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Properties

Label 1-161-161.45-r0-0-0
Degree $1$
Conductor $161$
Sign $-0.991 - 0.126i$
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  + (−0.5 + 0.866i)2-s + (0.5 + 0.866i)3-s + (−0.5 − 0.866i)4-s + (−0.5 + 0.866i)5-s − 6-s + 8-s + (−0.5 + 0.866i)9-s + (−0.5 − 0.866i)10-s + (0.5 + 0.866i)11-s + (0.5 − 0.866i)12-s − 13-s − 15-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)17-s + (−0.5 − 0.866i)18-s + (−0.5 + 0.866i)19-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s + (0.5 + 0.866i)3-s + (−0.5 − 0.866i)4-s + (−0.5 + 0.866i)5-s − 6-s + 8-s + (−0.5 + 0.866i)9-s + (−0.5 − 0.866i)10-s + (0.5 + 0.866i)11-s + (0.5 − 0.866i)12-s − 13-s − 15-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)17-s + (−0.5 − 0.866i)18-s + (−0.5 + 0.866i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(161\)    =    \(7 \cdot 23\)
Sign: $-0.991 - 0.126i$
Analytic conductor: \(0.747680\)
Root analytic conductor: \(0.747680\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{161} (45, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 161,\ (0:\ ),\ -0.991 - 0.126i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.04542247058 + 0.7157632899i\)
\(L(\frac12)\) \(\approx\) \(-0.04542247058 + 0.7157632899i\)
\(L(1)\) \(\approx\) \(0.4617546513 + 0.6187678456i\)
\(L(1)\) \(\approx\) \(0.4617546513 + 0.6187678456i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 + (0.5 + 0.866i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 - T \)
17 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + T \)
31 \( 1 + (0.5 + 0.866i)T \)
37 \( 1 + (0.5 - 0.866i)T \)
41 \( 1 - T \)
43 \( 1 - T \)
47 \( 1 + (0.5 - 0.866i)T \)
53 \( 1 + (0.5 + 0.866i)T \)
59 \( 1 + (0.5 + 0.866i)T \)
61 \( 1 + (-0.5 + 0.866i)T \)
67 \( 1 + (0.5 + 0.866i)T \)
71 \( 1 + T \)
73 \( 1 + (0.5 + 0.866i)T \)
79 \( 1 + (0.5 - 0.866i)T \)
83 \( 1 + T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + T \)
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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

−27.30256341395450853297747123267, −26.53333907170824843983922446602, −25.38164032536579689759387495077, −24.36837179734555731560233639176, −23.630012665317095831592313507285, −22.170692798280411545155330204558, −21.18626652745940692613188063455, −19.96545665846494880214867310871, −19.59506571182092239188412068400, −18.76512387510280436184681191540, −17.408033186724678024420109628479, −16.84396355760022859031328509461, −15.24339212451528903317642556186, −13.78984541014864668098899769455, −12.94330000624208561385027690798, −12.09400346536416243535042821363, −11.25111255224735892840836780504, −9.6208279243168800091636128290, −8.59178083465108247437191609766, −8.02652227132241609526646131060, −6.62246523533513873904138453301, −4.67108836594652404748758014423, −3.38715741054036388004882276944, −2.05310983629723574917329625559, −0.66867848583289897339235641525, 2.388116219197055132880575397778, 4.03007251085049152339270058153, 5.02716759816093810381197819580, 6.66862673296244085660082894614, 7.5673961566142606464092198363, 8.70426876425784899285251645316, 9.86079933369515724964612416611, 10.48681382763370943744038305867, 11.918898323458078263501280108735, 13.83787971536080408754154299337, 14.67981132533403132158166423711, 15.24311977073062701032079821033, 16.22591697205725854039841921018, 17.27265497253102526269234708449, 18.363387751482306940871440732184, 19.51959930233448977012723238560, 20.073481742818981566762942898231, 21.69493494590054649577518045836, 22.63498047159129305660402828109, 23.28658198020394412032915685614, 24.8865053762547872936626375202, 25.39887467580525856283826923721, 26.61161296430989814989061508423, 27.00367183672485836885081637398, 27.78265999480777823577250113839

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