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Properties

Label 2-429-429.194-c1-0-36
Degree $2$
Conductor $429$
Sign $0.862 + 0.506i$
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  + (−1.54 + 2.12i)2-s + (0.535 − 1.64i)3-s + (−1.51 − 4.65i)4-s + (3.60 − 2.61i)5-s + (2.67 + 3.67i)6-s + (7.22 + 2.34i)8-s + (−2.42 − 1.76i)9-s + 11.6i·10-s + (−2.32 + 2.36i)11-s − 8.47·12-s + (2.11 − 2.91i)13-s + (−2.38 − 7.33i)15-s + (−8.22 + 5.97i)16-s + (7.49 − 2.43i)18-s + (−17.6 − 12.8i)20-s + ⋯
L(s)  = 1  + (−1.09 + 1.50i)2-s + (0.309 − 0.951i)3-s + (−0.756 − 2.32i)4-s + (1.61 − 1.17i)5-s + (1.09 + 1.50i)6-s + (2.55 + 0.830i)8-s + (−0.809 − 0.587i)9-s + 3.69i·10-s + (−0.700 + 0.714i)11-s − 2.44·12-s + (0.587 − 0.809i)13-s + (−0.615 − 1.89i)15-s + (−2.05 + 1.49i)16-s + (1.76 − 0.573i)18-s + (−3.94 − 2.86i)20-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $0.862 + 0.506i$
Analytic conductor: \(3.42558\)
Root analytic conductor: \(1.85083\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{429} (194, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 429,\ (\ :1/2),\ 0.862 + 0.506i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.975267 - 0.265129i\)
\(L(\frac12)\) \(\approx\) \(0.975267 - 0.265129i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.32 - 2.36i)T \)
13 \( 1 + (-2.11 + 2.91i)T \)
good2 \( 1 + (1.54 - 2.12i)T + (-0.618 - 1.90i)T^{2} \)
5 \( 1 + (-3.60 + 2.61i)T + (1.54 - 4.75i)T^{2} \)
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 \( 1 + (9.64 + 3.13i)T + (33.1 + 24.0i)T^{2} \)
43 \( 1 + 0.0553iT - 43T^{2} \)
47 \( 1 + (0.816 - 2.51i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (-2.90 - 8.94i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-8.21 - 11.3i)T + (-18.8 + 58.0i)T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 + (-13.0 + 9.47i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-3.26 + 4.49i)T + (-24.4 - 75.1i)T^{2} \)
83 \( 1 + (4.66 + 6.41i)T + (-25.6 + 78.9i)T^{2} \)
89 \( 1 - 4.31T + 89T^{2} \)
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

−10.44728649518552806857773801971, −9.824892581010457317565010856830, −8.873266752318680162467484698446, −8.440481999577501422733846036022, −7.48465881295511490726373590366, −6.47599609189318813991209083722, −5.69650113364171065298145123778, −5.07088617932477524087519741090, −2.08884899655730498554734086369, −0.925836154199858621818724476920, 1.93276120662023855422499215399, 2.82522446730605939876443239820, 3.66522333865344385142514115017, 5.34212182284311481521272227570, 6.66554408622749837292453494552, 8.151713014698443222955823687903, 9.000845749688957653434743766191, 9.751758915920929355362712131228, 10.23574522813647940158258104297, 11.00956753241710731127643704670

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