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Properties

Label 2-429-143.21-c1-0-21
Degree $2$
Conductor $429$
Sign $0.281 + 0.959i$
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 + (2.46 − 2.22i)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 + (0.742 − 0.669i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.281 + 0.959i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.281 + 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $0.281 + 0.959i$
Analytic conductor: \(3.42558\)
Root analytic conductor: \(1.85083\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{429} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 429,\ (\ :1/2),\ 0.281 + 0.959i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.05005 - 1.53545i\)
\(L(\frac12)\) \(\approx\) \(2.05005 - 1.53545i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.46 + 2.22i)T \)
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

−11.13469491315187709819175236943, −10.52583007255307960623224115361, −9.713904569289001531451567234458, −8.402419522426388434729508649810, −6.63819293485152544728218613669, −5.95348933565090491712490722961, −5.16410249600627530400370981518, −3.91823629313193508319308142856, −2.73191998345355037972816460179, −1.69722727676700081628628987116, 1.80808505850435193901761767095, 4.28188710908535482279581305328, 4.57331691949590944182304990523, 5.57248139803585977820325328320, 6.65228130594504285502044578068, 7.00471996953543594123989476509, 8.553850390679500798132607242836, 9.248581242300013771919419983313, 10.58339109295445301488237153612, 11.66063867361291105024890572733

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