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Properties

Label 1-63-63.40-r1-0-0
Degree $1$
Conductor $63$
Sign $0.888 - 0.458i$
Analytic cond. $6.77029$
Root an. cond. $6.77029$
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  + 2-s + 4-s + (0.5 − 0.866i)5-s + 8-s + (0.5 − 0.866i)10-s + (−0.5 − 0.866i)11-s + (0.5 + 0.866i)13-s + 16-s + (0.5 − 0.866i)17-s + (0.5 + 0.866i)19-s + (0.5 − 0.866i)20-s + (−0.5 − 0.866i)22-s + (−0.5 + 0.866i)23-s + (−0.5 − 0.866i)25-s + (0.5 + 0.866i)26-s + ⋯
L(s)  = 1  + 2-s + 4-s + (0.5 − 0.866i)5-s + 8-s + (0.5 − 0.866i)10-s + (−0.5 − 0.866i)11-s + (0.5 + 0.866i)13-s + 16-s + (0.5 − 0.866i)17-s + (0.5 + 0.866i)19-s + (0.5 − 0.866i)20-s + (−0.5 − 0.866i)22-s + (−0.5 + 0.866i)23-s + (−0.5 − 0.866i)25-s + (0.5 + 0.866i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.888 - 0.458i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.888 - 0.458i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $0.888 - 0.458i$
Analytic conductor: \(6.77029\)
Root analytic conductor: \(6.77029\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (40, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 63,\ (1:\ ),\ 0.888 - 0.458i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.059012581 - 0.7418126486i\)
\(L(\frac12)\) \(\approx\) \(3.059012581 - 0.7418126486i\)
\(L(1)\) \(\approx\) \(2.073213055 - 0.2971169318i\)
\(L(1)\) \(\approx\) \(2.073213055 - 0.2971169318i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 + T \)
5 \( 1 + (0.5 - 0.866i)T \)
11 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + (0.5 + 0.866i)T \)
17 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (-0.5 + 0.866i)T \)
31 \( 1 - T \)
37 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + (0.5 + 0.866i)T \)
43 \( 1 + (-0.5 + 0.866i)T \)
47 \( 1 - T \)
53 \( 1 + (-0.5 + 0.866i)T \)
59 \( 1 - T \)
61 \( 1 - T \)
67 \( 1 + T \)
71 \( 1 + T \)
73 \( 1 + (0.5 - 0.866i)T \)
79 \( 1 + T \)
83 \( 1 + (0.5 - 0.866i)T \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 + (0.5 - 0.866i)T \)
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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

−32.33261155476458045267224810913, −30.74079694045858754012473744605, −30.37123643197133291331888600155, −29.13234726686480607195243914184, −28.03731381803955155847392930730, −26.13939358071978532003162555965, −25.548762629155654103453422691880, −24.20258662805533466329938492179, −22.97556236574444175030583273426, −22.24663199866920285897533006434, −21.0815095173360558916896934705, −20.042499403622263354321625511998, −18.51718579991861646061470421524, −17.26162938760909661064144142716, −15.60705987934156643708050893929, −14.78947558159727230259106434498, −13.57504247323292285531241344932, −12.52758968912776062803079877345, −10.99603930091906372719962683054, −10.05461166463026957629289341755, −7.751033910406363023576183196459, −6.477171390374483047820456285708, −5.258263908814165500852196519995, −3.510890437731257889655225325521, −2.13352049364479411492124644125, 1.58383548899691464539161993848, 3.46881558818835764372243455580, 5.06279904105178444604751993433, 6.05049240657365185180007576299, 7.787923018356822018680152708288, 9.43811140145275318813610774010, 11.08880352772153036084839562016, 12.29147317830405083890155632564, 13.476631102819157450096961212610, 14.26239526416895353007822646223, 16.07170765633500915651228949912, 16.553341477517949310499023529821, 18.38818487585423253931588011011, 19.935347129381804175224038193182, 21.01763767270609422819933150069, 21.67413628398963260965755127857, 23.17201641355782044327165598075, 24.114713527671545010090314890406, 25.01551314654732998802431370633, 26.14602127822975547186726734553, 27.85994857972581796900221069019, 29.097562584937171783187219128597, 29.6403694623016335550090965218, 31.321466245309495633999434870677, 31.806309246491337715615858272125

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