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Properties

Label 2-429-143.138-c1-0-0
Degree $2$
Conductor $429$
Sign $-0.583 - 0.811i$
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  + (0.396 + 0.201i)2-s + (−0.809 − 0.587i)3-s + (−1.05 − 1.45i)4-s + (−1.97 + 1.00i)5-s + (−0.201 − 0.396i)6-s + (0.380 + 0.0603i)7-s + (−0.264 − 1.67i)8-s + (0.309 + 0.951i)9-s − 0.986·10-s + (2.38 + 2.30i)11-s + 1.80i·12-s + (−3.31 + 1.42i)13-s + (0.138 + 0.100i)14-s + (2.19 + 0.347i)15-s + (−0.881 + 2.71i)16-s + (−0.807 + 2.48i)17-s + ⋯
L(s)  = 1  + (0.280 + 0.142i)2-s + (−0.467 − 0.339i)3-s + (−0.529 − 0.728i)4-s + (−0.884 + 0.450i)5-s + (−0.0824 − 0.161i)6-s + (0.143 + 0.0227i)7-s + (−0.0935 − 0.590i)8-s + (0.103 + 0.317i)9-s − 0.312·10-s + (0.720 + 0.693i)11-s + 0.520i·12-s + (−0.918 + 0.394i)13-s + (0.0370 + 0.0269i)14-s + (0.565 + 0.0896i)15-s + (−0.220 + 0.678i)16-s + (−0.195 + 0.602i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.583 - 0.811i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.583 - 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $-0.583 - 0.811i$
Analytic conductor: \(3.42558\)
Root analytic conductor: \(1.85083\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{429} (424, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 429,\ (\ :1/2),\ -0.583 - 0.811i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.162529 + 0.317123i\)
\(L(\frac12)\) \(\approx\) \(0.162529 + 0.317123i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.38 - 2.30i)T \)
13 \( 1 + (3.31 - 1.42i)T \)
good2 \( 1 + (-0.396 - 0.201i)T + (1.17 + 1.61i)T^{2} \)
5 \( 1 + (1.97 - 1.00i)T + (2.93 - 4.04i)T^{2} \)
7 \( 1 + (-0.380 - 0.0603i)T + (6.65 + 2.16i)T^{2} \)
17 \( 1 + (0.807 - 2.48i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (6.72 - 1.06i)T + (18.0 - 5.87i)T^{2} \)
23 \( 1 - 3.32iT - 23T^{2} \)
29 \( 1 + (0.151 + 0.207i)T + (-8.96 + 27.5i)T^{2} \)
31 \( 1 + (2.96 - 5.81i)T + (-18.2 - 25.0i)T^{2} \)
37 \( 1 + (-1.43 + 9.04i)T + (-35.1 - 11.4i)T^{2} \)
41 \( 1 + (-9.16 + 1.45i)T + (38.9 - 12.6i)T^{2} \)
43 \( 1 - 4.96T + 43T^{2} \)
47 \( 1 + (6.66 - 1.05i)T + (44.6 - 14.5i)T^{2} \)
53 \( 1 + (-0.308 - 0.948i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-1.40 + 8.88i)T + (-56.1 - 18.2i)T^{2} \)
61 \( 1 + (-4.89 - 1.58i)T + (49.3 + 35.8i)T^{2} \)
67 \( 1 + (4.47 + 4.47i)T + 67iT^{2} \)
71 \( 1 + (1.60 - 0.819i)T + (41.7 - 57.4i)T^{2} \)
73 \( 1 + (14.8 + 2.34i)T + (69.4 + 22.5i)T^{2} \)
79 \( 1 + (12.6 - 4.11i)T + (63.9 - 46.4i)T^{2} \)
83 \( 1 + (3.71 + 7.28i)T + (-48.7 + 67.1i)T^{2} \)
89 \( 1 + (11.7 - 11.7i)T - 89iT^{2} \)
97 \( 1 + (2.27 - 4.47i)T + (-57.0 - 78.4i)T^{2} \)
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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

−11.41842546640229681133391952918, −10.72795261472736111948077876089, −9.746391510019620426394169720305, −8.803310800834457817582550975470, −7.53762375690138227584020385797, −6.78512125088039683044398222581, −5.84082992029108439986743211200, −4.61306564149000876115244943949, −3.90881014897196201932048654666, −1.83260862955833963100650359162, 0.21889833107623770415887362229, 2.86049646765132520225683327350, 4.25940097655199849350616297176, 4.54195680794780783762931237046, 5.93164978939148566682071466684, 7.24206602986611471082865021220, 8.247683965364096221029185359362, 8.871490820928890422223490071895, 9.943349408462836000241736061006, 11.26335413311607561932636987687

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