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Properties

Label 2-429-13.4-c1-0-2
Degree $2$
Conductor $429$
Sign $0.455 - 0.890i$
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  + (−1.00 − 0.582i)2-s + (0.5 − 0.866i)3-s + (−0.322 − 0.558i)4-s + 1.30i·5-s + (−1.00 + 0.582i)6-s + (−2.07 + 1.19i)7-s + 3.07i·8-s + (−0.499 − 0.866i)9-s + (0.761 − 1.31i)10-s + (0.866 + 0.5i)11-s − 0.644·12-s + (−1.68 + 3.18i)13-s + 2.78·14-s + (1.13 + 0.653i)15-s + (1.14 − 1.98i)16-s + (1.71 + 2.96i)17-s + ⋯
L(s)  = 1  + (−0.712 − 0.411i)2-s + (0.288 − 0.499i)3-s + (−0.161 − 0.279i)4-s + 0.584i·5-s + (−0.411 + 0.237i)6-s + (−0.783 + 0.452i)7-s + 1.08i·8-s + (−0.166 − 0.288i)9-s + (0.240 − 0.416i)10-s + (0.261 + 0.150i)11-s − 0.186·12-s + (−0.466 + 0.884i)13-s + 0.744·14-s + (0.292 + 0.168i)15-s + (0.286 − 0.496i)16-s + (0.414 + 0.718i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.455 - 0.890i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.455 - 0.890i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $0.455 - 0.890i$
Analytic conductor: \(3.42558\)
Root analytic conductor: \(1.85083\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{429} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 429,\ (\ :1/2),\ 0.455 - 0.890i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.477823 + 0.292402i\)
\(L(\frac12)\) \(\approx\) \(0.477823 + 0.292402i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.68 - 3.18i)T \)
good2 \( 1 + (1.00 + 0.582i)T + (1 + 1.73i)T^{2} \)
5 \( 1 - 1.30iT - 5T^{2} \)
7 \( 1 + (2.07 - 1.19i)T + (3.5 - 6.06i)T^{2} \)
17 \( 1 + (-1.71 - 2.96i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (7.44 - 4.29i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.0396 - 0.0687i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.98 + 5.16i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 2.82iT - 31T^{2} \)
37 \( 1 + (4.65 + 2.68i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.35 - 0.781i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.24 - 5.62i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 12.9iT - 47T^{2} \)
53 \( 1 + 1.32T + 53T^{2} \)
59 \( 1 + (-4.25 + 2.45i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.0339 - 0.0587i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (13.0 + 7.53i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-3.10 + 1.79i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 2.67iT - 73T^{2} \)
79 \( 1 + 8.50T + 79T^{2} \)
83 \( 1 - 15.6iT - 83T^{2} \)
89 \( 1 + (6.46 + 3.73i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-8.29 + 4.78i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.10096682382319644738076485041, −10.31978586399937799389314242060, −9.525711062341620106221626352791, −8.760294297218727251088812890615, −7.87504532929840227017848855184, −6.57702495483472112942420048512, −5.99108838655799848793565534106, −4.33566461587460345334704110321, −2.80768912062132781714894801483, −1.72461055750210966331791034962, 0.43277171278020819579865440949, 2.95284225633210189904927717079, 4.05449913900442679020826967688, 5.12461426713249889817488140961, 6.63231149894259248958812879811, 7.39480820739015355246477442396, 8.635571552015240106129124116271, 8.896850289364357859828494642484, 10.01472926800496841283674835211, 10.51899139480320070323189285292

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