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Properties

Label 2-429-13.10-c1-0-23
Degree $2$
Conductor $429$
Sign $-0.373 + 0.927i$
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  + (2.44 − 1.41i)2-s + (−0.5 − 0.866i)3-s + (2.98 − 5.16i)4-s + 2.29i·5-s + (−2.44 − 1.41i)6-s + (−0.726 − 0.419i)7-s − 11.1i·8-s + (−0.499 + 0.866i)9-s + (3.23 + 5.60i)10-s + (−0.866 + 0.5i)11-s − 5.96·12-s + (1.30 − 3.36i)13-s − 2.36·14-s + (1.98 − 1.14i)15-s + (−9.82 − 17.0i)16-s + (−2.85 + 4.94i)17-s + ⋯
L(s)  = 1  + (1.72 − 0.997i)2-s + (−0.288 − 0.499i)3-s + (1.49 − 2.58i)4-s + 1.02i·5-s + (−0.997 − 0.576i)6-s + (−0.274 − 0.158i)7-s − 3.95i·8-s + (−0.166 + 0.288i)9-s + (1.02 + 1.77i)10-s + (−0.261 + 0.150i)11-s − 1.72·12-s + (0.361 − 0.932i)13-s − 0.632·14-s + (0.513 − 0.296i)15-s + (−2.45 − 4.25i)16-s + (−0.692 + 1.19i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.373 + 0.927i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.373 + 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $-0.373 + 0.927i$
Analytic conductor: \(3.42558\)
Root analytic conductor: \(1.85083\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{429} (166, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 429,\ (\ :1/2),\ -0.373 + 0.927i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.78457 - 2.64148i\)
\(L(\frac12)\) \(\approx\) \(1.78457 - 2.64148i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.30 + 3.36i)T \)
good2 \( 1 + (-2.44 + 1.41i)T + (1 - 1.73i)T^{2} \)
5 \( 1 - 2.29iT - 5T^{2} \)
7 \( 1 + (0.726 + 0.419i)T + (3.5 + 6.06i)T^{2} \)
17 \( 1 + (2.85 - 4.94i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-6.49 - 3.75i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.58 - 2.74i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.44 + 2.49i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 6.24iT - 31T^{2} \)
37 \( 1 + (-0.565 + 0.326i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.508 - 0.293i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.49 - 6.05i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 8.08iT - 47T^{2} \)
53 \( 1 + 0.0119T + 53T^{2} \)
59 \( 1 + (4.30 + 2.48i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.09 + 3.63i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.49 + 0.861i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (0.808 + 0.466i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 11.3iT - 73T^{2} \)
79 \( 1 - 4.38T + 79T^{2} \)
83 \( 1 + 1.36iT - 83T^{2} \)
89 \( 1 + (-7.14 + 4.12i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (3.89 + 2.24i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.01057235849173927125315526760, −10.54874515930848511967545133790, −9.720282045021266238757627709221, −7.66927962813385142783304329486, −6.63881579489389639022361567746, −5.98339058346110833574301699919, −5.04487930770158373324711486274, −3.59741368474788796513093332659, −2.94416640304668514466873478368, −1.52812699575508128806355242558, 2.71698837546474203676150904147, 3.97213049059980920204889339570, 4.90865884112783606988573506197, 5.36172006002603588983751665547, 6.54774746122088003839467185381, 7.32848632461934170072945398238, 8.603393873254051802524848661661, 9.289911666459230554631772503672, 11.15620473777708622313864277585, 11.70482578600626557239052904581

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