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Properties

Label 2-429-429.230-c1-0-33
Degree $2$
Conductor $429$
Sign $0.869 - 0.494i$
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  + (0.724 + 1.57i)3-s + (1 + 1.73i)4-s − 2.82i·5-s + (3.12 − 1.80i)7-s + (−1.94 + 2.28i)9-s + (3.11 − 1.14i)11-s + (−2 + 2.82i)12-s + (3.12 − 1.80i)13-s + (4.44 − 2.04i)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  + (0.418 + 0.908i)3-s + (0.5 + 0.866i)4-s − 1.26i·5-s + (1.18 − 0.681i)7-s + (−0.649 + 0.760i)9-s + (0.938 − 0.345i)11-s + (−0.577 + 0.816i)12-s + (0.866 − 0.499i)13-s + (1.14 − 0.529i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.869 - 0.494i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.869 - 0.494i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $0.869 - 0.494i$
Analytic conductor: \(3.42558\)
Root analytic conductor: \(1.85083\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{429} (230, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 429,\ (\ :1/2),\ 0.869 - 0.494i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.89494 + 0.500772i\)
\(L(\frac12)\) \(\approx\) \(1.89494 + 0.500772i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.724 - 1.57i)T \)
11 \( 1 + (-3.11 + 1.14i)T \)
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

−11.26408863378996542174381685324, −10.48820247417600445477252086263, −9.125052690439528879647916363550, −8.431502387652884506857931243553, −8.058582484783778486758595831076, −6.62606414980759346745050208688, −5.09240981749246338586608450479, −4.32457579862894571796930517625, −3.44584298042806483049951223756, −1.64386923129970370788706601215, 1.74392190924411688422833733555, 2.35741776147607944689051126225, 4.06453753758798973073707821425, 5.75569678485003744959185347471, 6.58800664428096279015314492227, 7.03923729808204186478804526717, 8.421022970832591347032313558700, 9.048580201049640688720068926963, 10.48225650399751837107838906389, 11.24160902126431458748605396292

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