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Properties

Label 2-429-429.428-c1-0-35
Degree $2$
Conductor $429$
Sign $-0.986 + 0.166i$
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  − 2.39i·2-s − 1.73·3-s − 3.73·4-s + 4.11·5-s + 4.14i·6-s + 4.14i·8-s + 2.99·9-s − 9.85i·10-s + (−0.551 − 3.27i)11-s + 6.46·12-s − 3.60i·13-s − 7.12·15-s + 2.46·16-s − 7.18i·18-s − 15.3·20-s + ⋯
L(s)  = 1  − 1.69i·2-s − 1.00·3-s − 1.86·4-s + 1.84·5-s + 1.69i·6-s + 1.46i·8-s + 0.999·9-s − 3.11i·10-s + (−0.166 − 0.986i)11-s + 1.86·12-s − 0.999i·13-s − 1.84·15-s + 0.616·16-s − 1.69i·18-s − 3.43·20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.986 + 0.166i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.986 + 0.166i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $-0.986 + 0.166i$
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: \((2,\ 429,\ (\ :1/2),\ -0.986 + 0.166i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0991463 - 1.18479i\)
\(L(\frac12)\) \(\approx\) \(0.0991463 - 1.18479i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 1.73T \)
11 \( 1 + (0.551 + 3.27i)T \)
13 \( 1 + 3.60iT \)
good2 \( 1 + 2.39iT - 2T^{2} \)
5 \( 1 - 4.11T + 5T^{2} \)
7 \( 1 + 7T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 - 7.82iT - 41T^{2} \)
43 \( 1 + 12.4iT - 43T^{2} \)
47 \( 1 + 9.33T + 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 - 15.3T + 59T^{2} \)
61 \( 1 + 7.21iT - 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 + 4.92T + 71T^{2} \)
73 \( 1 + 73T^{2} \)
79 \( 1 - 14.4iT - 79T^{2} \)
83 \( 1 - 12.6iT - 83T^{2} \)
89 \( 1 - 18.3T + 89T^{2} \)
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

−10.74007111277672260122753367553, −10.08926748836877124388729127005, −9.572841339010858030798942094990, −8.442803686753883875688063604963, −6.60484433269214728395970828137, −5.63723292544306819127508266032, −4.97320534310515017466767023897, −3.35953776803622668256751741464, −2.13151846616453730628057715606, −0.918782567530916970510357489167, 1.85444885260487459481646255009, 4.56249832888358429430801712619, 5.20396055195507200866315718850, 6.14095638585584024082862426120, 6.61945582258300538992877969503, 7.46718310758623506794487648845, 8.926290968793436222000228140326, 9.658400866853392951597485334137, 10.28949705082652558273070011512, 11.61152808777268856752223956617

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