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Properties

Label 1-167-167.3-r0-0-0
Degree $1$
Conductor $167$
Sign $0.759 - 0.651i$
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  + (−0.584 − 0.811i)2-s + (0.132 + 0.991i)3-s + (−0.316 + 0.948i)4-s + (−0.914 − 0.404i)5-s + (0.726 − 0.686i)6-s + (0.421 − 0.906i)7-s + (0.954 − 0.298i)8-s + (−0.965 + 0.261i)9-s + (0.206 + 0.978i)10-s + (0.614 − 0.788i)11-s + (−0.982 − 0.188i)12-s + (−0.455 + 0.890i)13-s + (−0.982 + 0.188i)14-s + (0.280 − 0.959i)15-s + (−0.800 − 0.599i)16-s + (0.997 + 0.0756i)17-s + ⋯
L(s)  = 1  + (−0.584 − 0.811i)2-s + (0.132 + 0.991i)3-s + (−0.316 + 0.948i)4-s + (−0.914 − 0.404i)5-s + (0.726 − 0.686i)6-s + (0.421 − 0.906i)7-s + (0.954 − 0.298i)8-s + (−0.965 + 0.261i)9-s + (0.206 + 0.978i)10-s + (0.614 − 0.788i)11-s + (−0.982 − 0.188i)12-s + (−0.455 + 0.890i)13-s + (−0.982 + 0.188i)14-s + (0.280 − 0.959i)15-s + (−0.800 − 0.599i)16-s + (0.997 + 0.0756i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 167 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.759 - 0.651i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 167 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.759 - 0.651i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(167\)
Sign: $0.759 - 0.651i$
Analytic conductor: \(0.775544\)
Root analytic conductor: \(0.775544\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{167} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 167,\ (0:\ ),\ 0.759 - 0.651i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7136704177 - 0.2641260511i\)
\(L(\frac12)\) \(\approx\) \(0.7136704177 - 0.2641260511i\)
\(L(1)\) \(\approx\) \(0.7455300789 - 0.1588288351i\)
\(L(1)\) \(\approx\) \(0.7455300789 - 0.1588288351i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad167 \( 1 \)
good2 \( 1 + (-0.584 - 0.811i)T \)
3 \( 1 + (0.132 + 0.991i)T \)
5 \( 1 + (-0.914 - 0.404i)T \)
7 \( 1 + (0.421 - 0.906i)T \)
11 \( 1 + (0.614 - 0.788i)T \)
13 \( 1 + (-0.455 + 0.890i)T \)
17 \( 1 + (0.997 + 0.0756i)T \)
19 \( 1 + (0.553 - 0.832i)T \)
23 \( 1 + (0.929 + 0.369i)T \)
29 \( 1 + (0.929 - 0.369i)T \)
31 \( 1 + (0.974 - 0.225i)T \)
37 \( 1 + (-0.965 - 0.261i)T \)
41 \( 1 + (0.898 - 0.438i)T \)
43 \( 1 + (-0.0944 + 0.995i)T \)
47 \( 1 + (-0.0189 - 0.999i)T \)
53 \( 1 + (-0.169 - 0.985i)T \)
59 \( 1 + (0.997 - 0.0756i)T \)
61 \( 1 + (-0.700 + 0.713i)T \)
67 \( 1 + (-0.914 + 0.404i)T \)
71 \( 1 + (-0.993 + 0.113i)T \)
73 \( 1 + (-0.800 + 0.599i)T \)
79 \( 1 + (0.351 - 0.936i)T \)
83 \( 1 + (-0.584 + 0.811i)T \)
89 \( 1 + (0.280 + 0.959i)T \)
97 \( 1 + (0.974 + 0.225i)T \)
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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

−27.61734467833834147303468494649, −26.83468590118206632629744961493, −25.433571538435451817731435341269, −25.02250260366581463209718899832, −24.129164772716199269182669520892, −23.00888001050226780390330006556, −22.58201054861341618700308716368, −20.53202656262119154470844116906, −19.474047119876362699508216083749, −18.82153426194690811923945310160, −17.98861862735128564006544107771, −17.13562610597517001239652165240, −15.73956520931230523278517462986, −14.796617221629203303742443220936, −14.24973233788621583547008552050, −12.48105646526222026554759678662, −11.81437024764081895297106415293, −10.35793513321680659603958320179, −8.91977117216429164599896128637, −7.94908726819614464738681771852, −7.3038313572444680278525770589, −6.15040131139466080355946194027, −4.93060274696129303051921652522, −2.91277341732541720484264776360, −1.26047613044035787036058610049, 0.96274072221063270482192342069, 3.09386819944482040350611983684, 4.03321519293055913965729939452, 4.87625255445000127095832422807, 7.208904675402092340421986466580, 8.32370661077290110268056662683, 9.18631978276550876101927249441, 10.275930128922257977714878680048, 11.37538632488789721076842865645, 11.83091487772370969292192375851, 13.52708251891048633500882089487, 14.51343053732771911799492234176, 16.01013182333326825350559335439, 16.706397428097495722576719369057, 17.46047487342959465889727508040, 19.384765184126167105733106613351, 19.48248079190732010680415853074, 20.8001832866225720715842630969, 21.28146037134071730620378400470, 22.48302440001228775192343251204, 23.43979558665168714557976004515, 24.7242615714504364830520487324, 26.26163968480659266677299555105, 26.77610443267112408784005039845, 27.47018901034760768458289161069

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