Properties

Label 1-161-161.110-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: $\chi_{161} (110, \cdot )$
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

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

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