↑

Properties

Label 2-429-11.3-c1-0-19
Degree $2$
Conductor $429$
Sign $0.523 + 0.851i$
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  + (2.25 − 1.63i)2-s + (0.309 + 0.951i)3-s + (1.77 − 5.47i)4-s + (2.25 + 1.63i)5-s + (2.25 + 1.63i)6-s + (−1.26 + 3.89i)7-s + (−3.23 − 9.94i)8-s + (−0.809 + 0.587i)9-s + 7.76·10-s + (−1.80 − 2.78i)11-s + 5.75·12-s + (0.809 − 0.587i)13-s + (3.52 + 10.8i)14-s + (−0.862 + 2.65i)15-s + (−14.2 − 10.3i)16-s + (0.435 + 0.316i)17-s + ⋯
L(s)  = 1  + (1.59 − 1.15i)2-s + (0.178 + 0.549i)3-s + (0.889 − 2.73i)4-s + (1.00 + 0.733i)5-s + (0.919 + 0.668i)6-s + (−0.478 + 1.47i)7-s + (−1.14 − 3.51i)8-s + (−0.269 + 0.195i)9-s + 2.45·10-s + (−0.543 − 0.839i)11-s + 1.66·12-s + (0.224 − 0.163i)13-s + (0.942 + 2.90i)14-s + (−0.222 + 0.685i)15-s + (−3.56 − 2.58i)16-s + (0.105 + 0.0766i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.523 + 0.851i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.523 + 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $0.523 + 0.851i$
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: \((2,\ 429,\ (\ :1/2),\ 0.523 + 0.851i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.09068 - 1.72811i\)
\(L(\frac12)\) \(\approx\) \(3.09068 - 1.72811i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.80 + 2.78i)T \)
13 \( 1 + (-0.809 + 0.587i)T \)
good2 \( 1 + (-2.25 + 1.63i)T + (0.618 - 1.90i)T^{2} \)
5 \( 1 + (-2.25 - 1.63i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (1.26 - 3.89i)T + (-5.66 - 4.11i)T^{2} \)
17 \( 1 + (-0.435 - 0.316i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (0.145 + 0.448i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 7.04T + 23T^{2} \)
29 \( 1 + (0.836 - 2.57i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (0.649 - 0.471i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (0.0991 - 0.305i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-1.02 - 3.15i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 3.75T + 43T^{2} \)
47 \( 1 + (1.87 + 5.78i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-10.3 + 7.53i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-0.00194 + 0.00598i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-7.97 - 5.79i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 4.14T + 67T^{2} \)
71 \( 1 + (-11.8 - 8.63i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (4.36 - 13.4i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-0.867 + 0.630i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (1.42 + 1.03i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 6.15T + 89T^{2} \)
97 \( 1 + (2.92 - 2.12i)T + (29.9 - 92.2i)T^{2} \)
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

−11.16865666673078639265702124101, −10.22951280453149000489607295118, −9.818245342711262524441381967506, −8.710106790626988004969321043382, −6.54083062399376873235319642056, −5.73499233569686270499048342223, −5.38718651476594508392554262770, −3.79253051253345421739007605434, −2.80502134699525936764232483961, −2.19431008114959468514212332353, 2.17934825377338968330605304480, 3.73051519533803109718868755102, 4.61313554073251440541352117313, 5.69963430635450781904556240864, 6.45708995523897838241974526684, 7.38492214648381550098405169587, 7.971167710788509388582035687826, 9.295529513487637409908607997390, 10.49219198849883893267998392030, 11.89610325728379549630336677695

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