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Properties

Label 1-161-161.66-r0-0-0
Degree $1$
Conductor $161$
Sign $-0.842 - 0.538i$
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  + (0.235 − 0.971i)2-s + (0.327 − 0.945i)3-s + (−0.888 − 0.458i)4-s + (0.928 + 0.371i)5-s + (−0.841 − 0.540i)6-s + (−0.654 + 0.755i)8-s + (−0.786 − 0.618i)9-s + (0.580 − 0.814i)10-s + (−0.235 − 0.971i)11-s + (−0.723 + 0.690i)12-s + (−0.415 − 0.909i)13-s + (0.654 − 0.755i)15-s + (0.580 + 0.814i)16-s + (0.0475 − 0.998i)17-s + (−0.786 + 0.618i)18-s + (0.0475 + 0.998i)19-s + ⋯
L(s)  = 1  + (0.235 − 0.971i)2-s + (0.327 − 0.945i)3-s + (−0.888 − 0.458i)4-s + (0.928 + 0.371i)5-s + (−0.841 − 0.540i)6-s + (−0.654 + 0.755i)8-s + (−0.786 − 0.618i)9-s + (0.580 − 0.814i)10-s + (−0.235 − 0.971i)11-s + (−0.723 + 0.690i)12-s + (−0.415 − 0.909i)13-s + (0.654 − 0.755i)15-s + (0.580 + 0.814i)16-s + (0.0475 − 0.998i)17-s + (−0.786 + 0.618i)18-s + (0.0475 + 0.998i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.842 - 0.538i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.842 - 0.538i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(161\)    =    \(7 \cdot 23\)
Sign: $-0.842 - 0.538i$
Analytic conductor: \(0.747680\)
Root analytic conductor: \(0.747680\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{161} (66, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 161,\ (0:\ ),\ -0.842 - 0.538i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3747940474 - 1.281986746i\)
\(L(\frac12)\) \(\approx\) \(0.3747940474 - 1.281986746i\)
\(L(1)\) \(\approx\) \(0.8190787655 - 0.9527034573i\)
\(L(1)\) \(\approx\) \(0.8190787655 - 0.9527034573i\)

Euler product

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

−28.08862487703530957290172079321, −26.93658515960763890787720265840, −25.93395232516861181674583614951, −25.59398390521308100564279746426, −24.42897394663961579789419812358, −23.42717686525508856393559860103, −22.146755753438099652083703265430, −21.60325598977623166838930458660, −20.69196130800234875886076782671, −19.39011431104283341043451170861, −17.80380295666280598295664562104, −17.13676850048837151280224529765, −16.21943609826121385228789549152, −15.189556674872761485458001853230, −14.36452501468146369843208917587, −13.45216299380925216388336199109, −12.32490784112192575497315717836, −10.48680682458984363099307573636, −9.42934223317054138575489040336, −8.826833791227891157974771460804, −7.38422607586281921299614203914, −6.03224304646265269849476402326, −4.938025043709756979235551949885, −4.132432646740369034508269854361, −2.35994995100766781706971209906, 1.09146675534962470458168084428, 2.491141608440553421400084503088, 3.24671051108692272513177966917, 5.3066850872453659895009317361, 6.198356201123890676823022916900, 7.79276519511243850742240030711, 8.99764452663650970039857839173, 10.08711502642804681236252496517, 11.180643268973680497153114869949, 12.348908625488585953832719028364, 13.27439210580964968179417099456, 13.98909245124977831014253861440, 14.815023038737077349284069977471, 16.80436069806911808160023751713, 18.188074986273333438998934990326, 18.33171839628106192021359584230, 19.58185825347650816339827728465, 20.4764864682420467501718236551, 21.44392733930379478991703485806, 22.44899532296181161163580706687, 23.33382673607778138398653773040, 24.52701501395852721785668404177, 25.30282238892801300742423597287, 26.52916094604086563820935241911, 27.4457492526309192670344838459

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