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Properties

Label 2-429-33.32-c1-0-47
Degree $2$
Conductor $429$
Sign $-0.920 + 0.391i$
Analytic cond. $3.42558$
Root an. cond. $1.85083$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.822·2-s + (−0.0778 − 1.73i)3-s − 1.32·4-s − 3.64i·5-s + (−0.0640 − 1.42i)6-s + 1.97i·7-s − 2.73·8-s + (−2.98 + 0.269i)9-s − 2.99i·10-s + (3.10 − 1.15i)11-s + (0.103 + 2.28i)12-s + i·13-s + 1.62i·14-s + (−6.29 + 0.283i)15-s + 0.397·16-s − 7.48·17-s + ⋯
L(s)  = 1  + 0.581·2-s + (−0.0449 − 0.998i)3-s − 0.661·4-s − 1.62i·5-s + (−0.0261 − 0.581i)6-s + 0.744i·7-s − 0.966·8-s + (−0.995 + 0.0898i)9-s − 0.946i·10-s + (0.936 − 0.349i)11-s + (0.0297 + 0.660i)12-s + 0.277i·13-s + 0.433i·14-s + (−1.62 + 0.0732i)15-s + 0.0994·16-s − 1.81·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.920 + 0.391i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.920 + 0.391i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $-0.920 + 0.391i$
Analytic conductor: \(3.42558\)
Root analytic conductor: \(1.85083\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{429} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 429,\ (\ :1/2),\ -0.920 + 0.391i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.222336 - 1.09133i\)
\(L(\frac12)\) \(\approx\) \(0.222336 - 1.09133i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.0778 + 1.73i)T \)
11 \( 1 + (-3.10 + 1.15i)T \)
13 \( 1 - iT \)
good2 \( 1 - 0.822T + 2T^{2} \)
5 \( 1 + 3.64iT - 5T^{2} \)
7 \( 1 - 1.97iT - 7T^{2} \)
17 \( 1 + 7.48T + 17T^{2} \)
19 \( 1 + 4.73iT - 19T^{2} \)
23 \( 1 + 1.40iT - 23T^{2} \)
29 \( 1 - 0.599T + 29T^{2} \)
31 \( 1 + 3.80T + 31T^{2} \)
37 \( 1 - 4.99T + 37T^{2} \)
41 \( 1 - 9.36T + 41T^{2} \)
43 \( 1 + 5.55iT - 43T^{2} \)
47 \( 1 + 11.4iT - 47T^{2} \)
53 \( 1 + 7.53iT - 53T^{2} \)
59 \( 1 - 1.46iT - 59T^{2} \)
61 \( 1 - 0.106iT - 61T^{2} \)
67 \( 1 - 7.88T + 67T^{2} \)
71 \( 1 - 5.03iT - 71T^{2} \)
73 \( 1 + 0.797iT - 73T^{2} \)
79 \( 1 - 0.729iT - 79T^{2} \)
83 \( 1 - 1.46T + 83T^{2} \)
89 \( 1 + 13.4iT - 89T^{2} \)
97 \( 1 + 2.10T + 97T^{2} \)
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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

−11.32007984155418965887522800084, −9.239365197446643520561347988959, −8.925789178672903802731607521961, −8.381279696547349932223407041055, −6.81222049506352358848379332795, −5.83591843946917062069287976134, −4.98906113450330360402868504136, −4.05982700485997251563191447970, −2.25192807658994405506436373770, −0.59969689486107175439682550744, 2.79578467912869652092388604339, 3.90192883201436581814165023288, 4.35454638820004242886916033889, 5.87962327207448420726117881926, 6.61569693090437795297305866416, 7.84245624692120738639327098919, 9.185972889993409617534448662605, 9.780668126381627384990175011246, 10.82187506097870106070295456996, 11.19628639748419134396751921124

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