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Properties

Label 2-429-429.350-c0-0-3
Degree $2$
Conductor $429$
Sign $-0.781 + 0.624i$
Analytic cond. $0.214098$
Root an. cond. $0.462708$
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  + (0.587 − 1.80i)2-s + (0.809 − 0.587i)3-s + (−2.11 − 1.53i)4-s + (0.363 + 1.11i)5-s + (−0.587 − 1.80i)6-s + (−2.48 + 1.80i)8-s + (0.309 − 0.951i)9-s + 2.23·10-s + (−0.951 + 0.309i)11-s − 2.61·12-s + (−0.309 + 0.951i)13-s + (0.951 + 0.690i)15-s + (1.00 + 3.07i)16-s + (−1.53 − 1.11i)18-s + (0.951 − 2.92i)20-s + ⋯
L(s)  = 1  + (0.587 − 1.80i)2-s + (0.809 − 0.587i)3-s + (−2.11 − 1.53i)4-s + (0.363 + 1.11i)5-s + (−0.587 − 1.80i)6-s + (−2.48 + 1.80i)8-s + (0.309 − 0.951i)9-s + 2.23·10-s + (−0.951 + 0.309i)11-s − 2.61·12-s + (−0.309 + 0.951i)13-s + (0.951 + 0.690i)15-s + (1.00 + 3.07i)16-s + (−1.53 − 1.11i)18-s + (0.951 − 2.92i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.781 + 0.624i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.781 + 0.624i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $-0.781 + 0.624i$
Analytic conductor: \(0.214098\)
Root analytic conductor: \(0.462708\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{429} (350, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 429,\ (\ :0),\ -0.781 + 0.624i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.233695733\)
\(L(\frac12)\) \(\approx\) \(1.233695733\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.809 + 0.587i)T \)
11 \( 1 + (0.951 - 0.309i)T \)
13 \( 1 + (0.309 - 0.951i)T \)
good2 \( 1 + (-0.587 + 1.80i)T + (-0.809 - 0.587i)T^{2} \)
5 \( 1 + (-0.363 - 1.11i)T + (-0.809 + 0.587i)T^{2} \)
7 \( 1 + (-0.309 - 0.951i)T^{2} \)
17 \( 1 + (0.809 - 0.587i)T^{2} \)
19 \( 1 + (-0.309 + 0.951i)T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + (-0.309 - 0.951i)T^{2} \)
31 \( 1 + (0.809 + 0.587i)T^{2} \)
37 \( 1 + (-0.309 - 0.951i)T^{2} \)
41 \( 1 + (0.309 - 0.951i)T^{2} \)
43 \( 1 + 1.61T + T^{2} \)
47 \( 1 + (-1.53 + 1.11i)T + (0.309 - 0.951i)T^{2} \)
53 \( 1 + (0.809 + 0.587i)T^{2} \)
59 \( 1 + (1.53 + 1.11i)T + (0.309 + 0.951i)T^{2} \)
61 \( 1 + (0.190 + 0.587i)T + (-0.809 + 0.587i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (-0.809 + 0.587i)T^{2} \)
73 \( 1 + (-0.309 - 0.951i)T^{2} \)
79 \( 1 + (0.190 - 0.587i)T + (-0.809 - 0.587i)T^{2} \)
83 \( 1 + (0.363 + 1.11i)T + (-0.809 + 0.587i)T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (0.809 + 0.587i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.07112914635642544512596210746, −10.25539082679420940330141780588, −9.625063661463710144896538820845, −8.671179532834960810511145191449, −7.33666265734262180258665168598, −6.19476176494610659380490603999, −4.78666277226373006225493738430, −3.55073369425341405866554275408, −2.64677487073326551684795938580, −1.90755163675949141804022711470, 3.06899231554797084728625985944, 4.38133279910081686622752697949, 5.15878342154403649132149615941, 5.77203694809446698676076164616, 7.29941588115205561415771861110, 8.080110184418893923874602965934, 8.660211528086965297391715130500, 9.446398202025009898992147011060, 10.45189091315671496983927600678, 12.32426096729215313894333303025

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