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Properties

Label 2-429-11.3-c1-0-11
Degree $2$
Conductor $429$
Sign $0.984 + 0.174i$
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  + (1.21 − 0.879i)2-s + (−0.309 − 0.951i)3-s + (0.0742 − 0.228i)4-s + (2.08 + 1.51i)5-s + (−1.21 − 0.879i)6-s + (−0.780 + 2.40i)7-s + (0.813 + 2.50i)8-s + (−0.809 + 0.587i)9-s + 3.85·10-s + (3.27 + 0.499i)11-s − 0.240·12-s + (0.809 − 0.587i)13-s + (1.16 + 3.59i)14-s + (0.796 − 2.45i)15-s + (3.57 + 2.59i)16-s + (−0.636 − 0.462i)17-s + ⋯
L(s)  = 1  + (0.856 − 0.622i)2-s + (−0.178 − 0.549i)3-s + (0.0371 − 0.114i)4-s + (0.932 + 0.677i)5-s + (−0.494 − 0.359i)6-s + (−0.295 + 0.908i)7-s + (0.287 + 0.885i)8-s + (−0.269 + 0.195i)9-s + 1.21·10-s + (0.988 + 0.150i)11-s − 0.0693·12-s + (0.224 − 0.163i)13-s + (0.312 + 0.961i)14-s + (0.205 − 0.632i)15-s + (0.894 + 0.649i)16-s + (−0.154 − 0.112i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 + 0.174i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.984 + 0.174i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $0.984 + 0.174i$
Analytic conductor: \(3.42558\)
Root analytic conductor: \(1.85083\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{429} (157, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 429,\ (\ :1/2),\ 0.984 + 0.174i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.25755 - 0.197997i\)
\(L(\frac12)\) \(\approx\) \(2.25755 - 0.197997i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.309 + 0.951i)T \)
11 \( 1 + (-3.27 - 0.499i)T \)
13 \( 1 + (-0.809 + 0.587i)T \)
good2 \( 1 + (-1.21 + 0.879i)T + (0.618 - 1.90i)T^{2} \)
5 \( 1 + (-2.08 - 1.51i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (0.780 - 2.40i)T + (-5.66 - 4.11i)T^{2} \)
17 \( 1 + (0.636 + 0.462i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (1.94 + 5.99i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 4.00T + 23T^{2} \)
29 \( 1 + (-1.37 + 4.24i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (5.52 - 4.01i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-1.79 + 5.52i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (0.313 + 0.966i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 6.60T + 43T^{2} \)
47 \( 1 + (0.938 + 2.88i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-4.00 + 2.90i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (0.304 - 0.938i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-7.67 - 5.57i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 1.72T + 67T^{2} \)
71 \( 1 + (5.68 + 4.12i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-2.33 + 7.19i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (3.55 - 2.58i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (11.6 + 8.46i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 12.7T + 89T^{2} \)
97 \( 1 + (8.68 - 6.30i)T + (29.9 - 92.2i)T^{2} \)
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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

−11.39233263293222397708148839445, −10.55254397301294757214003168398, −9.354923609861794628679423893622, −8.559699462290816788686352359235, −7.14593873832485135264162259767, −6.19480149879424626171448824304, −5.48930194075778659635217504727, −4.11827162165329685621091420598, −2.75175043686064128150207672145, −2.04278025727522896802144109606, 1.37494473303161638727798214457, 3.76918180955949196305338542369, 4.33632064919941586165522896227, 5.58842589334530327424399878554, 6.12870791775337165592104550625, 7.09937017225342060194856181611, 8.528291628788643320519598111509, 9.653554029684156211755228298041, 10.03213034219846508673040937892, 11.12053996970784255471389774322

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