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Properties

Label 2-429-143.32-c1-0-24
Degree $2$
Conductor $429$
Sign $-0.926 + 0.377i$
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  + (−0.391 + 0.104i)2-s + (0.5 − 0.866i)3-s + (−1.58 + 0.917i)4-s + (2.00 − 2.00i)5-s + (−0.104 + 0.391i)6-s + (−3.95 − 1.05i)7-s + (1.09 − 1.09i)8-s + (−0.499 − 0.866i)9-s + (−0.576 + 0.997i)10-s + (−2.31 + 2.37i)11-s + 1.83i·12-s + (−3.35 − 1.30i)13-s + 1.65·14-s + (−0.735 − 2.74i)15-s + (1.52 − 2.63i)16-s + (−0.407 − 0.705i)17-s + ⋯
L(s)  = 1  + (−0.276 + 0.0741i)2-s + (0.288 − 0.499i)3-s + (−0.794 + 0.458i)4-s + (0.898 − 0.898i)5-s + (−0.0428 + 0.159i)6-s + (−1.49 − 0.400i)7-s + (0.388 − 0.388i)8-s + (−0.166 − 0.288i)9-s + (−0.182 + 0.315i)10-s + (−0.699 + 0.714i)11-s + 0.529i·12-s + (−0.931 − 0.362i)13-s + 0.443·14-s + (−0.189 − 0.708i)15-s + (0.380 − 0.658i)16-s + (−0.0988 − 0.171i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.926 + 0.377i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.926 + 0.377i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $-0.926 + 0.377i$
Analytic conductor: \(3.42558\)
Root analytic conductor: \(1.85083\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{429} (175, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 429,\ (\ :1/2),\ -0.926 + 0.377i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0894680 - 0.457083i\)
\(L(\frac12)\) \(\approx\) \(0.0894680 - 0.457083i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.31 - 2.37i)T \)
13 \( 1 + (3.35 + 1.30i)T \)
good2 \( 1 + (0.391 - 0.104i)T + (1.73 - i)T^{2} \)
5 \( 1 + (-2.00 + 2.00i)T - 5iT^{2} \)
7 \( 1 + (3.95 + 1.05i)T + (6.06 + 3.5i)T^{2} \)
17 \( 1 + (0.407 + 0.705i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.953 - 3.55i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (3.35 + 1.93i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.70 + 2.71i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4.88 + 4.88i)T - 31iT^{2} \)
37 \( 1 + (2.24 - 0.602i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (-6.89 + 1.84i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (4.75 + 8.23i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (5.58 + 5.58i)T + 47iT^{2} \)
53 \( 1 + 3.97T + 53T^{2} \)
59 \( 1 + (2.78 - 10.4i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-9.30 + 5.37i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.10 - 4.11i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (0.994 + 0.266i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (-9.57 + 9.57i)T - 73iT^{2} \)
79 \( 1 + 4.76iT - 79T^{2} \)
83 \( 1 + (-5.63 - 5.63i)T + 83iT^{2} \)
89 \( 1 + (-14.1 + 3.78i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (8.44 + 2.26i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.07843374587302394906513871734, −9.887849434033809629062228328882, −9.077264026450524977739774741899, −8.042953310094519284989317988463, −7.23852206226937632381471755972, −6.05138591918943487866445145388, −4.96419952573090336874644285460, −3.73112075651695239560381439369, −2.28661963502000730637966063522, −0.29603603176804692859945598514, 2.41351892544532622532775506602, 3.36451252280211783367138590832, 4.90525600638949361598466352388, 5.91504720968191349318202846428, 6.69639203561730012799857154453, 8.144116911456126187629892370543, 9.318266931283799605869161834407, 9.653000401688691256142158580413, 10.34388822203368917284799779082, 11.14672023777984271009025105042

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