Properties

Label 2-429-11.3-c1-0-7
Degree $2$
Conductor $429$
Sign $0.958 - 0.283i$
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  + (−0.972 + 0.706i)2-s + (−0.309 − 0.951i)3-s + (−0.171 + 0.527i)4-s + (−1.69 − 1.23i)5-s + (0.972 + 0.706i)6-s + (−0.640 + 1.97i)7-s + (−0.949 − 2.92i)8-s + (−0.809 + 0.587i)9-s + 2.51·10-s + (3.15 − 1.01i)11-s + 0.555·12-s + (0.809 − 0.587i)13-s + (−0.769 − 2.36i)14-s + (−0.647 + 1.99i)15-s + (2.08 + 1.51i)16-s + (0.922 + 0.669i)17-s + ⋯
L(s)  = 1  + (−0.687 + 0.499i)2-s + (−0.178 − 0.549i)3-s + (−0.0857 + 0.263i)4-s + (−0.757 − 0.550i)5-s + (0.397 + 0.288i)6-s + (−0.242 + 0.744i)7-s + (−0.335 − 1.03i)8-s + (−0.269 + 0.195i)9-s + 0.796·10-s + (0.951 − 0.306i)11-s + 0.160·12-s + (0.224 − 0.163i)13-s + (−0.205 − 0.633i)14-s + (−0.167 + 0.514i)15-s + (0.522 + 0.379i)16-s + (0.223 + 0.162i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.958 - 0.283i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.958 - 0.283i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $0.958 - 0.283i$
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: \((2,\ 429,\ (\ :1/2),\ 0.958 - 0.283i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.743456 + 0.107613i\)
\(L(\frac12)\) \(\approx\) \(0.743456 + 0.107613i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.15 + 1.01i)T \)
13 \( 1 + (-0.809 + 0.587i)T \)
good2 \( 1 + (0.972 - 0.706i)T + (0.618 - 1.90i)T^{2} \)
5 \( 1 + (1.69 + 1.23i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (0.640 - 1.97i)T + (-5.66 - 4.11i)T^{2} \)
17 \( 1 + (-0.922 - 0.669i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-1.10 - 3.40i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 8.83T + 23T^{2} \)
29 \( 1 + (-0.795 + 2.44i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-4.77 + 3.46i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (1.41 - 4.36i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-2.27 - 7.00i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 5.87T + 43T^{2} \)
47 \( 1 + (3.44 + 10.6i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-3.70 + 2.69i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (1.80 - 5.56i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-10.3 - 7.54i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 9.04T + 67T^{2} \)
71 \( 1 + (3.60 + 2.62i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-2.38 + 7.34i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-1.64 + 1.19i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (4.55 + 3.31i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 15.7T + 89T^{2} \)
97 \( 1 + (-13.4 + 9.76i)T + (29.9 - 92.2i)T^{2} \)
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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

−11.54851249104208386813281128718, −10.04701602069433510851569924876, −8.973228864902484751523143357534, −8.494782448310299588887678970018, −7.67530091443516560011279103122, −6.70766823384437561212564502465, −5.80873294945378745257177300421, −4.31091771515166420482159383570, −3.11141731629640903546378435361, −0.932340204146043758028936901997, 0.966239149922050835953435630128, 2.97424989398067211527771541242, 4.09842042175263562187929120293, 5.20295358156832115785379766994, 6.64123329171943257045388915209, 7.41701141617003646533806318187, 8.806711459970764621696733917845, 9.367307065600546332065509269740, 10.32988693712863454761711944588, 11.06876647387453219094237374994

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