Properties

Label 1-429-429.350-r1-0-0
Degree $1$
Conductor $429$
Sign $-0.0457 + 0.998i$
Analytic cond. $46.1024$
Root an. cond. $46.1024$
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 + (−0.809 + 0.587i)5-s + (−0.309 − 0.951i)7-s + (0.309 − 0.951i)8-s + 10-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 + (−0.809 − 0.587i)20-s − 23-s + (0.309 − 0.951i)25-s + (0.809 − 0.587i)28-s + (−0.309 − 0.951i)29-s + ⋯
L(s)  = 1  + (−0.809 − 0.587i)2-s + (0.309 + 0.951i)4-s + (−0.809 + 0.587i)5-s + (−0.309 − 0.951i)7-s + (0.309 − 0.951i)8-s + 10-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 + (−0.809 − 0.587i)20-s − 23-s + (0.309 − 0.951i)25-s + (0.809 − 0.587i)28-s + (−0.309 − 0.951i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0457 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0457 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $-0.0457 + 0.998i$
Analytic conductor: \(46.1024\)
Root analytic conductor: \(46.1024\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{429} (350, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 429,\ (1:\ ),\ -0.0457 + 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2226903527 + 0.2331210515i\)
\(L(\frac12)\) \(\approx\) \(0.2226903527 + 0.2331210515i\)
\(L(1)\) \(\approx\) \(0.5402889161 - 0.1049431347i\)
\(L(1)\) \(\approx\) \(0.5402889161 - 0.1049431347i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.809 - 0.587i)T \)
5 \( 1 + (-0.809 + 0.587i)T \)
7 \( 1 + (-0.309 - 0.951i)T \)
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

−24.029261283663249348286565594708, −23.12481420982784714672609321082, −22.09891749914273475439819524856, −20.948995359374800521906805496381, −19.96244983832791804940616776981, −19.26028736737591999046241890767, −18.61096446153805024246088378168, −17.58481910092343770955876923886, −16.66540492261968432388082378066, −15.87984737123267850209429581511, −15.2875281218309030317666584352, −14.41499270220519875424465790489, −13.0203225409362487434169799339, −12.05631758693992850930152297218, −11.24617243037123119238116909147, −10.04849009531831687852780599006, −9.10266486161088682418819431980, −8.380825735668847032234239764445, −7.5981233344700724491634846317, −6.400029171502507417039845495094, −5.50130262239441297690314126755, −4.43848061330561706471024408916, −2.89476993869930541992513924271, −1.48131236301462595999791301850, −0.146288949971700986694101042142, 0.93496123109748676620155463053, 2.45267091942875487683375125626, 3.60216914739325012409938914147, 4.18987495900871680610237151023, 6.131356948146316251259980411415, 7.37415525336967663815879235946, 7.70058130369774872649031251796, 8.90757044907378004454701111658, 10.24079944930170167393835592408, 10.45571268292710913608585785915, 11.73808849001022986261018691763, 12.25548735459031904349615557494, 13.53514354671027410824216660665, 14.45692335891177715927281974634, 15.77432186231560262932703968330, 16.383922174997393735922199749437, 17.30254105373785727572291216114, 18.27989674270615709476128159665, 19.08826882538807942648821490015, 19.682646943873311107945174031522, 20.549925658447721188024540892709, 21.31090818982285494407518077518, 22.639163352496920218129040704194, 22.972486261881411450311647891395, 24.163468356066990954034219800128

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