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Properties

Label 2-429-429.263-c1-0-48
Degree $2$
Conductor $429$
Sign $-0.999 + 0.00594i$
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  + (−1.72 − 0.158i)3-s + (1 − 1.73i)4-s − 2.82i·5-s + (−3.12 − 1.80i)7-s + (2.94 + 0.548i)9-s + (−0.562 + 3.26i)11-s + (−2 + 2.82i)12-s + (−3.12 − 1.80i)13-s + (−0.449 + 4.87i)15-s + (−1.99 − 3.46i)16-s + (−2.54 + 4.41i)17-s + (6.24 + 3.60i)19-s + (−4.89 − 2.82i)20-s + (5.09 + 3.60i)21-s + (−1.22 + 0.707i)23-s + ⋯
L(s)  = 1  + (−0.995 − 0.0917i)3-s + (0.5 − 0.866i)4-s − 1.26i·5-s + (−1.18 − 0.681i)7-s + (0.983 + 0.182i)9-s + (−0.169 + 0.985i)11-s + (−0.577 + 0.816i)12-s + (−0.866 − 0.499i)13-s + (−0.116 + 1.25i)15-s + (−0.499 − 0.866i)16-s + (−0.618 + 1.07i)17-s + (1.43 + 0.827i)19-s + (−1.09 − 0.632i)20-s + (1.11 + 0.786i)21-s + (−0.255 + 0.147i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.00594i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.00594i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $-0.999 + 0.00594i$
Analytic conductor: \(3.42558\)
Root analytic conductor: \(1.85083\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{429} (263, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 429,\ (\ :1/2),\ -0.999 + 0.00594i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00169339 - 0.569352i\)
\(L(\frac12)\) \(\approx\) \(0.00169339 - 0.569352i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.72 + 0.158i)T \)
11 \( 1 + (0.562 - 3.26i)T \)
13 \( 1 + (3.12 + 1.80i)T \)
good2 \( 1 + (-1 + 1.73i)T^{2} \)
5 \( 1 + 2.82iT - 5T^{2} \)
7 \( 1 + (3.12 + 1.80i)T + (3.5 + 6.06i)T^{2} \)
17 \( 1 + (2.54 - 4.41i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-6.24 - 3.60i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.22 - 0.707i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.54 + 4.41i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - T + 31T^{2} \)
37 \( 1 + (4 + 6.92i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (5.09 + 8.83i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.12 - 1.80i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + 9.89iT - 47T^{2} \)
53 \( 1 - 4.24iT - 53T^{2} \)
59 \( 1 + (3.67 + 2.12i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.12 + 1.80i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.5 - 11.2i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-2.44 - 1.41i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 10.8iT - 73T^{2} \)
79 \( 1 - 3.60iT - 79T^{2} \)
83 \( 1 + 5.09T + 83T^{2} \)
89 \( 1 + (-2.44 + 1.41i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-5.5 + 9.52i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.42562333882064802232152971000, −10.05357000131143266600460995613, −9.298494742675282883921122763051, −7.61425281220202034399162837160, −6.88831672849265148513168737874, −5.78950583657860183493568946833, −5.14702706892100967251821211383, −4.01173915255173045493345171300, −1.80139215713748479751806087694, −0.38771309251265738167860752431, 2.72230569829365717372483381548, 3.30877193684492185928301777750, 4.99866506532959088931104009007, 6.31518322655139844187293936273, 6.78814390401612768204692394836, 7.53178069266353664534866798935, 9.114716289566780964719273703890, 9.906416413709523470896409251657, 11.02899011294720725706509692048, 11.52481283391196278256142469175

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