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Properties

Label 2-429-33.32-c1-0-36
Degree $2$
Conductor $429$
Sign $-0.784 - 0.620i$
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  − 0.664·2-s + (−1.72 + 0.153i)3-s − 1.55·4-s − 2.45i·5-s + (1.14 − 0.101i)6-s − 3.30i·7-s + 2.36·8-s + (2.95 − 0.528i)9-s + 1.63i·10-s + (−2.27 + 2.40i)11-s + (2.68 − 0.238i)12-s + i·13-s + 2.19i·14-s + (0.376 + 4.24i)15-s + 1.54·16-s − 5.05·17-s + ⋯
L(s)  = 1  − 0.470·2-s + (−0.996 + 0.0883i)3-s − 0.778·4-s − 1.10i·5-s + (0.468 − 0.0415i)6-s − 1.24i·7-s + 0.836·8-s + (0.984 − 0.176i)9-s + 0.517i·10-s + (−0.687 + 0.726i)11-s + (0.775 − 0.0688i)12-s + 0.277i·13-s + 0.586i·14-s + (0.0972 + 1.09i)15-s + 0.385·16-s − 1.22·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.784 - 0.620i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.784 - 0.620i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $-0.784 - 0.620i$
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: \((2,\ 429,\ (\ :1/2),\ -0.784 - 0.620i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0179603 + 0.0516677i\)
\(L(\frac12)\) \(\approx\) \(0.0179603 + 0.0516677i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.72 - 0.153i)T \)
11 \( 1 + (2.27 - 2.40i)T \)
13 \( 1 - iT \)
good2 \( 1 + 0.664T + 2T^{2} \)
5 \( 1 + 2.45iT - 5T^{2} \)
7 \( 1 + 3.30iT - 7T^{2} \)
17 \( 1 + 5.05T + 17T^{2} \)
19 \( 1 + 5.14iT - 19T^{2} \)
23 \( 1 - 6.01iT - 23T^{2} \)
29 \( 1 + 7.11T + 29T^{2} \)
31 \( 1 + 7.67T + 31T^{2} \)
37 \( 1 - 4.17T + 37T^{2} \)
41 \( 1 - 7.49T + 41T^{2} \)
43 \( 1 - 7.74iT - 43T^{2} \)
47 \( 1 - 3.38iT - 47T^{2} \)
53 \( 1 - 11.3iT - 53T^{2} \)
59 \( 1 - 6.98iT - 59T^{2} \)
61 \( 1 + 1.08iT - 61T^{2} \)
67 \( 1 + 2.35T + 67T^{2} \)
71 \( 1 + 12.4iT - 71T^{2} \)
73 \( 1 + 11.1iT - 73T^{2} \)
79 \( 1 - 11.4iT - 79T^{2} \)
83 \( 1 + 15.8T + 83T^{2} \)
89 \( 1 + 6.36iT - 89T^{2} \)
97 \( 1 - 0.274T + 97T^{2} \)
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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

−10.72823881860189486290119143467, −9.471910197700006429766149371799, −9.219641460751570184941741652099, −7.69578566992965261779376314410, −7.15725542487250035602182044784, −5.58732285586435816727702220762, −4.59533405127763556614524879538, −4.22495106701028383945550966585, −1.36960328274242501689224395571, −0.05038536075927224286784156090, 2.26340423508119489251018049533, 3.88528050803899406119912847881, 5.30528565887038238369414760912, 5.92990352423047289678314651606, 7.05085553229635660475697345522, 8.135333801567252011011669847664, 9.020574231346988482677185954579, 10.08148192774143154570265852919, 10.81142573446166240027932036791, 11.38259470153402226181391071818

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