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Properties

Label 1-165-165.104-r1-0-0
Degree $1$
Conductor $165$
Sign $-0.935 + 0.352i$
Analytic cond. $17.7317$
Root an. cond. $17.7317$
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  + (−0.809 + 0.587i)2-s + (0.309 − 0.951i)4-s + (−0.309 + 0.951i)7-s + (0.309 + 0.951i)8-s + (0.809 − 0.587i)13-s + (−0.309 − 0.951i)14-s + (−0.809 − 0.587i)16-s + (−0.809 − 0.587i)17-s + (0.309 + 0.951i)19-s + 23-s + (−0.309 + 0.951i)26-s + (0.809 + 0.587i)28-s + (−0.309 + 0.951i)29-s + (−0.809 + 0.587i)31-s + 32-s + ⋯
L(s)  = 1  + (−0.809 + 0.587i)2-s + (0.309 − 0.951i)4-s + (−0.309 + 0.951i)7-s + (0.309 + 0.951i)8-s + (0.809 − 0.587i)13-s + (−0.309 − 0.951i)14-s + (−0.809 − 0.587i)16-s + (−0.809 − 0.587i)17-s + (0.309 + 0.951i)19-s + 23-s + (−0.309 + 0.951i)26-s + (0.809 + 0.587i)28-s + (−0.309 + 0.951i)29-s + (−0.809 + 0.587i)31-s + 32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.935 + 0.352i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.935 + 0.352i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(165\)    =    \(3 \cdot 5 \cdot 11\)
Sign: $-0.935 + 0.352i$
Analytic conductor: \(17.7317\)
Root analytic conductor: \(17.7317\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{165} (104, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 165,\ (1:\ ),\ -0.935 + 0.352i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1141487527 + 0.6274404439i\)
\(L(\frac12)\) \(\approx\) \(0.1141487527 + 0.6274404439i\)
\(L(1)\) \(\approx\) \(0.5882515645 + 0.2756882404i\)
\(L(1)\) \(\approx\) \(0.5882515645 + 0.2756882404i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.809 + 0.587i)T \)
7 \( 1 + (-0.309 + 0.951i)T \)
13 \( 1 + (0.809 - 0.587i)T \)
17 \( 1 + (-0.809 - 0.587i)T \)
19 \( 1 + (0.309 + 0.951i)T \)
23 \( 1 + T \)
29 \( 1 + (-0.309 + 0.951i)T \)
31 \( 1 + (-0.809 + 0.587i)T \)
37 \( 1 + (-0.309 + 0.951i)T \)
41 \( 1 + (-0.309 - 0.951i)T \)
43 \( 1 - T \)
47 \( 1 + (0.309 + 0.951i)T \)
53 \( 1 + (-0.809 + 0.587i)T \)
59 \( 1 + (-0.309 + 0.951i)T \)
61 \( 1 + (-0.809 - 0.587i)T \)
67 \( 1 - T \)
71 \( 1 + (0.809 + 0.587i)T \)
73 \( 1 + (-0.309 + 0.951i)T \)
79 \( 1 + (-0.809 + 0.587i)T \)
83 \( 1 + (-0.809 - 0.587i)T \)
89 \( 1 - T \)
97 \( 1 + (0.809 - 0.587i)T \)
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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

−26.81448192847598631600247546395, −26.385175364803681098718605962553, −25.42254289565428350588451940316, −24.17869839814882233894033155095, −23.06868865519878345711574761540, −21.93667446678877078060218921174, −20.95160198368179826807483363567, −20.03631856427200197728581625519, −19.27644357792080134171292484157, −18.199673677124974653074153418071, −17.19485860413190914624050925084, −16.43040990807027727429972059511, −15.29992435864191493623307957768, −13.586790749215890731409998375142, −12.96291509240260335446657212522, −11.4249236633988036228758212480, −10.81053170308617125517711974268, −9.60630503352334009530913919001, −8.67771170501189553226511112129, −7.39416893689253892737036531219, −6.47134503893866051391434611656, −4.386853267285972901438145330127, −3.31676344806048835327156334702, −1.742671014756052888910236809151, −0.31322995384889749134766877544, 1.48291489354496454678767711815, 3.05324098151383933075942574310, 5.10516213109961780561479852070, 6.06094347112708715771768560564, 7.18632076670254091578242902358, 8.507037558908550243767350643302, 9.183908156869066872218202824107, 10.42858264819383116135397786774, 11.46717312717525959870745035645, 12.80468524783268296961616138228, 14.132983582245589461388101455831, 15.29945064129031728889803606480, 15.9167346683936803174063158998, 16.995593346672095976843945017329, 18.25091081017194975706654863553, 18.64102311444904553698057314683, 19.903870090057880241058034355369, 20.81217823743017814596644526546, 22.2432839399046283395672011587, 23.140251844929290059796285319267, 24.30482130455727553220154049241, 25.20846943613617210940893021657, 25.72424149324767427403676677347, 27.03664295646039374939842414093, 27.66292748054481909424788965074

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