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Properties

Label 2-429-143.25-c1-0-4
Degree $2$
Conductor $429$
Sign $-0.999 + 0.0369i$
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  + (1.59 + 2.18i)2-s + (−0.309 − 0.951i)3-s + (−1.64 + 5.05i)4-s + (−0.714 + 0.983i)5-s + (1.59 − 2.18i)6-s + (−1.83 − 0.595i)7-s + (−8.54 + 2.77i)8-s + (−0.809 + 0.587i)9-s − 3.28·10-s + (−0.579 + 3.26i)11-s + 5.31·12-s + (2.74 + 2.33i)13-s + (−1.61 − 4.96i)14-s + (1.15 + 0.375i)15-s + (−11.0 − 8.02i)16-s + (−2.21 − 1.61i)17-s + ⋯
L(s)  = 1  + (1.12 + 1.54i)2-s + (−0.178 − 0.549i)3-s + (−0.821 + 2.52i)4-s + (−0.319 + 0.439i)5-s + (0.649 − 0.893i)6-s + (−0.693 − 0.225i)7-s + (−3.01 + 0.981i)8-s + (−0.269 + 0.195i)9-s − 1.03·10-s + (−0.174 + 0.984i)11-s + 1.53·12-s + (0.760 + 0.648i)13-s + (−0.430 − 1.32i)14-s + (0.298 + 0.0969i)15-s + (−2.76 − 2.00i)16-s + (−0.537 − 0.390i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0369i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0369i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $-0.999 + 0.0369i$
Analytic conductor: \(3.42558\)
Root analytic conductor: \(1.85083\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{429} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 429,\ (\ :1/2),\ -0.999 + 0.0369i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0311440 - 1.68322i\)
\(L(\frac12)\) \(\approx\) \(0.0311440 - 1.68322i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.309 + 0.951i)T \)
11 \( 1 + (0.579 - 3.26i)T \)
13 \( 1 + (-2.74 - 2.33i)T \)
good2 \( 1 + (-1.59 - 2.18i)T + (-0.618 + 1.90i)T^{2} \)
5 \( 1 + (0.714 - 0.983i)T + (-1.54 - 4.75i)T^{2} \)
7 \( 1 + (1.83 + 0.595i)T + (5.66 + 4.11i)T^{2} \)
17 \( 1 + (2.21 + 1.61i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-2.24 + 0.730i)T + (15.3 - 11.1i)T^{2} \)
23 \( 1 - 0.986T + 23T^{2} \)
29 \( 1 + (-1.88 + 5.78i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-6.17 - 8.49i)T + (-9.57 + 29.4i)T^{2} \)
37 \( 1 + (2.39 + 0.776i)T + (29.9 + 21.7i)T^{2} \)
41 \( 1 + (1.41 - 0.459i)T + (33.1 - 24.0i)T^{2} \)
43 \( 1 - 9.90T + 43T^{2} \)
47 \( 1 + (-7.47 + 2.42i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (7.61 - 5.53i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-3.65 - 1.18i)T + (47.7 + 34.6i)T^{2} \)
61 \( 1 + (-1.34 - 0.976i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 15.2iT - 67T^{2} \)
71 \( 1 + (3.16 - 4.36i)T + (-21.9 - 67.5i)T^{2} \)
73 \( 1 + (-7.00 - 2.27i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (-2.93 + 2.13i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (7.83 - 10.7i)T + (-25.6 - 78.9i)T^{2} \)
89 \( 1 - 7.77iT - 89T^{2} \)
97 \( 1 + (0.673 + 0.926i)T + (-29.9 + 92.2i)T^{2} \)
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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

−12.10833430647580600748414323610, −11.04270499950166830031765390715, −9.488115787959806789094465639267, −8.455794424472726122579285074084, −7.38821030352446768689443478598, −6.88920586962887241918408682749, −6.22377361491471760513550505753, −5.03539320481267916437290158469, −4.04341966888703790219753774831, −2.88536580320362172546327460793, 0.78764967933979814726164833918, 2.76028060930910496141996359789, 3.59696197889503091864613303497, 4.51808691640499996843997286935, 5.61193264173524540216484449315, 6.25157611806939798907017632298, 8.414830953972812787901548294807, 9.263051141608778913388488470471, 10.21862292496856626282324298208, 10.88682926033740327917936703690

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