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Properties

Label 1-167-167.122-r0-0-0
Degree $1$
Conductor $167$
Sign $0.823 - 0.567i$
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  + (−0.965 − 0.261i)2-s + (0.988 + 0.150i)3-s + (0.862 + 0.505i)4-s + (−0.644 − 0.764i)5-s + (−0.914 − 0.404i)6-s + (−0.584 + 0.811i)7-s + (−0.700 − 0.713i)8-s + (0.954 + 0.298i)9-s + (0.421 + 0.906i)10-s + (0.726 − 0.686i)11-s + (0.776 + 0.629i)12-s + (0.132 − 0.991i)13-s + (0.776 − 0.629i)14-s + (−0.521 − 0.853i)15-s + (0.489 + 0.872i)16-s + (0.553 − 0.832i)17-s + ⋯
L(s)  = 1  + (−0.965 − 0.261i)2-s + (0.988 + 0.150i)3-s + (0.862 + 0.505i)4-s + (−0.644 − 0.764i)5-s + (−0.914 − 0.404i)6-s + (−0.584 + 0.811i)7-s + (−0.700 − 0.713i)8-s + (0.954 + 0.298i)9-s + (0.421 + 0.906i)10-s + (0.726 − 0.686i)11-s + (0.776 + 0.629i)12-s + (0.132 − 0.991i)13-s + (0.776 − 0.629i)14-s + (−0.521 − 0.853i)15-s + (0.489 + 0.872i)16-s + (0.553 − 0.832i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 167 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.823 - 0.567i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 167 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.823 - 0.567i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(167\)
Sign: $0.823 - 0.567i$
Analytic conductor: \(0.775544\)
Root analytic conductor: \(0.775544\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{167} (122, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 167,\ (0:\ ),\ 0.823 - 0.567i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8990999592 - 0.2798252214i\)
\(L(\frac12)\) \(\approx\) \(0.8990999592 - 0.2798252214i\)
\(L(1)\) \(\approx\) \(0.8849567154 - 0.1562815540i\)
\(L(1)\) \(\approx\) \(0.8849567154 - 0.1562815540i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad167 \( 1 \)
good2 \( 1 + (-0.965 - 0.261i)T \)
3 \( 1 + (0.988 + 0.150i)T \)
5 \( 1 + (-0.644 - 0.764i)T \)
7 \( 1 + (-0.584 + 0.811i)T \)
11 \( 1 + (0.726 - 0.686i)T \)
13 \( 1 + (0.132 - 0.991i)T \)
17 \( 1 + (0.553 - 0.832i)T \)
19 \( 1 + (0.974 + 0.225i)T \)
23 \( 1 + (0.206 + 0.978i)T \)
29 \( 1 + (0.206 - 0.978i)T \)
31 \( 1 + (-0.982 + 0.188i)T \)
37 \( 1 + (0.954 - 0.298i)T \)
41 \( 1 + (0.929 - 0.369i)T \)
43 \( 1 + (-0.942 - 0.334i)T \)
47 \( 1 + (-0.243 + 0.969i)T \)
53 \( 1 + (-0.800 - 0.599i)T \)
59 \( 1 + (0.553 + 0.832i)T \)
61 \( 1 + (0.614 + 0.788i)T \)
67 \( 1 + (-0.644 + 0.764i)T \)
71 \( 1 + (-0.0944 - 0.995i)T \)
73 \( 1 + (0.489 - 0.872i)T \)
79 \( 1 + (-0.999 - 0.0378i)T \)
83 \( 1 + (-0.965 + 0.261i)T \)
89 \( 1 + (-0.521 + 0.853i)T \)
97 \( 1 + (-0.982 - 0.188i)T \)
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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

−27.36160179736318694626984333558, −26.530869761568462773413714643728, −26.073330406002001832533591230478, −25.200394410812317094675310876029, −24.008907201909368263752925731410, −23.22597911580802970972113332061, −21.78390058717221640556018412927, −20.27019827708504961097109199282, −19.84867287217713459610060162022, −18.951550606307309739340798447005, −18.21430566312764670803201851126, −16.7843600191582060382137686256, −15.92952006152261060046050741944, −14.70988788621345222621073268031, −14.26747811649761729677393742452, −12.60642709581286548754244699791, −11.308903283913755232355636419517, −10.131421808713600819792003340629, −9.34735908196443255803812533296, −8.11800024086427829975968290228, −7.10017366389075007057929999872, −6.618288429152887873575123094422, −4.10803908776529219694607944612, −2.99815033299990704229639524294, −1.47880840736466630147620899361, 1.14941592935894783669816355201, 2.87455195340129937365409161280, 3.674209118374792514625672022294, 5.63951950877711326696029938831, 7.37445237790410790878357012681, 8.205523148649066537925085707, 9.16216570604719958847855284921, 9.73411301121248260263821646001, 11.379627179213245161101582536702, 12.314659317678841797268739522775, 13.33896441091978538865824448061, 14.93500382564276455173548281947, 15.91791994287465807565086731556, 16.38614506734044505422712086510, 17.95180098049227182354548701019, 19.032339800114206869244886648207, 19.620367033348891718927734197648, 20.44082443070230488681733972486, 21.31136081895677776916296350260, 22.46933853927526151562389910384, 24.192516955485598047432805727559, 25.09491335785396741348457324770, 25.410185315775594493511907652670, 26.93192998142683984965766708130, 27.31545171030452708191668579101

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