↑

Properties

Label 2-429-13.3-c1-0-3
Degree $2$
Conductor $429$
Sign $-0.697 - 0.716i$
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  + (0.149 + 0.258i)2-s + (0.5 + 0.866i)3-s + (0.955 − 1.65i)4-s − 3.44·5-s + (−0.149 + 0.258i)6-s + (−2.46 + 4.27i)7-s + 1.16·8-s + (−0.499 + 0.866i)9-s + (−0.514 − 0.891i)10-s + (−0.5 − 0.866i)11-s + 1.91·12-s + (1.73 + 3.15i)13-s − 1.47·14-s + (−1.72 − 2.98i)15-s + (−1.73 − 3.00i)16-s + (−2.78 + 4.82i)17-s + ⋯
L(s)  = 1  + (0.105 + 0.183i)2-s + (0.288 + 0.499i)3-s + (0.477 − 0.827i)4-s − 1.53·5-s + (−0.0610 + 0.105i)6-s + (−0.933 + 1.61i)7-s + 0.413·8-s + (−0.166 + 0.288i)9-s + (−0.162 − 0.281i)10-s + (−0.150 − 0.261i)11-s + 0.551·12-s + (0.482 + 0.876i)13-s − 0.394·14-s + (−0.444 − 0.769i)15-s + (−0.433 − 0.751i)16-s + (−0.676 + 1.17i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.697 - 0.716i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.697 - 0.716i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $-0.697 - 0.716i$
Analytic conductor: \(3.42558\)
Root analytic conductor: \(1.85083\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{429} (133, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 429,\ (\ :1/2),\ -0.697 - 0.716i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.325527 + 0.770517i\)
\(L(\frac12)\) \(\approx\) \(0.325527 + 0.770517i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + (-1.73 - 3.15i)T \)
good2 \( 1 + (-0.149 - 0.258i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + 3.44T + 5T^{2} \)
7 \( 1 + (2.46 - 4.27i)T + (-3.5 - 6.06i)T^{2} \)
17 \( 1 + (2.78 - 4.82i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.61 - 4.53i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.56 + 2.71i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.54 + 2.67i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 8.42T + 31T^{2} \)
37 \( 1 + (-0.853 - 1.47i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.234 + 0.405i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.77 - 6.54i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 2.55T + 47T^{2} \)
53 \( 1 - 6.18T + 53T^{2} \)
59 \( 1 + (5.17 - 8.96i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.79 + 11.7i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.847 - 1.46i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-3.42 + 5.93i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 1.86T + 73T^{2} \)
79 \( 1 - 2.12T + 79T^{2} \)
83 \( 1 + 3.77T + 83T^{2} \)
89 \( 1 + (-1.55 - 2.68i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (4.65 - 8.06i)T + (-48.5 - 84.0i)T^{2} \)
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

−11.56091745721498902782492285402, −10.64263256381252470873040778003, −9.697868115057611178872842171353, −8.645295705908398924104233420544, −8.105331319992697304025932292002, −6.51352803869269284039910987299, −6.04720433131300257783886801509, −4.62697034977416555338742249396, −3.60456376776804243474256013699, −2.27379008427393523283063438479, 0.47479116155542178815150191892, 2.88272348919605216028888290474, 3.66590972786746741075053979033, 4.49113104904822610793042917886, 6.72022636443108942481286934421, 7.17268081674632478352284356392, 7.82921917474558382770202665788, 8.698720083261986000826337577677, 10.14995100810337426180617517473, 11.09070564524998436261761071709

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