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Properties

Label 1-161-161.125-r0-0-0
Degree $1$
Conductor $161$
Sign $-0.993 + 0.117i$
Analytic cond. $0.747680$
Root an. cond. $0.747680$
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  + (−0.142 + 0.989i)2-s + (−0.415 − 0.909i)3-s + (−0.959 − 0.281i)4-s + (−0.654 − 0.755i)5-s + (0.959 − 0.281i)6-s + (0.415 − 0.909i)8-s + (−0.654 + 0.755i)9-s + (0.841 − 0.540i)10-s + (0.142 + 0.989i)11-s + (0.142 + 0.989i)12-s + (−0.841 + 0.540i)13-s + (−0.415 + 0.909i)15-s + (0.841 + 0.540i)16-s + (−0.959 + 0.281i)17-s + (−0.654 − 0.755i)18-s + (−0.959 − 0.281i)19-s + ⋯
L(s)  = 1  + (−0.142 + 0.989i)2-s + (−0.415 − 0.909i)3-s + (−0.959 − 0.281i)4-s + (−0.654 − 0.755i)5-s + (0.959 − 0.281i)6-s + (0.415 − 0.909i)8-s + (−0.654 + 0.755i)9-s + (0.841 − 0.540i)10-s + (0.142 + 0.989i)11-s + (0.142 + 0.989i)12-s + (−0.841 + 0.540i)13-s + (−0.415 + 0.909i)15-s + (0.841 + 0.540i)16-s + (−0.959 + 0.281i)17-s + (−0.654 − 0.755i)18-s + (−0.959 − 0.281i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.993 + 0.117i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.993 + 0.117i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(161\)    =    \(7 \cdot 23\)
Sign: $-0.993 + 0.117i$
Analytic conductor: \(0.747680\)
Root analytic conductor: \(0.747680\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{161} (125, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 161,\ (0:\ ),\ -0.993 + 0.117i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.006486596197 + 0.1100018344i\)
\(L(\frac12)\) \(\approx\) \(0.006486596197 + 0.1100018344i\)
\(L(1)\) \(\approx\) \(0.4795664676 + 0.09571239176i\)
\(L(1)\) \(\approx\) \(0.4795664676 + 0.09571239176i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (-0.142 + 0.989i)T \)
3 \( 1 + (-0.415 - 0.909i)T \)
5 \( 1 + (-0.654 - 0.755i)T \)
11 \( 1 + (0.142 + 0.989i)T \)
13 \( 1 + (-0.841 + 0.540i)T \)
17 \( 1 + (-0.959 + 0.281i)T \)
19 \( 1 + (-0.959 - 0.281i)T \)
29 \( 1 + (-0.959 + 0.281i)T \)
31 \( 1 + (-0.415 + 0.909i)T \)
37 \( 1 + (0.654 - 0.755i)T \)
41 \( 1 + (0.654 + 0.755i)T \)
43 \( 1 + (-0.415 - 0.909i)T \)
47 \( 1 - T \)
53 \( 1 + (-0.841 - 0.540i)T \)
59 \( 1 + (-0.841 + 0.540i)T \)
61 \( 1 + (0.415 - 0.909i)T \)
67 \( 1 + (0.142 - 0.989i)T \)
71 \( 1 + (-0.142 + 0.989i)T \)
73 \( 1 + (0.959 + 0.281i)T \)
79 \( 1 + (-0.841 + 0.540i)T \)
83 \( 1 + (-0.654 + 0.755i)T \)
89 \( 1 + (0.415 + 0.909i)T \)
97 \( 1 + (-0.654 - 0.755i)T \)
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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

−27.30464655283692036750236269921, −26.80302971241013852121719029522, −25.95010702738754136641908609425, −24.108398918069377767544200209290, −22.95011754509293759249566291287, −22.25300187181666153213782947285, −21.62671822897309131933157142943, −20.45216013549772600411739535816, −19.56056628045960991296124679021, −18.61812820111568853372192040055, −17.49970914012310304697758137183, −16.527360363922787516012794329664, −15.20992013225295886304710823358, −14.39193825435354688871248544186, −12.95311128160485972771400685002, −11.621117978094311363485384007275, −11.08171492150444909946407302895, −10.18741829947209059765251202827, −9.07371530131618891219739456644, −7.878996544026428347422494519868, −6.112239714128754202554582258166, −4.65405688638775595061284481191, −3.672462332866434666542270406812, −2.62435548932904998943152550837, −0.09934223928819960012113600107, 1.78798758659667125592421039866, 4.291804222111550917470659923378, 5.15822117226317780602991855191, 6.60892825008611585137773148810, 7.36124191999752510348384235760, 8.39725726810566432677762943918, 9.429827232435993012860762724571, 11.11930140691644797917462486490, 12.50864278818727891529000962962, 12.963670427450400178766845507002, 14.38215590589180317616325800744, 15.36574143409852549732263878007, 16.58065215811011636205136047486, 17.23571771170684953555521507640, 18.136948409614652071963731636305, 19.34923287035793942047678244138, 19.94699741815344330975538098080, 21.78975756441726006136491757945, 22.88901006896955417976773397444, 23.63084987354221026601067157988, 24.342900940098948665132119926634, 25.08108706771879612691420987906, 26.13634393941474043628291990363, 27.338857567989709768295959232906, 28.196059267761122126133470946392

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