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Properties

Label 2-429-13.9-c1-0-18
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

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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: $\chi_{429} (100, \cdot )$
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 \( 1 + (4.65 + 8.06i)T + (-48.5 + 84.0i)T^{2} \)
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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

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

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