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Properties

Label 1-161-161.61-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: $\chi_{161} (61, \cdot )$
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

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

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