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Properties

Label 2-429-33.32-c1-0-39
Degree $2$
Conductor $429$
Sign $0.543 + 0.839i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.543 + 0.839i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.543 + 0.839i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $0.543 + 0.839i$
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: \((2,\ 429,\ (\ :1/2),\ 0.543 + 0.839i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.87296 - 1.01793i\)
\(L(\frac12)\) \(\approx\) \(1.87296 - 1.01793i\)
\(L(\frac{3}{2})\) 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

−11.41325116956969928051928792134, −10.16616508070091523883742814406, −9.466079484275353753065631364117, −8.310554280116417010931538661924, −6.82737810357117557662911171908, −5.97677936436226310672849960357, −4.81242734138912502863503075004, −4.31431910806547571188344005779, −3.70981082712373527488599934766, −1.02936436809903096025034709533, 2.39311993904243419529965888232, 3.06208097545356499954989360787, 4.67643862331723426055555163913, 5.82052939024811994425560312780, 6.29468542261686023736767839630, 7.00598535226509728250525311739, 8.413534409564874032635958481602, 9.611216391538360321737632535707, 11.09753957986422954164978172371, 11.51464878613953087321798536624

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