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Properties

Label 2-429-13.10-c1-0-8
Degree $2$
Conductor $429$
Sign $0.284 - 0.958i$
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  + (0.884 − 0.510i)2-s + (0.5 + 0.866i)3-s + (−0.478 + 0.828i)4-s + 0.892i·5-s + (0.884 + 0.510i)6-s + (−0.417 − 0.241i)7-s + 3.02i·8-s + (−0.499 + 0.866i)9-s + (0.455 + 0.789i)10-s + (0.866 − 0.5i)11-s − 0.956·12-s + (−0.982 + 3.46i)13-s − 0.493·14-s + (−0.772 + 0.446i)15-s + (0.586 + 1.01i)16-s + (−0.274 + 0.475i)17-s + ⋯
L(s)  = 1  + (0.625 − 0.361i)2-s + (0.288 + 0.499i)3-s + (−0.239 + 0.414i)4-s + 0.398i·5-s + (0.361 + 0.208i)6-s + (−0.157 − 0.0912i)7-s + 1.06i·8-s + (−0.166 + 0.288i)9-s + (0.144 + 0.249i)10-s + (0.261 − 0.150i)11-s − 0.276·12-s + (−0.272 + 0.962i)13-s − 0.131·14-s + (−0.199 + 0.115i)15-s + (0.146 + 0.253i)16-s + (−0.0665 + 0.115i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.284 - 0.958i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.284 - 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $0.284 - 0.958i$
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: \((2,\ 429,\ (\ :1/2),\ 0.284 - 0.958i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.44638 + 1.07919i\)
\(L(\frac12)\) \(\approx\) \(1.44638 + 1.07919i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.982 - 3.46i)T \)
good2 \( 1 + (-0.884 + 0.510i)T + (1 - 1.73i)T^{2} \)
5 \( 1 - 0.892iT - 5T^{2} \)
7 \( 1 + (0.417 + 0.241i)T + (3.5 + 6.06i)T^{2} \)
17 \( 1 + (0.274 - 0.475i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.384 + 0.222i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.19 - 2.07i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.76 - 3.06i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 3.95iT - 31T^{2} \)
37 \( 1 + (1.79 - 1.03i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-8.20 + 4.73i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.68 + 9.84i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 9.18iT - 47T^{2} \)
53 \( 1 + 0.984T + 53T^{2} \)
59 \( 1 + (1.03 + 0.596i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.82 + 10.0i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.39 + 3.69i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (3.93 + 2.27i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 5.22iT - 73T^{2} \)
79 \( 1 - 14.8T + 79T^{2} \)
83 \( 1 + 9.60iT - 83T^{2} \)
89 \( 1 + (9.68 - 5.59i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (10.4 + 6.05i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.33878837843860862990332364641, −10.69039664593664615128124746766, −9.442804864109483112303996319165, −8.810947647883074745977331515892, −7.70561905984007778433339388116, −6.65002973922189424060203937403, −5.29542644409292757876215611604, −4.28658088037349657525081746702, −3.45673942265250993787168019330, −2.32887820620573015365557715058, 0.989007669199370596172097044085, 2.85831762192634592059936565698, 4.26215725213057058862280403734, 5.23965485604255659115856823517, 6.19133027199561440036437872480, 7.07711610367555670602979836368, 8.179256224153125536721902944836, 9.145426170313096058662910606834, 9.971150180937587250556764600421, 10.97778703707684683520332136027

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