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Properties

Label 2-429-13.10-c1-0-22
Degree $2$
Conductor $429$
Sign $-0.540 + 0.841i$
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  + (1.80 − 1.04i)2-s + (−0.5 − 0.866i)3-s + (1.18 − 2.04i)4-s − 2.92i·5-s + (−1.80 − 1.04i)6-s + (0.684 + 0.395i)7-s − 0.756i·8-s + (−0.499 + 0.866i)9-s + (−3.05 − 5.28i)10-s + (−0.866 + 0.5i)11-s − 2.36·12-s + (1.90 − 3.05i)13-s + 1.65·14-s + (−2.52 + 1.46i)15-s + (1.57 + 2.72i)16-s + (0.549 − 0.952i)17-s + ⋯
L(s)  = 1  + (1.27 − 0.738i)2-s + (−0.288 − 0.499i)3-s + (0.590 − 1.02i)4-s − 1.30i·5-s + (−0.738 − 0.426i)6-s + (0.258 + 0.149i)7-s − 0.267i·8-s + (−0.166 + 0.288i)9-s + (−0.964 − 1.67i)10-s + (−0.261 + 0.150i)11-s − 0.681·12-s + (0.529 − 0.848i)13-s + 0.441·14-s + (−0.653 + 0.377i)15-s + (0.392 + 0.680i)16-s + (0.133 − 0.231i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.540 + 0.841i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.540 + 0.841i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $-0.540 + 0.841i$
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: \((2,\ 429,\ (\ :1/2),\ -0.540 + 0.841i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.16304 - 2.12817i\)
\(L(\frac12)\) \(\approx\) \(1.16304 - 2.12817i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.90 + 3.05i)T \)
good2 \( 1 + (-1.80 + 1.04i)T + (1 - 1.73i)T^{2} \)
5 \( 1 + 2.92iT - 5T^{2} \)
7 \( 1 + (-0.684 - 0.395i)T + (3.5 + 6.06i)T^{2} \)
17 \( 1 + (-0.549 + 0.952i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (6.90 + 3.98i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.75 - 4.76i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.25 - 7.37i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 2.39iT - 31T^{2} \)
37 \( 1 + (0.737 - 0.426i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-5.62 + 3.24i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.929 + 1.60i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 2.87iT - 47T^{2} \)
53 \( 1 - 12.8T + 53T^{2} \)
59 \( 1 + (-4.73 - 2.73i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.570 + 0.988i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.866 - 0.500i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (6.39 + 3.69i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 12.6iT - 73T^{2} \)
79 \( 1 + 1.98T + 79T^{2} \)
83 \( 1 - 17.9iT - 83T^{2} \)
89 \( 1 + (9.33 - 5.38i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (13.5 + 7.81i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.15279441945450059132752755689, −10.42871111941673176432428222684, −8.858153665747053358858518709158, −8.293691480282425915270771915516, −6.86477804520475474566590204012, −5.48493278790425122778420649478, −5.11986592392397474508310416469, −4.03060675656086517940403166182, −2.60424981929083899436793390470, −1.22075518094857614316444239480, 2.66062417120485557992846120346, 3.93013633220181935074025796586, 4.54208162028353323185277242632, 6.07620189468580488791218599595, 6.31975924876710791506172641369, 7.38108869338702844834687692073, 8.511144822618067296025237549901, 9.982485709577513291856753109798, 10.70445005315194741890150357494, 11.47670550579461955790252934057

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