Properties

Degree 1
Conductor 167
Sign $0.236 - 0.971i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.942 − 0.334i)2-s + (−0.800 − 0.599i)3-s + (0.776 + 0.629i)4-s + (0.974 − 0.225i)5-s + (0.553 + 0.832i)6-s + (−0.0944 − 0.995i)7-s + (−0.521 − 0.853i)8-s + (0.280 + 0.959i)9-s + (−0.993 − 0.113i)10-s + (0.997 − 0.0756i)11-s + (−0.243 − 0.969i)12-s + (−0.169 + 0.985i)13-s + (−0.243 + 0.969i)14-s + (−0.914 − 0.404i)15-s + (0.206 + 0.978i)16-s + (0.862 + 0.505i)17-s + ⋯
L(s,χ)  = 1  + (−0.942 − 0.334i)2-s + (−0.800 − 0.599i)3-s + (0.776 + 0.629i)4-s + (0.974 − 0.225i)5-s + (0.553 + 0.832i)6-s + (−0.0944 − 0.995i)7-s + (−0.521 − 0.853i)8-s + (0.280 + 0.959i)9-s + (−0.993 − 0.113i)10-s + (0.997 − 0.0756i)11-s + (−0.243 − 0.969i)12-s + (−0.169 + 0.985i)13-s + (−0.243 + 0.969i)14-s + (−0.914 − 0.404i)15-s + (0.206 + 0.978i)16-s + (0.862 + 0.505i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 167 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.236 - 0.971i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 167 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.236 - 0.971i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(167\)
\( \varepsilon \)  =  $0.236 - 0.971i$
motivic weight  =  \(0\)
character  :  $\chi_{167} (16, \cdot )$
Sato-Tate  :  $\mu(83)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 167,\ (0:\ ),\ 0.236 - 0.971i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.5586681606 - 0.4390256849i$
$L(\frac12,\chi)$  $\approx$  $0.5586681606 - 0.4390256849i$
$L(\chi,1)$  $\approx$  0.6406908223 - 0.2831982878i
$L(1,\chi)$  $\approx$  0.6406908223 - 0.2831982878i

Euler product

\[\begin{aligned} L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]
\[\begin{aligned} L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−27.74643724277749443753112231478, −27.13259276936798120323410476646, −25.87580492336015467454554834310, −25.10390340673924401867420500583, −24.35981289244904421908467005454, −22.78423262386869929762958016346, −22.09801659032776687581257683287, −21.035634911442621212984195157068, −20.05158126872193780534056792818, −18.44309298343062936532879422139, −18.08029752866406981503471117109, −16.99573792961610064257200473495, −16.26571801880831777552726461432, −15.12335339046169063512916618674, −14.32605900771308435568692192177, −12.36704887341332353192517935325, −11.52848238715998108948636395239, −10.1645998344353210688395820995, −9.70911847879779430130258398276, −8.626830056651061159752565845401, −6.96646382147099744877986474736, −5.85904141004363866717202896279, −5.33782855117194939143906120880, −3.062931840980177358603610110567, −1.40624952961623164308400335286, 1.04896877846010665491245863816, 2.00067357799700248105406624286, 3.99075250721802736449935087904, 5.81200745115553833896919527183, 6.7886708661633070824929550314, 7.68889751159879806629349711639, 9.28528375992090600515671367331, 10.06255227420269703559171976630, 11.20292626023495523529047598276, 12.06622904943761883914810336528, 13.2366340664140975771951857573, 14.18807944113166560504327307556, 16.21440031950630339218552146569, 17.01278317277391449605021419810, 17.368794392906472269793852373177, 18.517399476405514453733574293784, 19.430459101819196350082986669287, 20.430045917811828145404987128565, 21.60288527785794132765762626207, 22.366395854403708699646758810134, 23.882050620246782550163830269658, 24.54446873817957010289584589965, 25.64132622299281402677482694480, 26.461788969179455638862178191962, 27.6908672603952057845647268022

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