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Properties

Label 1-429-429.17-r0-0-0
Degree $1$
Conductor $429$
Sign $0.984 - 0.173i$
Analytic cond. $1.99226$
Root an. cond. $1.99226$
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  + (−0.913 − 0.406i)2-s + (0.669 + 0.743i)4-s + (−0.809 − 0.587i)5-s + (0.669 + 0.743i)7-s + (−0.309 − 0.951i)8-s + (0.5 + 0.866i)10-s + (−0.309 − 0.951i)14-s + (−0.104 + 0.994i)16-s + (0.913 − 0.406i)17-s + (−0.978 − 0.207i)19-s + (−0.104 − 0.994i)20-s + (0.5 + 0.866i)23-s + (0.309 + 0.951i)25-s + (−0.104 + 0.994i)28-s + (−0.978 + 0.207i)29-s + ⋯
L(s)  = 1  + (−0.913 − 0.406i)2-s + (0.669 + 0.743i)4-s + (−0.809 − 0.587i)5-s + (0.669 + 0.743i)7-s + (−0.309 − 0.951i)8-s + (0.5 + 0.866i)10-s + (−0.309 − 0.951i)14-s + (−0.104 + 0.994i)16-s + (0.913 − 0.406i)17-s + (−0.978 − 0.207i)19-s + (−0.104 − 0.994i)20-s + (0.5 + 0.866i)23-s + (0.309 + 0.951i)25-s + (−0.104 + 0.994i)28-s + (−0.978 + 0.207i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.984 - 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.984 - 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $0.984 - 0.173i$
Analytic conductor: \(1.99226\)
Root analytic conductor: \(1.99226\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{429} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 429,\ (0:\ ),\ 0.984 - 0.173i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7798639213 - 0.06826744750i\)
\(L(\frac12)\) \(\approx\) \(0.7798639213 - 0.06826744750i\)
\(L(1)\) \(\approx\) \(0.6944384580 - 0.08919857231i\)
\(L(1)\) \(\approx\) \(0.6944384580 - 0.08919857231i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.913 - 0.406i)T \)
5 \( 1 + (-0.809 - 0.587i)T \)
7 \( 1 + (0.669 + 0.743i)T \)
17 \( 1 + (0.913 - 0.406i)T \)
19 \( 1 + (-0.978 - 0.207i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 + (-0.978 + 0.207i)T \)
31 \( 1 + (0.809 - 0.587i)T \)
37 \( 1 + (0.978 - 0.207i)T \)
41 \( 1 + (-0.669 + 0.743i)T \)
43 \( 1 + (0.5 - 0.866i)T \)
47 \( 1 + (0.309 + 0.951i)T \)
53 \( 1 + (0.809 - 0.587i)T \)
59 \( 1 + (0.669 + 0.743i)T \)
61 \( 1 + (-0.913 + 0.406i)T \)
67 \( 1 + (0.5 + 0.866i)T \)
71 \( 1 + (0.913 - 0.406i)T \)
73 \( 1 + (0.309 - 0.951i)T \)
79 \( 1 + (0.809 - 0.587i)T \)
83 \( 1 + (0.809 + 0.587i)T \)
89 \( 1 + (-0.5 - 0.866i)T \)
97 \( 1 + (0.104 + 0.994i)T \)
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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

−24.16376009343217643813434558603, −23.425188638276438818550638497246, −22.8785775061857850200383166388, −21.370369646773991251620589899125, −20.50765613912172404415737952386, −19.6726679246783226785756509433, −18.87558468183610018437462194076, −18.23197773200601672549414735227, −17.062529162590401419924027247968, −16.62953955411677503504117397994, −15.38670235120443032937860690088, −14.77298596996865281991126453461, −14.03118086355161396465441100453, −12.473916835994625585519757944199, −11.38451409424229172250718729364, −10.71672243425645780953114061694, −10.00314991714624387179895280499, −8.57361149497018114482853396065, −7.923171967229757030600288948368, −7.122221158682622541204167024293, −6.22152741619715583073870679862, −4.83912506268597745562837608600, −3.661874823572144635096806851875, −2.24680712313424513793155423083, −0.83404260539574940810396909150, 0.97214977997389988805163042096, 2.20167233301032276472650113321, 3.42651661480834110568965117643, 4.59379919747351922513470203693, 5.8202087709476371071842882299, 7.296966413703214550488908113219, 8.01709207260761729427758454041, 8.81469656735462538240083742753, 9.60635650405002569013358695881, 10.92067534843919367147223445207, 11.62705474513945241771235809494, 12.29707789220307346781786905098, 13.23335188201465134263446243434, 14.87576416674496384171706397531, 15.46896190119956240253878924981, 16.511831234387620848959746046149, 17.169246073189001651684721476479, 18.22170949612229852452702994555, 18.992745580739424709792296677163, 19.64147346057218619637378586167, 20.75311741045766977580119195306, 21.13091479134541695216143932244, 22.21186027649756139104248386317, 23.446418721248034521100356772740, 24.263065881820973145490364742459

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