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Properties

Label 2-429-429.194-c1-0-19
Degree $2$
Conductor $429$
Sign $0.760 - 0.648i$
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  + (−1.65 + 2.28i)2-s + (−0.535 + 1.64i)3-s + (−1.84 − 5.67i)4-s + (−2.37 + 1.72i)5-s + (−2.87 − 3.95i)6-s + (10.6 + 3.45i)8-s + (−2.42 − 1.76i)9-s − 8.29i·10-s + (−1.47 − 2.96i)11-s + 10.3·12-s + (2.11 − 2.91i)13-s + (−1.57 − 4.84i)15-s + (−15.8 + 11.5i)16-s + (8.05 − 2.61i)18-s + (14.1 + 10.3i)20-s + ⋯
L(s)  = 1  + (−1.17 + 1.61i)2-s + (−0.309 + 0.951i)3-s + (−0.921 − 2.83i)4-s + (−1.06 + 0.772i)5-s + (−1.17 − 1.61i)6-s + (3.76 + 1.22i)8-s + (−0.809 − 0.587i)9-s − 2.62i·10-s + (−0.445 − 0.895i)11-s + 2.98·12-s + (0.587 − 0.809i)13-s + (−0.406 − 1.24i)15-s + (−3.97 + 2.88i)16-s + (1.89 − 0.616i)18-s + (3.17 + 2.30i)20-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(1)\) \(\approx\) \(0.257515 + 0.0948722i\)
\(L(\frac12)\) \(\approx\) \(0.257515 + 0.0948722i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.47 + 2.96i)T \)
13 \( 1 + (-2.11 + 2.91i)T \)
good2 \( 1 + (1.65 - 2.28i)T + (-0.618 - 1.90i)T^{2} \)
5 \( 1 + (2.37 - 1.72i)T + (1.54 - 4.75i)T^{2} \)
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 \( 1 - 7.66iT - 43T^{2} \)
47 \( 1 + (-0.508 + 1.56i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (3.92 + 12.0i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (1.35 + 1.86i)T + (-18.8 + 58.0i)T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 + (-3.98 + 2.89i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-10.4 + 14.3i)T + (-24.4 - 75.1i)T^{2} \)
83 \( 1 + (5.06 + 6.96i)T + (-25.6 + 78.9i)T^{2} \)
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

−10.76035812022483856811340150598, −10.35771277870126049240575814233, −9.238811508370142389989838944946, −8.336816654060918413166028942729, −7.79505831713904167446140864465, −6.64642028458115366140684167681, −5.82842545387085133668400706368, −4.88348192508369429489200271427, −3.50616206849810208908346713467, −0.32463278345443298804735821644, 1.13429859170229279608004978609, 2.32267608123293903326978400994, 3.80105644657510473438254703400, 4.80065521802622091972701169401, 6.99432816334917537330967309784, 7.75296084552967754525627742948, 8.469947985132011192472773476570, 9.171724689324529380244252306811, 10.37408277937491875051189192178, 11.19994124358339194245902274700

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