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Properties

Label 1-167-167.16-r0-0-0
Degree $1$
Conductor $167$
Sign $0.236 - 0.971i$
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  + (−0.942 − 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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(167\)
Sign: $0.236 - 0.971i$
Analytic conductor: \(0.775544\)
Root analytic conductor: \(0.775544\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{167} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 167,\ (0:\ ),\ 0.236 - 0.971i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5586681606 - 0.4390256849i\)
\(L(\frac12)\) \(\approx\) \(0.5586681606 - 0.4390256849i\)
\(L(1)\) \(\approx\) \(0.6406908223 - 0.2831982878i\)
\(L(1)\) \(\approx\) \(0.6406908223 - 0.2831982878i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad167 \( 1 \)
good2 \( 1 + (-0.942 - 0.334i)T \)
3 \( 1 + (-0.800 - 0.599i)T \)
5 \( 1 + (0.974 - 0.225i)T \)
7 \( 1 + (-0.0944 - 0.995i)T \)
11 \( 1 + (0.997 - 0.0756i)T \)
13 \( 1 + (-0.169 + 0.985i)T \)
17 \( 1 + (0.862 + 0.505i)T \)
19 \( 1 + (0.822 - 0.569i)T \)
23 \( 1 + (-0.881 + 0.472i)T \)
29 \( 1 + (-0.881 - 0.472i)T \)
31 \( 1 + (-0.0189 - 0.999i)T \)
37 \( 1 + (0.280 - 0.959i)T \)
41 \( 1 + (-0.999 + 0.0378i)T \)
43 \( 1 + (0.614 - 0.788i)T \)
47 \( 1 + (0.132 + 0.991i)T \)
53 \( 1 + (0.929 + 0.369i)T \)
59 \( 1 + (0.862 - 0.505i)T \)
61 \( 1 + (-0.752 - 0.658i)T \)
67 \( 1 + (0.974 + 0.225i)T \)
71 \( 1 + (-0.700 + 0.713i)T \)
73 \( 1 + (0.206 - 0.978i)T \)
79 \( 1 + (-0.584 - 0.811i)T \)
83 \( 1 + (-0.942 + 0.334i)T \)
89 \( 1 + (-0.914 + 0.404i)T \)
97 \( 1 + (-0.0189 + 0.999i)T \)
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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

−27.74643724277749443753112231478, −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