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Properties

Label 1-161-161.101-r1-0-0
Degree $1$
Conductor $161$
Sign $0.683 + 0.729i$
Analytic cond. $17.3018$
Root an. cond. $17.3018$
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  + (0.0475 + 0.998i)2-s + (−0.928 + 0.371i)3-s + (−0.995 + 0.0950i)4-s + (−0.235 + 0.971i)5-s + (−0.415 − 0.909i)6-s + (−0.142 − 0.989i)8-s + (0.723 − 0.690i)9-s + (−0.981 − 0.189i)10-s + (0.0475 − 0.998i)11-s + (0.888 − 0.458i)12-s + (0.654 − 0.755i)13-s + (−0.142 − 0.989i)15-s + (0.981 − 0.189i)16-s + (−0.580 + 0.814i)17-s + (0.723 + 0.690i)18-s + (−0.580 − 0.814i)19-s + ⋯
L(s)  = 1  + (0.0475 + 0.998i)2-s + (−0.928 + 0.371i)3-s + (−0.995 + 0.0950i)4-s + (−0.235 + 0.971i)5-s + (−0.415 − 0.909i)6-s + (−0.142 − 0.989i)8-s + (0.723 − 0.690i)9-s + (−0.981 − 0.189i)10-s + (0.0475 − 0.998i)11-s + (0.888 − 0.458i)12-s + (0.654 − 0.755i)13-s + (−0.142 − 0.989i)15-s + (0.981 − 0.189i)16-s + (−0.580 + 0.814i)17-s + (0.723 + 0.690i)18-s + (−0.580 − 0.814i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.683 + 0.729i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.683 + 0.729i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(161\)    =    \(7 \cdot 23\)
Sign: $0.683 + 0.729i$
Analytic conductor: \(17.3018\)
Root analytic conductor: \(17.3018\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{161} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 161,\ (1:\ ),\ 0.683 + 0.729i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8280182191 + 0.3587401910i\)
\(L(\frac12)\) \(\approx\) \(0.8280182191 + 0.3587401910i\)
\(L(1)\) \(\approx\) \(0.6116955632 + 0.3813146642i\)
\(L(1)\) \(\approx\) \(0.6116955632 + 0.3813146642i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (0.0475 + 0.998i)T \)
3 \( 1 + (-0.928 + 0.371i)T \)
5 \( 1 + (-0.235 + 0.971i)T \)
11 \( 1 + (0.0475 - 0.998i)T \)
13 \( 1 + (0.654 - 0.755i)T \)
17 \( 1 + (-0.580 + 0.814i)T \)
19 \( 1 + (-0.580 - 0.814i)T \)
29 \( 1 + (0.415 + 0.909i)T \)
31 \( 1 + (0.786 - 0.618i)T \)
37 \( 1 + (0.723 - 0.690i)T \)
41 \( 1 + (0.959 - 0.281i)T \)
43 \( 1 + (-0.142 + 0.989i)T \)
47 \( 1 + (0.5 - 0.866i)T \)
53 \( 1 + (-0.327 + 0.945i)T \)
59 \( 1 + (-0.981 - 0.189i)T \)
61 \( 1 + (-0.928 - 0.371i)T \)
67 \( 1 + (-0.888 - 0.458i)T \)
71 \( 1 + (0.841 - 0.540i)T \)
73 \( 1 + (0.995 - 0.0950i)T \)
79 \( 1 + (-0.327 - 0.945i)T \)
83 \( 1 + (0.959 + 0.281i)T \)
89 \( 1 + (0.786 + 0.618i)T \)
97 \( 1 + (0.959 - 0.281i)T \)
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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

−27.74669958664969011005421358017, −26.98589225594906885100217941607, −25.36933744663426186133022900056, −24.20051186047557060561745288304, −23.25542386663563229605256505496, −22.71470286874205588925976487707, −21.392137949807540985069906385347, −20.66035663484648420494714852100, −19.58227144655218345859675332269, −18.59501535205131594948464034638, −17.62932113763071114163602246343, −16.79007622000318402605187179231, −15.6271556807201795845849896935, −13.89178981945016931001843988932, −12.9330926783311365046984009646, −12.11662184929315711874521726830, −11.43190360234638729994199887461, −10.18379091577663108511188120258, −9.11041664943295687557259311798, −7.843103925274306974546431269174, −6.24017671989116527656271715624, −4.84020145058971203546816402978, −4.21039677174653023052967928921, −2.05901606366629317169637720268, −0.94071450954699615654481158473, 0.51729939015016439121861678531, 3.342166409309727438920517609733, 4.484185867298433687268781045641, 5.96292700526280549297113526772, 6.42969236607937876459352102806, 7.76227839993655625296374849848, 9.01823497620988622207086739610, 10.48892440208042345997696281808, 11.12337375315968980477220521595, 12.659548654305080079662652051653, 13.768906312014329543473760064483, 15.069566710317690329140541270699, 15.63580801349092159273575892725, 16.67774617056534224470873571019, 17.70644355505526138844928114773, 18.373579546318955075775611094201, 19.48188686504644606776018237242, 21.462388886649943918682116538387, 22.00641130560288305422338447284, 22.9845387043204904166208591143, 23.6218032051431047759073195436, 24.614399874809124429631267774998, 25.97933671264638450360259361974, 26.605016803204485994658066171214, 27.505497383770431260896431876952

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