↑

Properties

Label 2-429-11.4-c1-0-11
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

Related objects

Downloads

Learn more

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: $\chi_{429} (235, \cdot )$
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 \( 1 + (7.51 + 5.45i)T + (29.9 + 92.2i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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