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Properties

Label 2-429-429.428-c1-0-18
Degree $2$
Conductor $429$
Sign $0.265 - 0.964i$
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  + 0.662i·2-s + (−1.28 + 1.16i)3-s + 1.56·4-s + 2.33·5-s + (−0.772 − 0.848i)6-s + 2.35i·8-s + (0.280 − 2.98i)9-s + 1.54i·10-s + (2.33 + 2.35i)11-s + (−2 + 1.82i)12-s + (0.772 − 3.52i)13-s + (−2.98 + 2.71i)15-s + 1.56·16-s + 6.41·17-s + (1.97 + 0.185i)18-s − 7.04·19-s + ⋯
L(s)  = 1  + 0.468i·2-s + (−0.739 + 0.673i)3-s + 0.780·4-s + 1.04·5-s + (−0.315 − 0.346i)6-s + 0.833i·8-s + (0.0935 − 0.995i)9-s + 0.488i·10-s + (0.703 + 0.711i)11-s + (−0.577 + 0.525i)12-s + (0.214 − 0.976i)13-s + (−0.771 + 0.702i)15-s + 0.390·16-s + 1.55·17-s + (0.466 + 0.0438i)18-s − 1.61·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.265 - 0.964i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.265 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $0.265 - 0.964i$
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: \((2,\ 429,\ (\ :1/2),\ 0.265 - 0.964i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.29727 + 0.988780i\)
\(L(\frac12)\) \(\approx\) \(1.29727 + 0.988780i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.28 - 1.16i)T \)
11 \( 1 + (-2.33 - 2.35i)T \)
13 \( 1 + (-0.772 + 3.52i)T \)
good2 \( 1 - 0.662iT - 2T^{2} \)
5 \( 1 - 2.33T + 5T^{2} \)
7 \( 1 + 7T^{2} \)
17 \( 1 - 6.41T + 17T^{2} \)
19 \( 1 + 7.04T + 19T^{2} \)
23 \( 1 + 2.33iT - 23T^{2} \)
29 \( 1 + 6.41T + 29T^{2} \)
31 \( 1 - 9.68iT - 31T^{2} \)
37 \( 1 + 5.43iT - 37T^{2} \)
41 \( 1 - 4.34iT - 41T^{2} \)
43 \( 1 + 1.54iT - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 4.66iT - 53T^{2} \)
59 \( 1 - 1.30T + 59T^{2} \)
61 \( 1 + 7.04iT - 61T^{2} \)
67 \( 1 + 9.68iT - 67T^{2} \)
71 \( 1 + 5.97T + 71T^{2} \)
73 \( 1 - 8.58T + 73T^{2} \)
79 \( 1 + 12.5iT - 79T^{2} \)
83 \( 1 + 9.80iT - 83T^{2} \)
89 \( 1 + 14.2T + 89T^{2} \)
97 \( 1 + 13.9iT - 97T^{2} \)
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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

−11.12884525320462662603145325303, −10.41628032317139642559050306275, −9.809411724895302592422064164218, −8.696588226326163324178931048525, −7.43097058947433492248790696945, −6.35197681106178667136829091041, −5.85107595483679586314824202338, −4.90505858875340842969521417787, −3.36349143035023687668116184285, −1.72793089964006953017260676647, 1.36898559824603942816776935627, 2.25847024438095658048998501496, 3.89068675563019178208917458577, 5.66158667359727380052884340362, 6.17621789627775329047049690189, 6.97968663278369496964399144570, 8.116057510720678571696231264280, 9.483056693497369218478037079360, 10.22383639904007627263373220660, 11.26132456032850081317273665160

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