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Properties

Label 2-429-429.272-c1-0-37
Degree $2$
Conductor $429$
Sign $0.506 + 0.862i$
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  + (0.618 + 0.850i)2-s + (−0.535 − 1.64i)3-s + (0.276 − 0.850i)4-s + (0.319 + 0.232i)5-s + (1.07 − 1.47i)6-s + (2.89 − 0.940i)8-s + (−2.42 + 1.76i)9-s + 0.415i·10-s + (2.36 − 2.32i)11-s − 1.54·12-s + (−2.11 − 2.91i)13-s + (0.211 − 0.651i)15-s + (1.14 + 0.830i)16-s + (−3.00 − 0.975i)18-s + (0.285 − 0.207i)20-s + ⋯
L(s)  = 1  + (0.437 + 0.601i)2-s + (−0.309 − 0.951i)3-s + (0.138 − 0.425i)4-s + (0.143 + 0.103i)5-s + (0.437 − 0.601i)6-s + (1.02 − 0.332i)8-s + (−0.809 + 0.587i)9-s + 0.131i·10-s + (0.714 − 0.700i)11-s − 0.446·12-s + (−0.587 − 0.809i)13-s + (0.0546 − 0.168i)15-s + (0.285 + 0.207i)16-s + (−0.707 − 0.229i)18-s + (0.0639 − 0.0464i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.506 + 0.862i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.506 + 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $0.506 + 0.862i$
Analytic conductor: \(3.42558\)
Root analytic conductor: \(1.85083\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{429} (272, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 429,\ (\ :1/2),\ 0.506 + 0.862i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.45441 - 0.832667i\)
\(L(\frac12)\) \(\approx\) \(1.45441 - 0.832667i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.535 + 1.64i)T \)
11 \( 1 + (-2.36 + 2.32i)T \)
13 \( 1 + (2.11 + 2.91i)T \)
good2 \( 1 + (-0.618 - 0.850i)T + (-0.618 + 1.90i)T^{2} \)
5 \( 1 + (-0.319 - 0.232i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (-5.66 - 4.11i)T^{2} \)
17 \( 1 + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-9.57 + 29.4i)T^{2} \)
37 \( 1 + (29.9 + 21.7i)T^{2} \)
41 \( 1 + (-7.44 + 2.41i)T + (33.1 - 24.0i)T^{2} \)
43 \( 1 - 0.0553iT - 43T^{2} \)
47 \( 1 + (-4.15 - 12.7i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (3.75 - 11.5i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (8.21 - 11.3i)T + (-18.8 - 58.0i)T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 + (-3.98 - 2.89i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (3.26 + 4.49i)T + (-24.4 + 75.1i)T^{2} \)
83 \( 1 + (9.64 - 13.2i)T + (-25.6 - 78.9i)T^{2} \)
89 \( 1 - 18.3T + 89T^{2} \)
97 \( 1 + (-29.9 + 92.2i)T^{2} \)
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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

−11.03391356796967773038989390558, −10.33271576961322165415732487127, −9.078340058072406062949523572858, −7.87076892692377878489809206983, −7.17274588946690693865020530272, −6.10344389819023993189144379512, −5.72294133036788258129629328105, −4.41603384121380204107631471906, −2.62628612251833137613855149610, −1.04571158564762353752651691965, 2.05977639282217870160474157668, 3.48844238747366427922121314801, 4.33096391247918015027268311121, 5.16455349338143769983732310255, 6.55906579876787024498629788054, 7.59080262843378539421656976975, 8.917442918936881792561879984530, 9.627509938120873424100379257446, 10.55254499525732652522850275433, 11.47757901034143718113723376497

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