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Properties

Label 2-429-429.428-c1-0-28
Degree $2$
Conductor $429$
Sign $0.162 + 0.986i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.162 + 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.162 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $0.162 + 0.986i$
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: \((2,\ 429,\ (\ :1/2),\ 0.162 + 0.986i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.743533 - 0.631017i\)
\(L(\frac12)\) \(\approx\) \(0.743533 - 0.631017i\)
\(L(\frac{3}{2})\) 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

−11.08814305869077033442405744318, −10.27137260510944787391547088746, −9.587733077121738794538580654632, −7.924376567322431610349422604341, −7.46986260096459444792240704994, −5.98309316715332258375385519378, −5.26237582625287262359265934667, −3.71226339409945453303969815990, −3.07991080699262754868527672428, −0.70213241302438095693760636330, 1.63555134784168466664413674975, 3.26325697755599570948849357896, 4.94603935153668270090491045111, 5.71510097382530947668671500461, 7.01887776895394925802570453342, 7.47172690505102403461811273812, 8.035255326862639313375443124461, 9.671775968113202284428431804798, 10.73789772415423475974754511807, 11.67364051840014193085023139910

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