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Properties

Label 1-161-161.12-r1-0-0
Degree $1$
Conductor $161$
Sign $-0.993 - 0.118i$
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  + (−0.995 + 0.0950i)2-s + (−0.723 + 0.690i)3-s + (0.981 − 0.189i)4-s + (0.888 + 0.458i)5-s + (0.654 − 0.755i)6-s + (−0.959 + 0.281i)8-s + (0.0475 − 0.998i)9-s + (−0.928 − 0.371i)10-s + (−0.995 − 0.0950i)11-s + (−0.580 + 0.814i)12-s + (0.142 + 0.989i)13-s + (−0.959 + 0.281i)15-s + (0.928 − 0.371i)16-s + (0.327 + 0.945i)17-s + (0.0475 + 0.998i)18-s + (0.327 − 0.945i)19-s + ⋯
L(s)  = 1  + (−0.995 + 0.0950i)2-s + (−0.723 + 0.690i)3-s + (0.981 − 0.189i)4-s + (0.888 + 0.458i)5-s + (0.654 − 0.755i)6-s + (−0.959 + 0.281i)8-s + (0.0475 − 0.998i)9-s + (−0.928 − 0.371i)10-s + (−0.995 − 0.0950i)11-s + (−0.580 + 0.814i)12-s + (0.142 + 0.989i)13-s + (−0.959 + 0.281i)15-s + (0.928 − 0.371i)16-s + (0.327 + 0.945i)17-s + (0.0475 + 0.998i)18-s + (0.327 − 0.945i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.993 - 0.118i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.993 - 0.118i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(161\)    =    \(7 \cdot 23\)
Sign: $-0.993 - 0.118i$
Analytic conductor: \(17.3018\)
Root analytic conductor: \(17.3018\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{161} (12, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 161,\ (1:\ ),\ -0.993 - 0.118i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02599978162 + 0.4388034136i\)
\(L(\frac12)\) \(\approx\) \(0.02599978162 + 0.4388034136i\)
\(L(1)\) \(\approx\) \(0.4893385911 + 0.2454255792i\)
\(L(1)\) \(\approx\) \(0.4893385911 + 0.2454255792i\)

Euler product

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

−27.29729989605566718156599543640, −25.94457717139005696085407134329, −25.07668571192700989920714124967, −24.49649363685223898793424865659, −23.31172440862327284301586436701, −22.12923579083620384255573708569, −20.85429762446621545288094319701, −20.22826382497348297254641291053, −18.5561194069359357667593339811, −18.33834810676294642191271319519, −17.20418046107774505386761150692, −16.56049090663263031263302565787, −15.418432507829197193171266966477, −13.63236124912938285126770826933, −12.725610762651313908481831970206, −11.72658526273431517863190049119, −10.48386826394495935505350621411, −9.77307349746795040855467152864, −8.25425887123672231771929859728, −7.42816033064797161321145741390, −6.04142764055135609115591388079, −5.28468061303295198055902091050, −2.72565770475354207974007881724, −1.50536233574974633617165979404, −0.24472014361701469563304748854, 1.632325004917914632718113556775, 3.17300035288292880432132642517, 5.1383894249787226499353840712, 6.16316886179082701726000737054, 7.13314258092810079968177159596, 8.79340042837360650938610257652, 9.719124848095787040200748806521, 10.616212952045430079338882456248, 11.27805206289801636234802122549, 12.71047090137597666539117540850, 14.32145973766752734417182787254, 15.40016030787568045721734989663, 16.34990854174079285286482427455, 17.220576328770626062469768781420, 18.05025744581719225352022860504, 18.82940050279854300182606302352, 20.27750446830683466696290530599, 21.39378940735928839970253530191, 21.760152318617598971628490109807, 23.371124436882320595843408451314, 24.15007485314082715522615977939, 25.61883327775861932869908472356, 26.25306158213407121896843952601, 26.91612200449243702279127211220, 28.30949012078627775211156593852

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