Properties

Label 2-429-143.109-c1-0-10
Degree $2$
Conductor $429$
Sign $-0.379 + 0.925i$
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  + (−1.67 − 1.67i)2-s − 3-s + 3.58i·4-s + (2.12 − 2.12i)5-s + (1.67 + 1.67i)6-s + (−1.37 + 1.37i)7-s + (2.64 − 2.64i)8-s + 9-s − 7.10·10-s + (2.22 + 2.46i)11-s − 3.58i·12-s + (−0.554 − 3.56i)13-s + 4.58·14-s + (−2.12 + 2.12i)15-s − 1.68·16-s + 6.69·17-s + ⋯
L(s)  = 1  + (−1.18 − 1.18i)2-s − 0.577·3-s + 1.79i·4-s + (0.950 − 0.950i)5-s + (0.682 + 0.682i)6-s + (−0.518 + 0.518i)7-s + (0.936 − 0.936i)8-s + 0.333·9-s − 2.24·10-s + (0.669 + 0.742i)11-s − 1.03i·12-s + (−0.153 − 0.988i)13-s + 1.22·14-s + (−0.548 + 0.548i)15-s − 0.420·16-s + 1.62·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.379 + 0.925i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.379 + 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $-0.379 + 0.925i$
Analytic conductor: \(3.42558\)
Root analytic conductor: \(1.85083\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{429} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 429,\ (\ :1/2),\ -0.379 + 0.925i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.400502 - 0.597093i\)
\(L(\frac12)\) \(\approx\) \(0.400502 - 0.597093i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T \)
11 \( 1 + (-2.22 - 2.46i)T \)
13 \( 1 + (0.554 + 3.56i)T \)
good2 \( 1 + (1.67 + 1.67i)T + 2iT^{2} \)
5 \( 1 + (-2.12 + 2.12i)T - 5iT^{2} \)
7 \( 1 + (1.37 - 1.37i)T - 7iT^{2} \)
17 \( 1 - 6.69T + 17T^{2} \)
19 \( 1 + (-2.46 - 2.46i)T + 19iT^{2} \)
23 \( 1 + 3.28iT - 23T^{2} \)
29 \( 1 + 3.36iT - 29T^{2} \)
31 \( 1 + (-0.273 + 0.273i)T - 31iT^{2} \)
37 \( 1 + (5.74 + 5.74i)T + 37iT^{2} \)
41 \( 1 + (0.977 + 0.977i)T + 41iT^{2} \)
43 \( 1 - 2.55T + 43T^{2} \)
47 \( 1 + (5.10 + 5.10i)T + 47iT^{2} \)
53 \( 1 - 9.35T + 53T^{2} \)
59 \( 1 + (2.90 + 2.90i)T + 59iT^{2} \)
61 \( 1 + 6.86iT - 61T^{2} \)
67 \( 1 + (5.85 - 5.85i)T - 67iT^{2} \)
71 \( 1 + (-11.6 + 11.6i)T - 71iT^{2} \)
73 \( 1 + (-2.50 + 2.50i)T - 73iT^{2} \)
79 \( 1 + 0.392iT - 79T^{2} \)
83 \( 1 + (-11.1 - 11.1i)T + 83iT^{2} \)
89 \( 1 + (5.52 + 5.52i)T + 89iT^{2} \)
97 \( 1 + (8.21 - 8.21i)T - 97iT^{2} \)
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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

−10.58386532527453166559290619857, −9.841410844373535405817906101801, −9.517601453132148636378223633535, −8.535608699122837196388633437032, −7.52271551471256260461014807061, −5.99262545043996642607899464677, −5.18676869830693993015492013820, −3.49185016790531365725859813535, −2.05047489527011647113449972127, −0.892568324736893238073973323327, 1.25329216584913031266242097081, 3.37991494605112863648924341490, 5.34641796314851078783664116216, 6.19060911105199547738358676420, 6.81526685227295134520161974809, 7.45504757103582042779438157322, 8.805130423380231996173799251131, 9.738821734396592171322476567185, 10.09070038266560547583067817372, 11.06711303024636775892485942388

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