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Properties

Label 2-429-429.428-c1-0-20
Degree $2$
Conductor $429$
Sign $0.719 - 0.694i$
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  − 0.168i·2-s + (1.20 + 1.24i)3-s + 1.97·4-s + (0.209 − 0.202i)6-s − 2.38·7-s − 0.668i·8-s + (−0.106 + 2.99i)9-s + 3.31i·11-s + (2.37 + 2.45i)12-s + 3.60·13-s + 0.400i·14-s + 3.83·16-s + (0.504 + 0.0179i)18-s + 2.62·19-s + (−2.86 − 2.96i)21-s + 0.557·22-s + ⋯
L(s)  = 1  − 0.118i·2-s + (0.694 + 0.719i)3-s + 0.985·4-s + (0.0855 − 0.0825i)6-s − 0.899·7-s − 0.236i·8-s + (−0.0355 + 0.999i)9-s + 1.00i·11-s + (0.684 + 0.709i)12-s + 1.00·13-s + 0.107i·14-s + 0.957·16-s + (0.118 + 0.00423i)18-s + 0.601·19-s + (−0.624 − 0.647i)21-s + 0.118·22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.719 - 0.694i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.719 - 0.694i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $0.719 - 0.694i$
Analytic conductor: \(3.42558\)
Root analytic conductor: \(1.85083\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{429} (428, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 429,\ (\ :1/2),\ 0.719 - 0.694i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.86077 + 0.751437i\)
\(L(\frac12)\) \(\approx\) \(1.86077 + 0.751437i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.20 - 1.24i)T \)
11 \( 1 - 3.31iT \)
13 \( 1 - 3.60T \)
good2 \( 1 + 0.168iT - 2T^{2} \)
5 \( 1 + 5T^{2} \)
7 \( 1 + 2.38T + 7T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 - 2.62T + 19T^{2} \)
23 \( 1 + 5.04iT - 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 + 12.1iT - 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 6.26iT - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 14.3T + 73T^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 - 10.1iT - 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 - 97T^{2} \)
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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

−11.09232168302487403846125904488, −10.25785161613110853489830560365, −9.682004179767402467331024352809, −8.640173249622734480859072645824, −7.57383473141583466646285616802, −6.68662016811052599809654313177, −5.59581637431412321961072834878, −4.10605030178366789691129430305, −3.17411487826031384983575243352, −2.03269741379948272648138893998, 1.38643573331377849453371839820, 2.93140085780579425130498497074, 3.58252927108295612463343559362, 5.87393510006596306760032712249, 6.31704006505831173586818048869, 7.38841497842960150246173904717, 8.130965995323226577816938857581, 9.141793954883373746323457034793, 10.08054235537105747041679746696, 11.31373143504556146300241726740

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