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Properties

Label 1-161-161.108-r1-0-0
Degree $1$
Conductor $161$
Sign $0.440 + 0.897i$
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  + (0.928 + 0.371i)2-s + (0.995 − 0.0950i)3-s + (0.723 + 0.690i)4-s + (0.327 + 0.945i)5-s + (0.959 + 0.281i)6-s + (0.415 + 0.909i)8-s + (0.981 − 0.189i)9-s + (−0.0475 + 0.998i)10-s + (0.928 − 0.371i)11-s + (0.786 + 0.618i)12-s + (−0.841 − 0.540i)13-s + (0.415 + 0.909i)15-s + (0.0475 + 0.998i)16-s + (−0.235 − 0.971i)17-s + (0.981 + 0.189i)18-s + (−0.235 + 0.971i)19-s + ⋯
L(s)  = 1  + (0.928 + 0.371i)2-s + (0.995 − 0.0950i)3-s + (0.723 + 0.690i)4-s + (0.327 + 0.945i)5-s + (0.959 + 0.281i)6-s + (0.415 + 0.909i)8-s + (0.981 − 0.189i)9-s + (−0.0475 + 0.998i)10-s + (0.928 − 0.371i)11-s + (0.786 + 0.618i)12-s + (−0.841 − 0.540i)13-s + (0.415 + 0.909i)15-s + (0.0475 + 0.998i)16-s + (−0.235 − 0.971i)17-s + (0.981 + 0.189i)18-s + (−0.235 + 0.971i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.440 + 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.440 + 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(161\)    =    \(7 \cdot 23\)
Sign: $0.440 + 0.897i$
Analytic conductor: \(17.3018\)
Root analytic conductor: \(17.3018\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{161} (108, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 161,\ (1:\ ),\ 0.440 + 0.897i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.156056741 + 2.589950282i\)
\(L(\frac12)\) \(\approx\) \(4.156056741 + 2.589950282i\)
\(L(1)\) \(\approx\) \(2.520160422 + 0.9725819726i\)
\(L(1)\) \(\approx\) \(2.520160422 + 0.9725819726i\)

Euler product

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

−27.618886941407147798799364568122, −26.15969524252864385712888265707, −25.18009680785718395411292003897, −24.38603029829704159697818657177, −23.78007424412893886765479296399, −22.07427543128486182694793852918, −21.5885553317112559954439723240, −20.44792286090678003175462344895, −19.85785446907671612546488506153, −19.06683122480636499929618720516, −17.28247637259695095017837225436, −16.18717615928486486891608629572, −14.97216809707180658244764284500, −14.351435053734013575696985413773, −13.13515002084994002444060422916, −12.61521340941293415440603344359, −11.28644271736477364558106280904, −9.735545692432612197303578218613, −9.114575497546624071415021335611, −7.55841733416937112678641684756, −6.25591691318562584753793623890, −4.694701011967161733709639186989, −4.02740171308669419998963974599, −2.4269378513393974532969801401, −1.419298516419956846712287494088, 2.04358437821342322316046683444, 3.08240329202989892183507098370, 4.05776947197782622850419111102, 5.68991721721212571960082784118, 6.938938438312092787670521306920, 7.64705112127102041281615668987, 9.095161101295159629682340379, 10.38481662864961408981363867267, 11.71741686358654180693897179433, 12.87489094485002418861691462846, 13.99727436577887815750121571074, 14.50164845509284017052778138569, 15.30652070205579918722755586435, 16.5504964016649493192704368009, 17.82400117429048769847873738126, 19.04309612804813945585476815677, 20.028489826367198737780840085399, 21.003060866993008711005004323995, 22.04385966876620299446244484564, 22.62719813241753665859129257887, 23.98228201896467628694873999515, 25.080348804021384795041962566777, 25.32867603205780562682268231115, 26.65107557972610445720241474764, 27.17285591674477528693090787493

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