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Properties

Label 2-429-429.428-c1-0-6
Degree $2$
Conductor $429$
Sign $-0.984 + 0.173i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.984 + 0.173i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.984 + 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $-0.984 + 0.173i$
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: \((2,\ 429,\ (\ :1/2),\ -0.984 + 0.173i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0482069 - 0.550982i\)
\(L(\frac12)\) \(\approx\) \(0.0482069 - 0.550982i\)
\(L(\frac{3}{2})\) 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

−11.43853166774927135833530736681, −10.97231165813657361290869336381, −10.06049863965188274515132722355, −8.770784878772577816245330255533, −7.894069340302938421543772250232, −6.72272803646708374075247852128, −6.29077775422613759280849689601, −4.72879523442242027363547043884, −4.15607177320934728683932990453, −2.38299531335441636860065280850, 0.35234643187767701587351697959, 2.15581000120584173975613536725, 3.44345404584525960289834025881, 4.84609921779727077429460892754, 6.15347948209504701628813099326, 6.87111073120599705330786018726, 7.87863325281646840784820476171, 8.556921355996711031450965698543, 10.45155765658281867220963316211, 10.86092615423082237348482919511

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