↑

Properties

Label 2-429-13.4-c1-0-4
Degree $2$
Conductor $429$
Sign $0.518 - 0.854i$
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  + (−2.08 − 1.20i)2-s + (−0.5 + 0.866i)3-s + (1.88 + 3.27i)4-s + 3.20i·5-s + (2.08 − 1.20i)6-s + (2.13 − 1.22i)7-s − 4.27i·8-s + (−0.499 − 0.866i)9-s + (3.84 − 6.66i)10-s + (0.866 + 0.5i)11-s − 3.77·12-s + (1.91 − 3.05i)13-s − 5.91·14-s + (−2.77 − 1.60i)15-s + (−1.35 + 2.34i)16-s + (3.11 + 5.39i)17-s + ⋯
L(s)  = 1  + (−1.47 − 0.849i)2-s + (−0.288 + 0.499i)3-s + (0.944 + 1.63i)4-s + 1.43i·5-s + (0.849 − 0.490i)6-s + (0.805 − 0.464i)7-s − 1.50i·8-s + (−0.166 − 0.288i)9-s + (1.21 − 2.10i)10-s + (0.261 + 0.150i)11-s − 1.09·12-s + (0.529 − 0.848i)13-s − 1.58·14-s + (−0.716 − 0.413i)15-s + (−0.338 + 0.586i)16-s + (0.755 + 1.30i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.518 - 0.854i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.518 - 0.854i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $0.518 - 0.854i$
Analytic conductor: \(3.42558\)
Root analytic conductor: \(1.85083\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{429} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 429,\ (\ :1/2),\ 0.518 - 0.854i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.545060 + 0.306717i\)
\(L(\frac12)\) \(\approx\) \(0.545060 + 0.306717i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.5 - 0.866i)T \)
11 \( 1 + (-0.866 - 0.5i)T \)
13 \( 1 + (-1.91 + 3.05i)T \)
good2 \( 1 + (2.08 + 1.20i)T + (1 + 1.73i)T^{2} \)
5 \( 1 - 3.20iT - 5T^{2} \)
7 \( 1 + (-2.13 + 1.22i)T + (3.5 - 6.06i)T^{2} \)
17 \( 1 + (-3.11 - 5.39i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.83 - 1.63i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.43 - 2.48i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.43 + 4.22i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 8.70iT - 31T^{2} \)
37 \( 1 + (-0.232 - 0.134i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (4.60 + 2.65i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.37 - 7.58i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 9.26iT - 47T^{2} \)
53 \( 1 - 6.70T + 53T^{2} \)
59 \( 1 + (12.1 - 7.00i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.80 - 4.86i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.79 - 2.77i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (3.58 - 2.07i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 5.12iT - 73T^{2} \)
79 \( 1 + 4.19T + 79T^{2} \)
83 \( 1 - 5.40iT - 83T^{2} \)
89 \( 1 + (11.9 + 6.90i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (2.64 - 1.52i)T + (48.5 - 84.0i)T^{2} \)
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

−10.90785388017844220121506761790, −10.36964724393341682236613322129, −10.04072900407507475847466363682, −8.602338366561115679569349845041, −7.955206858765473639935499712008, −6.99307412194921560601758041104, −5.80996018445552555539600301983, −3.95064397388624823479734108100, −2.98663296597457963455624038736, −1.49936351972136573759768400376, 0.74046098938276750982013295186, 1.85648664163634752261168938629, 4.60077531647583318297517958273, 5.56455934291265096254698227104, 6.52586492973012478466070884502, 7.59857917320504960746439554734, 8.379857742491370739695878879127, 8.945584022405378007698478051497, 9.630454359669456375935640573006, 10.96219680114294829959673650792

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