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Properties

Label 2-429-11.9-c1-0-14
Degree $2$
Conductor $429$
Sign $-0.296 + 0.955i$
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  + (0.622 − 1.91i)2-s + (−0.809 + 0.587i)3-s + (−1.66 − 1.21i)4-s + (−0.155 − 0.479i)5-s + (0.622 + 1.91i)6-s + (2.43 + 1.77i)7-s + (−0.0985 + 0.0715i)8-s + (0.309 − 0.951i)9-s − 1.01·10-s + (2.03 − 2.61i)11-s + 2.06·12-s + (0.309 − 0.951i)13-s + (4.91 − 3.57i)14-s + (0.407 + 0.296i)15-s + (−1.19 − 3.68i)16-s + (−0.836 − 2.57i)17-s + ⋯
L(s)  = 1  + (0.440 − 1.35i)2-s + (−0.467 + 0.339i)3-s + (−0.833 − 0.605i)4-s + (−0.0696 − 0.214i)5-s + (0.254 + 0.782i)6-s + (0.921 + 0.669i)7-s + (−0.0348 + 0.0253i)8-s + (0.103 − 0.317i)9-s − 0.320·10-s + (0.613 − 0.789i)11-s + 0.594·12-s + (0.0857 − 0.263i)13-s + (1.31 − 0.954i)14-s + (0.105 + 0.0764i)15-s + (−0.299 − 0.921i)16-s + (−0.202 − 0.624i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.296 + 0.955i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.296 + 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $-0.296 + 0.955i$
Analytic conductor: \(3.42558\)
Root analytic conductor: \(1.85083\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{429} (196, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 429,\ (\ :1/2),\ -0.296 + 0.955i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.992610 - 1.34698i\)
\(L(\frac12)\) \(\approx\) \(0.992610 - 1.34698i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.809 - 0.587i)T \)
11 \( 1 + (-2.03 + 2.61i)T \)
13 \( 1 + (-0.309 + 0.951i)T \)
good2 \( 1 + (-0.622 + 1.91i)T + (-1.61 - 1.17i)T^{2} \)
5 \( 1 + (0.155 + 0.479i)T + (-4.04 + 2.93i)T^{2} \)
7 \( 1 + (-2.43 - 1.77i)T + (2.16 + 6.65i)T^{2} \)
17 \( 1 + (0.836 + 2.57i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (1.38 - 1.00i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 - 1.83T + 23T^{2} \)
29 \( 1 + (2.23 + 1.62i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (3.02 - 9.30i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (0.248 + 0.180i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-4.08 + 2.97i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 1.76T + 43T^{2} \)
47 \( 1 + (-9.78 + 7.11i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (3.19 - 9.82i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (0.0976 + 0.0709i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-4.54 - 13.9i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + 8.35T + 67T^{2} \)
71 \( 1 + (-3.61 - 11.1i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (4.71 + 3.42i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-2.54 + 7.82i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-0.961 - 2.95i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + 6.94T + 89T^{2} \)
97 \( 1 + (4.98 - 15.3i)T + (-78.4 - 57.0i)T^{2} \)
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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

−10.93264073922522124927846589627, −10.52798190505202813392945792275, −9.207043210143688648484912178590, −8.577958356908823460462349516094, −7.10902323545511426649292638621, −5.68065727216262789689203804470, −4.85059082205007485004949863583, −3.86405436923810460459612843331, −2.63109929224007887527843666819, −1.16934506337872315744584265369, 1.73107866375407406828243860948, 4.09776330198370098943143548511, 4.80549282375027990663607715457, 5.91044643382969779589243073106, 6.83932252941785893219531211514, 7.42024476207205238808694943944, 8.240068273027069291351860253218, 9.424917043258119819543356376152, 10.88688487716189320787535199615, 11.22676778935685079354245026725

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