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Properties

Label 1-161-161.60-r1-0-0
Degree $1$
Conductor $161$
Sign $-0.993 - 0.109i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.993 - 0.109i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.993 - 0.109i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(161\)    =    \(7 \cdot 23\)
Sign: $-0.993 - 0.109i$
Analytic conductor: \(17.3018\)
Root analytic conductor: \(17.3018\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{161} (60, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 161,\ (1:\ ),\ -0.993 - 0.109i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.07439378387 + 1.356056573i\)
\(L(\frac12)\) \(\approx\) \(-0.07439378387 + 1.356056573i\)
\(L(1)\) \(\approx\) \(0.8169266064 + 0.7330581806i\)
\(L(1)\) \(\approx\) \(0.8169266064 + 0.7330581806i\)

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

−27.24293830509889000083688050611, −26.45451978075802750859933035344, −25.00424260032345742487833374157, −24.271107427830097337693102608461, −22.93808180817952337598600578000, −21.78477313572594856970134528198, −20.9918948029682395848966109339, −20.20267171969550899044945699460, −19.50598890533511122869662423768, −18.57277141768010038836582768414, −17.15110628701749285504144179663, −16.04123637528285437605462972134, −14.830249743212414814071617887154, −13.68371869026063138304174135712, −12.97491440511190914697684272746, −11.86688391377618388650594000736, −10.59029857162849968837448822587, −9.53477193688878537280766323377, −8.64452915530648648700755935269, −7.848035620100856830245505926944, −5.44798260855053201237751228004, −4.37529505320567802022036603965, −3.3167461336816494399357108096, −2.046411787262328514152302100897, −0.42544490738729164540450027814, 2.06446199097644397603781773457, 3.55737681764909693499016512988, 4.7117645666971287916633946122, 6.72935463657086125907419326160, 6.99616438443519277810647392012, 8.24080973505929185068064797321, 9.286644398327826318291516566365, 10.39586992366052977968294307995, 12.21358340271074168785316471081, 13.25794102238680136900317266986, 14.37123901882630611707852376115, 14.92761953212542031801800595774, 15.72309035736316801150880085886, 17.30107258121378361085168833974, 18.12541424454009817912418951007, 19.100153755758249249843255954642, 19.86928403983331138672804942510, 21.492677117404005059012584068918, 22.29542401820065933485053882645, 23.57791907751732686028876329876, 24.0993572324038661713981482848, 25.46511070645497540832357740607, 25.892853453154732801274840673706, 26.70626978570675165486561489290, 27.59515688976688073908077779874

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