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Properties

Label 2-429-143.21-c1-0-27
Degree $2$
Conductor $429$
Sign $-0.999 + 0.00747i$
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  + (1.92 − 1.92i)2-s − 3-s − 5.42i·4-s + (−1.59 − 1.59i)5-s + (−1.92 + 1.92i)6-s + (1.66 + 1.66i)7-s + (−6.60 − 6.60i)8-s + 9-s − 6.16·10-s + (−2.25 − 2.43i)11-s + 5.42i·12-s + (−1.55 + 3.25i)13-s + 6.42·14-s + (1.59 + 1.59i)15-s − 14.6·16-s + 4.73·17-s + ⋯
L(s)  = 1  + (1.36 − 1.36i)2-s − 0.577·3-s − 2.71i·4-s + (−0.714 − 0.714i)5-s + (−0.786 + 0.786i)6-s + (0.630 + 0.630i)7-s + (−2.33 − 2.33i)8-s + 0.333·9-s − 1.94·10-s + (−0.680 − 0.733i)11-s + 1.56i·12-s + (−0.431 + 0.901i)13-s + 1.71·14-s + (0.412 + 0.412i)15-s − 3.65·16-s + 1.14·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.00747i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.00747i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $-0.999 + 0.00747i$
Analytic conductor: \(3.42558\)
Root analytic conductor: \(1.85083\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{429} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 429,\ (\ :1/2),\ -0.999 + 0.00747i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00736193 - 1.96929i\)
\(L(\frac12)\) \(\approx\) \(0.00736193 - 1.96929i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T \)
11 \( 1 + (2.25 + 2.43i)T \)
13 \( 1 + (1.55 - 3.25i)T \)
good2 \( 1 + (-1.92 + 1.92i)T - 2iT^{2} \)
5 \( 1 + (1.59 + 1.59i)T + 5iT^{2} \)
7 \( 1 + (-1.66 - 1.66i)T + 7iT^{2} \)
17 \( 1 - 4.73T + 17T^{2} \)
19 \( 1 + (-3.03 + 3.03i)T - 19iT^{2} \)
23 \( 1 + 1.62iT - 23T^{2} \)
29 \( 1 + 3.51iT - 29T^{2} \)
31 \( 1 + (-4.40 - 4.40i)T + 31iT^{2} \)
37 \( 1 + (3.16 - 3.16i)T - 37iT^{2} \)
41 \( 1 + (-4.68 + 4.68i)T - 41iT^{2} \)
43 \( 1 - 4.69T + 43T^{2} \)
47 \( 1 + (6.86 - 6.86i)T - 47iT^{2} \)
53 \( 1 + 0.475T + 53T^{2} \)
59 \( 1 + (-8.18 + 8.18i)T - 59iT^{2} \)
61 \( 1 + 9.65iT - 61T^{2} \)
67 \( 1 + (-5.61 - 5.61i)T + 67iT^{2} \)
71 \( 1 + (-5.96 - 5.96i)T + 71iT^{2} \)
73 \( 1 + (-1.78 - 1.78i)T + 73iT^{2} \)
79 \( 1 - 1.07iT - 79T^{2} \)
83 \( 1 + (-10.3 + 10.3i)T - 83iT^{2} \)
89 \( 1 + (13.0 - 13.0i)T - 89iT^{2} \)
97 \( 1 + (-8.39 - 8.39i)T + 97iT^{2} \)
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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

−11.22244973505267416476370168096, −10.23397660004010183578299502148, −9.259351664833584545195651821576, −8.046252556502595367730663195455, −6.44921173373848674026570665046, −5.22417031608235060049290265321, −4.94624911637785533718827794270, −3.78467704394450030343281852942, −2.48015722755010025566725843645, −0.926048420521276908981509060598, 3.06011490169210464376075953950, 4.05369662473186253871204008574, 5.08094181863933914370573717081, 5.74465762250477540318913837946, 7.05672027431090264983459700301, 7.59799420800358858435013288342, 8.029177157779279395117959116265, 9.973713783075101474860236405628, 11.03659925302629852492628006846, 11.94629573433182530639686152647

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