Properties

Label 2-429-13.12-c1-0-18
Degree $2$
Conductor $429$
Sign $-0.204 + 0.978i$
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  − 1.42i·2-s + 3-s − 0.0397·4-s − 0.0606i·5-s − 1.42i·6-s − 1.70i·7-s − 2.79i·8-s + 9-s − 0.0866·10-s + i·11-s − 0.0397·12-s + (3.52 + 0.738i)13-s − 2.43·14-s − 0.0606i·15-s − 4.07·16-s − 3.75·17-s + ⋯
L(s)  = 1  − 1.00i·2-s + 0.577·3-s − 0.0198·4-s − 0.0271i·5-s − 0.583i·6-s − 0.644i·7-s − 0.989i·8-s + 0.333·9-s − 0.0273·10-s + 0.301i·11-s − 0.0114·12-s + (0.978 + 0.204i)13-s − 0.651·14-s − 0.0156i·15-s − 1.01·16-s − 0.910·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.204 + 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.204 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $-0.204 + 0.978i$
Analytic conductor: \(3.42558\)
Root analytic conductor: \(1.85083\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{429} (298, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 429,\ (\ :1/2),\ -0.204 + 0.978i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.17348 - 1.44444i\)
\(L(\frac12)\) \(\approx\) \(1.17348 - 1.44444i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T \)
11 \( 1 - iT \)
13 \( 1 + (-3.52 - 0.738i)T \)
good2 \( 1 + 1.42iT - 2T^{2} \)
5 \( 1 + 0.0606iT - 5T^{2} \)
7 \( 1 + 1.70iT - 7T^{2} \)
17 \( 1 + 3.75T + 17T^{2} \)
19 \( 1 + 2.02iT - 19T^{2} \)
23 \( 1 + 0.704T + 23T^{2} \)
29 \( 1 + 0.0346T + 29T^{2} \)
31 \( 1 + 1.85iT - 31T^{2} \)
37 \( 1 - 8.82iT - 37T^{2} \)
41 \( 1 + 3.32iT - 41T^{2} \)
43 \( 1 + 5.29T + 43T^{2} \)
47 \( 1 - 6.04iT - 47T^{2} \)
53 \( 1 + 10.6T + 53T^{2} \)
59 \( 1 - 3.34iT - 59T^{2} \)
61 \( 1 - 3.26T + 61T^{2} \)
67 \( 1 - 10.9iT - 67T^{2} \)
71 \( 1 + 1.96iT - 71T^{2} \)
73 \( 1 - 15.3iT - 73T^{2} \)
79 \( 1 - 10.5T + 79T^{2} \)
83 \( 1 + 4.19iT - 83T^{2} \)
89 \( 1 - 14.3iT - 89T^{2} \)
97 \( 1 + 5.86iT - 97T^{2} \)
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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

−10.93238376607560255482925802290, −10.18467038853085726941499780358, −9.280927799198286336048733680358, −8.381457916197158578566280326651, −7.14210775191071345324748184735, −6.43491688752738045578483380283, −4.60536327221685497106649791888, −3.69133482550734492073584201251, −2.59497172826488486684952458292, −1.28293401292494700422053799570, 2.06132777468751451878608717052, 3.37923803072449927445123238955, 4.88988733306491757804696045746, 5.98089640186134828577182169959, 6.69937063470640473028116709772, 7.79528606982338850504969311243, 8.573558917508861084022221161173, 9.138511178296992250131426622046, 10.58380355004422770085359126018, 11.31103253308247141360558306092

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