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Properties

Label 2-429-33.32-c1-0-17
Degree $2$
Conductor $429$
Sign $0.885 + 0.464i$
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  − 1.96·2-s + (−1.16 + 1.27i)3-s + 1.85·4-s − 3.33i·5-s + (2.29 − 2.51i)6-s + 4.49i·7-s + 0.292·8-s + (−0.275 − 2.98i)9-s + 6.55i·10-s + (−3.20 + 0.840i)11-s + (−2.16 + 2.36i)12-s − i·13-s − 8.82i·14-s + (4.27 + 3.89i)15-s − 4.27·16-s + 0.227·17-s + ⋯
L(s)  = 1  − 1.38·2-s + (−0.673 + 0.738i)3-s + 0.925·4-s − 1.49i·5-s + (0.935 − 1.02i)6-s + 1.70i·7-s + 0.103·8-s + (−0.0918 − 0.995i)9-s + 2.07i·10-s + (−0.967 + 0.253i)11-s + (−0.623 + 0.683i)12-s − 0.277i·13-s − 2.35i·14-s + (1.10 + 1.00i)15-s − 1.06·16-s + 0.0551·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.885 + 0.464i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.885 + 0.464i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $0.885 + 0.464i$
Analytic conductor: \(3.42558\)
Root analytic conductor: \(1.85083\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{429} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 429,\ (\ :1/2),\ 0.885 + 0.464i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.419646 - 0.103387i\)
\(L(\frac12)\) \(\approx\) \(0.419646 - 0.103387i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.16 - 1.27i)T \)
11 \( 1 + (3.20 - 0.840i)T \)
13 \( 1 + iT \)
good2 \( 1 + 1.96T + 2T^{2} \)
5 \( 1 + 3.33iT - 5T^{2} \)
7 \( 1 - 4.49iT - 7T^{2} \)
17 \( 1 - 0.227T + 17T^{2} \)
19 \( 1 + 4.91iT - 19T^{2} \)
23 \( 1 + 0.962iT - 23T^{2} \)
29 \( 1 - 0.116T + 29T^{2} \)
31 \( 1 - 9.05T + 31T^{2} \)
37 \( 1 - 10.9T + 37T^{2} \)
41 \( 1 - 5.48T + 41T^{2} \)
43 \( 1 + 2.48iT - 43T^{2} \)
47 \( 1 + 2.09iT - 47T^{2} \)
53 \( 1 + 11.2iT - 53T^{2} \)
59 \( 1 - 5.88iT - 59T^{2} \)
61 \( 1 + 10.6iT - 61T^{2} \)
67 \( 1 - 8.34T + 67T^{2} \)
71 \( 1 + 11.2iT - 71T^{2} \)
73 \( 1 + 4.41iT - 73T^{2} \)
79 \( 1 + 0.735iT - 79T^{2} \)
83 \( 1 - 9.78T + 83T^{2} \)
89 \( 1 - 11.4iT - 89T^{2} \)
97 \( 1 + 6.53T + 97T^{2} \)
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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

−10.94381351077713455505943966134, −9.809538936989545548178301474300, −9.323755824794544358225693199524, −8.609636029579207829139317367017, −7.972564750876115855318861755631, −6.31145638803299200311554550330, −5.19728607390583953373933210691, −4.68742283413422485704731894404, −2.45497900917531647570720585694, −0.61470715962693333294134931931, 1.02003458576865405601265736829, 2.63036932975221059625728833948, 4.31477573988134370070027006919, 6.10488739311198770654685878947, 6.91894833820349286119401160293, 7.66230554811879444762736436412, 8.011098317386144955513952763058, 9.861359850094958491013019226964, 10.37908195138208814374070440016, 10.90493678628307454361889848619

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