Properties

Label 2-429-429.389-c0-0-1
Degree $2$
Conductor $429$
Sign $0.606 + 0.794i$
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  + (−0.951 − 0.690i)2-s + (−0.309 + 0.951i)3-s + (0.118 + 0.363i)4-s + (1.53 − 1.11i)5-s + (0.951 − 0.690i)6-s + (−0.224 + 0.690i)8-s + (−0.809 − 0.587i)9-s − 2.23·10-s + (−0.587 − 0.809i)11-s − 0.381·12-s + (0.809 + 0.587i)13-s + (0.587 + 1.80i)15-s + (0.999 − 0.726i)16-s + (0.363 + 1.11i)18-s + (0.587 + 0.427i)20-s + ⋯
L(s)  = 1  + (−0.951 − 0.690i)2-s + (−0.309 + 0.951i)3-s + (0.118 + 0.363i)4-s + (1.53 − 1.11i)5-s + (0.951 − 0.690i)6-s + (−0.224 + 0.690i)8-s + (−0.809 − 0.587i)9-s − 2.23·10-s + (−0.587 − 0.809i)11-s − 0.381·12-s + (0.809 + 0.587i)13-s + (0.587 + 1.80i)15-s + (0.999 − 0.726i)16-s + (0.363 + 1.11i)18-s + (0.587 + 0.427i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.606 + 0.794i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.606 + 0.794i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $0.606 + 0.794i$
Analytic conductor: \(0.214098\)
Root analytic conductor: \(0.462708\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{429} (389, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 429,\ (\ :0),\ 0.606 + 0.794i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5794626791\)
\(L(\frac12)\) \(\approx\) \(0.5794626791\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.309 - 0.951i)T \)
11 \( 1 + (0.587 + 0.809i)T \)
13 \( 1 + (-0.809 - 0.587i)T \)
good2 \( 1 + (0.951 + 0.690i)T + (0.309 + 0.951i)T^{2} \)
5 \( 1 + (-1.53 + 1.11i)T + (0.309 - 0.951i)T^{2} \)
7 \( 1 + (0.809 - 0.587i)T^{2} \)
17 \( 1 + (-0.309 + 0.951i)T^{2} \)
19 \( 1 + (0.809 + 0.587i)T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + (0.809 - 0.587i)T^{2} \)
31 \( 1 + (-0.309 - 0.951i)T^{2} \)
37 \( 1 + (0.809 - 0.587i)T^{2} \)
41 \( 1 + (-0.809 - 0.587i)T^{2} \)
43 \( 1 - 0.618T + T^{2} \)
47 \( 1 + (0.363 - 1.11i)T + (-0.809 - 0.587i)T^{2} \)
53 \( 1 + (-0.309 - 0.951i)T^{2} \)
59 \( 1 + (-0.363 - 1.11i)T + (-0.809 + 0.587i)T^{2} \)
61 \( 1 + (1.30 - 0.951i)T + (0.309 - 0.951i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (0.309 - 0.951i)T^{2} \)
73 \( 1 + (0.809 - 0.587i)T^{2} \)
79 \( 1 + (1.30 + 0.951i)T + (0.309 + 0.951i)T^{2} \)
83 \( 1 + (1.53 - 1.11i)T + (0.309 - 0.951i)T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (-0.309 - 0.951i)T^{2} \)
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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

−10.88108725916123347196898584287, −10.27313225566652335663711525976, −9.441544221589262475221545055156, −8.962458077530065192437702136688, −8.285656851553297498226264612011, −6.08292241057315267795428473586, −5.59053938126484071494115898510, −4.54004502837107880212446274385, −2.77380936128371061368570598996, −1.30437242615509866641677873868, 1.77684000250295276032352506307, 3.05892434731011445562152044693, 5.42748429448858997062667933421, 6.24896461850620929455597854968, 6.89856689822672206107379900204, 7.66967343643274434130268235562, 8.652615700436890498185208447759, 9.748273535454978068409905105211, 10.36383123347516927102883972620, 11.21767897935330953325198753794

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