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Properties

Label 2-429-11.3-c1-0-22
Degree $2$
Conductor $429$
Sign $-0.483 + 0.875i$
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  + (2.01 − 1.46i)2-s + (−0.309 − 0.951i)3-s + (1.29 − 3.99i)4-s + (1.02 + 0.741i)5-s + (−2.01 − 1.46i)6-s + (0.590 − 1.81i)7-s + (−1.69 − 5.21i)8-s + (−0.809 + 0.587i)9-s + 3.13·10-s + (−1.53 + 2.94i)11-s − 4.20·12-s + (0.809 − 0.587i)13-s + (−1.47 − 4.52i)14-s + (0.389 − 1.19i)15-s + (−4.24 − 3.08i)16-s + (−1.67 − 1.21i)17-s + ⋯
L(s)  = 1  + (1.42 − 1.03i)2-s + (−0.178 − 0.549i)3-s + (0.649 − 1.99i)4-s + (0.456 + 0.331i)5-s + (−0.822 − 0.597i)6-s + (0.223 − 0.686i)7-s + (−0.598 − 1.84i)8-s + (−0.269 + 0.195i)9-s + 0.992·10-s + (−0.462 + 0.886i)11-s − 1.21·12-s + (0.224 − 0.163i)13-s + (−0.392 − 1.20i)14-s + (0.100 − 0.309i)15-s + (−1.06 − 0.771i)16-s + (−0.407 − 0.295i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.483 + 0.875i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.483 + 0.875i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $-0.483 + 0.875i$
Analytic conductor: \(3.42558\)
Root analytic conductor: \(1.85083\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{429} (157, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 429,\ (\ :1/2),\ -0.483 + 0.875i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.47818 - 2.50596i\)
\(L(\frac12)\) \(\approx\) \(1.47818 - 2.50596i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.309 + 0.951i)T \)
11 \( 1 + (1.53 - 2.94i)T \)
13 \( 1 + (-0.809 + 0.587i)T \)
good2 \( 1 + (-2.01 + 1.46i)T + (0.618 - 1.90i)T^{2} \)
5 \( 1 + (-1.02 - 0.741i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (-0.590 + 1.81i)T + (-5.66 - 4.11i)T^{2} \)
17 \( 1 + (1.67 + 1.21i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-1.50 - 4.64i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 0.161T + 23T^{2} \)
29 \( 1 + (1.72 - 5.29i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (1.29 - 0.944i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (0.577 - 1.77i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (1.17 + 3.60i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 6.60T + 43T^{2} \)
47 \( 1 + (-0.598 - 1.84i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-2.08 + 1.51i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (2.07 - 6.39i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (11.7 + 8.50i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 8.27T + 67T^{2} \)
71 \( 1 + (-11.3 - 8.26i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-3.42 + 10.5i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-5.58 + 4.05i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-10.8 - 7.85i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 6.56T + 89T^{2} \)
97 \( 1 + (7.51 - 5.45i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.81855332058439137828755962699, −10.58078285362564587812436865125, −9.511742669223362928369144921666, −7.83528747436624740227856331833, −6.82277537031522261656865510155, −5.82994931437627597485954696747, −4.91557323005776997668951087399, −3.85015017827850586284962500487, −2.58557732147066819923288251342, −1.51500056393653012721319309376, 2.65603927338598413439359509182, 3.88230021830887701725730055854, 4.96913247401464805607443245360, 5.63183344533689422294285591279, 6.31299432552061529538356018230, 7.56182099082670470529155402751, 8.595669495008787577207940340061, 9.408432167079578859144194633274, 10.90559580943354698952057178239, 11.63346812777998518677289924257

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