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Properties

Label 2-429-11.5-c1-0-13
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  + (−0.264 − 0.813i)2-s + (−0.809 − 0.587i)3-s + (1.02 − 0.745i)4-s + (−0.893 + 2.75i)5-s + (−0.264 + 0.813i)6-s + (3.14 − 2.28i)7-s + (−2.26 − 1.64i)8-s + (0.309 + 0.951i)9-s + 2.47·10-s + (2.00 + 2.64i)11-s − 1.26·12-s + (−0.309 − 0.951i)13-s + (−2.68 − 1.95i)14-s + (2.33 − 1.69i)15-s + (0.0447 − 0.137i)16-s + (2.19 − 6.75i)17-s + ⋯
L(s)  = 1  + (−0.186 − 0.575i)2-s + (−0.467 − 0.339i)3-s + (0.513 − 0.372i)4-s + (−0.399 + 1.23i)5-s + (−0.107 + 0.332i)6-s + (1.18 − 0.863i)7-s + (−0.799 − 0.580i)8-s + (0.103 + 0.317i)9-s + 0.782·10-s + (0.603 + 0.797i)11-s − 0.366·12-s + (−0.0857 − 0.263i)13-s + (−0.718 − 0.522i)14-s + (0.604 − 0.438i)15-s + (0.0111 − 0.0344i)16-s + (0.532 − 1.63i)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: $\chi_{429} (313, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 429,\ (\ :1/2),\ 0.284 + 0.958i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.06495 - 0.795089i\)
\(L(\frac12)\) \(\approx\) \(1.06495 - 0.795089i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.809 + 0.587i)T \)
11 \( 1 + (-2.00 - 2.64i)T \)
13 \( 1 + (0.309 + 0.951i)T \)
good2 \( 1 + (0.264 + 0.813i)T + (-1.61 + 1.17i)T^{2} \)
5 \( 1 + (0.893 - 2.75i)T + (-4.04 - 2.93i)T^{2} \)
7 \( 1 + (-3.14 + 2.28i)T + (2.16 - 6.65i)T^{2} \)
17 \( 1 + (-2.19 + 6.75i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-5.46 - 3.97i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + 3.83T + 23T^{2} \)
29 \( 1 + (-5.55 + 4.03i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (2.44 + 7.51i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (0.468 - 0.340i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-1.34 - 0.974i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 8.65T + 43T^{2} \)
47 \( 1 + (0.114 + 0.0829i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-1.50 - 4.62i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-4.33 + 3.15i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (2.81 - 8.66i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 - 4.18T + 67T^{2} \)
71 \( 1 + (2.72 - 8.37i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-5.56 + 4.04i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-3.79 - 11.6i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (1.53 - 4.73i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 - 5.94T + 89T^{2} \)
97 \( 1 + (-2.40 - 7.39i)T + (-78.4 + 57.0i)T^{2} \)
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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

−11.17172213112203050704487679669, −10.18240879536478150257420181638, −9.752860847242480203995637653018, −7.76568963336182103577473748062, −7.35438839063491576739780562845, −6.48377842364711865247578581978, −5.23243135886629271848415147894, −3.84537279505733380428554871708, −2.50586115246967801227396265363, −1.13772951562492910939661910485, 1.52721775138849212305863556069, 3.48295271092961733262223339481, 4.87969591652170357476159064590, 5.54950321484855539185743556396, 6.59686642178241913450262919104, 7.955133575132589404717291453921, 8.526008366785083265867436431697, 9.081229281112704062169413273659, 10.63405026196082216300490541519, 11.67761714901008427046528368768

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