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Properties

Label 2-429-429.428-c1-0-12
Degree $2$
Conductor $429$
Sign $-0.691 - 0.722i$
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  + i·2-s + 1.73i·3-s + 4-s + 3.16·5-s − 1.73·6-s − 3.46·7-s + 3i·8-s − 2.99·9-s + 3.16i·10-s + (−3.16 − i)11-s + 1.73i·12-s + (−1.73 + 3.16i)13-s − 3.46i·14-s + 5.47i·15-s − 16-s + 5.47·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + 0.999i·3-s + 0.5·4-s + 1.41·5-s − 0.707·6-s − 1.30·7-s + 1.06i·8-s − 0.999·9-s + 1.00i·10-s + (−0.953 − 0.301i)11-s + 0.499i·12-s + (−0.480 + 0.877i)13-s − 0.925i·14-s + 1.41i·15-s − 0.250·16-s + 1.32·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.691 - 0.722i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.691 - 0.722i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $-0.691 - 0.722i$
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: \((2,\ 429,\ (\ :1/2),\ -0.691 - 0.722i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.653861 + 1.53077i\)
\(L(\frac12)\) \(\approx\) \(0.653861 + 1.53077i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 1.73iT \)
11 \( 1 + (3.16 + i)T \)
13 \( 1 + (1.73 - 3.16i)T \)
good2 \( 1 - iT - 2T^{2} \)
5 \( 1 - 3.16T + 5T^{2} \)
7 \( 1 + 3.46T + 7T^{2} \)
17 \( 1 - 5.47T + 17T^{2} \)
19 \( 1 - 3.46T + 19T^{2} \)
23 \( 1 - 6.92iT - 23T^{2} \)
29 \( 1 - 5.47T + 29T^{2} \)
31 \( 1 + 5.47iT - 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 + 2iT - 41T^{2} \)
43 \( 1 + 9.48iT - 43T^{2} \)
47 \( 1 - 6.32T + 47T^{2} \)
53 \( 1 + 6.92iT - 53T^{2} \)
59 \( 1 - 6.32T + 59T^{2} \)
61 \( 1 + 12.6iT - 61T^{2} \)
67 \( 1 - 5.47iT - 67T^{2} \)
71 \( 1 + 6.32T + 71T^{2} \)
73 \( 1 - 6.92T + 73T^{2} \)
79 \( 1 + 3.16iT - 79T^{2} \)
83 \( 1 - 2iT - 83T^{2} \)
89 \( 1 - 3.16T + 89T^{2} \)
97 \( 1 + 10.9iT - 97T^{2} \)
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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

−11.35198012377929920574333628112, −10.12493297211075936001386763201, −9.886532309311760695404841948919, −9.013626895289371552046926145534, −7.70851702315243312919994589241, −6.61753827473710211240832930768, −5.65862297782407298765784955235, −5.32975212600159920172135149651, −3.38943196111494385344942694132, −2.38675107297204637427583589064, 1.08088981180730255606093189827, 2.67657685307763313167054515563, 2.91334775514817237785793579861, 5.38650753307483782436748667287, 6.16879912377203412418444252921, 6.94160050809103668556387921843, 7.927949118517179706023419315699, 9.372041754519405413626908772971, 10.19130871802535883930115025447, 10.51494909281823014796089045830

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