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 167 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.818 + 0.574i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(167\)
Sign: $0.818 + 0.574i$
Analytic conductor: \(0.775544\)
Root analytic conductor: \(0.775544\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{167} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 167,\ (0:\ ),\ 0.818 + 0.574i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.960302251 + 0.6198644832i\)
\(L(\frac12)\) \(\approx\) \(1.960302251 + 0.6198644832i\)
\(L(1)\) \(\approx\) \(1.760029446 + 0.4555507778i\)
\(L(1)\) \(\approx\) \(1.760029446 + 0.4555507778i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad167 \( 1 \)
good2 \( 1 + (0.614 + 0.788i)T \)
3 \( 1 + (0.929 - 0.369i)T \)
5 \( 1 + (0.822 - 0.569i)T \)
7 \( 1 + (-0.700 - 0.713i)T \)
11 \( 1 + (-0.316 + 0.948i)T \)
13 \( 1 + (0.898 + 0.438i)T \)
17 \( 1 + (0.776 - 0.629i)T \)
19 \( 1 + (-0.843 + 0.537i)T \)
23 \( 1 + (-0.965 + 0.261i)T \)
29 \( 1 + (-0.965 - 0.261i)T \)
31 \( 1 + (-0.455 + 0.890i)T \)
37 \( 1 + (0.726 + 0.686i)T \)
41 \( 1 + (-0.584 - 0.811i)T \)
43 \( 1 + (-0.752 - 0.658i)T \)
47 \( 1 + (-0.169 + 0.985i)T \)
53 \( 1 + (-0.999 + 0.0378i)T \)
59 \( 1 + (0.776 + 0.629i)T \)
61 \( 1 + (-0.644 - 0.764i)T \)
67 \( 1 + (0.822 + 0.569i)T \)
71 \( 1 + (-0.521 - 0.853i)T \)
73 \( 1 + (-0.881 + 0.472i)T \)
79 \( 1 + (-0.0944 - 0.995i)T \)
83 \( 1 + (0.614 - 0.788i)T \)
89 \( 1 + (0.553 + 0.832i)T \)
97 \( 1 + (-0.455 - 0.890i)T \)
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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

−27.79909268778780140758532825624, −26.40465484051420256385454387704, −25.6830613617109039939088558723, −24.76068853255620341094534283978, −23.53096510426803226474054515285, −22.2031268461252639120706967945, −21.672419031726608481118852914, −20.96085901426347841614067632450, −19.83206542601065856062027879649, −18.80682276417610236917535635138, −18.35650454066925590404674990093, −16.37524393592336489712313210621, −15.213531455127020216134077906758, −14.50084961438538423578476841337, −13.34479599183190969266724171234, −12.930893642698672777001879842071, −11.17203995149749356225655789178, −10.26254894012350876389727727449, −9.407959277028387429969572811997, −8.3480219643825131642480248493, −6.31732173202605427390363481240, −5.52720329025826051260301587467, −3.741971530500535871800134801714, −2.95436994827860447979022792567, −1.92182057638445547624167192397, 1.846247571159574702704480937453, 3.38462827197969992755955153156, 4.45354279535256863403374010363, 5.97209153366150068558678490813, 6.96692324471509997677119783375, 8.00534687260139086429714368575, 9.15218145707289375604976959209, 10.05173690340333170387073578645, 12.24649116949366899504297078794, 13.05486703934869206584172021134, 13.732888668062159047935917937937, 14.54939937088068420886462852852, 15.82082912920270335372782838641, 16.670121670947954957631169186392, 17.77012397336792176629914011853, 18.78035097585204868338865792900, 20.42260387093084389724465995157, 20.7158302842470144367814016889, 21.92217629172690781514187239994, 23.27087371743963529149135997082, 23.81887643977305734727951124644, 25.109633483876124784328580640365, 25.65338174064474318735513626571, 26.11942028083633616679253803144, 27.47876703709680924450497505883

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