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Properties

Label 2-429-429.272-c1-0-21
Degree $2$
Conductor $429$
Sign $0.648 - 0.760i$
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  + (0.111 + 0.152i)2-s + (0.535 + 1.64i)3-s + (0.607 − 1.86i)4-s + (2.72 + 1.98i)5-s + (−0.192 + 0.264i)6-s + (0.712 − 0.231i)8-s + (−2.42 + 1.76i)9-s + 0.636i·10-s + (2.96 + 1.47i)11-s + 3.40·12-s + (−2.11 − 2.91i)13-s + (−1.80 + 5.55i)15-s + (−3.06 − 2.22i)16-s + (−0.538 − 0.175i)18-s + (5.35 − 3.89i)20-s + ⋯
L(s)  = 1  + (0.0785 + 0.108i)2-s + (0.309 + 0.951i)3-s + (0.303 − 0.934i)4-s + (1.21 + 0.886i)5-s + (−0.0785 + 0.108i)6-s + (0.251 − 0.0818i)8-s + (−0.809 + 0.587i)9-s + 0.201i·10-s + (0.895 + 0.445i)11-s + 0.982·12-s + (−0.587 − 0.809i)13-s + (−0.465 + 1.43i)15-s + (−0.765 − 0.556i)16-s + (−0.127 − 0.0412i)18-s + (1.19 − 0.870i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.648 - 0.760i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.648 - 0.760i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $0.648 - 0.760i$
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: \((2,\ 429,\ (\ :1/2),\ 0.648 - 0.760i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.83014 + 0.844694i\)
\(L(\frac12)\) \(\approx\) \(1.83014 + 0.844694i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.96 - 1.47i)T \)
13 \( 1 + (2.11 + 2.91i)T \)
good2 \( 1 + (-0.111 - 0.152i)T + (-0.618 + 1.90i)T^{2} \)
5 \( 1 + (-2.72 - 1.98i)T + (1.54 + 4.75i)T^{2} \)
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 \( 1 + 7.66iT - 43T^{2} \)
47 \( 1 + (-4.20 - 12.9i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (-2.67 + 8.22i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-1.35 + 1.86i)T + (-18.8 - 58.0i)T^{2} \)
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 \( 1 + (10.4 + 14.3i)T + (-24.4 + 75.1i)T^{2} \)
83 \( 1 + (-9.43 + 12.9i)T + (-25.6 - 78.9i)T^{2} \)
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

−10.84990679558104471460761336564, −10.28269105299977932323990487708, −9.728026714779761807921515704461, −8.991159560566921413234659500979, −7.41505310753246819495714789979, −6.36040110090418693519204083295, −5.62700978604393670325096570495, −4.63732641225685534886016145502, −3.06617313498104059715331219802, −1.95540815090089519686781405421, 1.51608481896284903461680715597, 2.52341351274665785820154480109, 3.96831841483028648423893736225, 5.42130041841959963240335851125, 6.52196481848561867870836716716, 7.21457344193270771580578000085, 8.548120828694033530050935285999, 8.888868228969641703454598049247, 9.939434191287379451738744395767, 11.48066138164775784710520136056

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