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Properties

Label 2-429-143.21-c1-0-2
Degree $2$
Conductor $429$
Sign $-0.912 - 0.409i$
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  + (−1.43 + 1.43i)2-s + 3-s − 2.12i·4-s + (−2.46 − 2.46i)5-s + (−1.43 + 1.43i)6-s + (0.806 + 0.806i)7-s + (0.184 + 0.184i)8-s + 9-s + 7.09·10-s + (0.500 + 3.27i)11-s − 2.12i·12-s + (−0.994 + 3.46i)13-s − 2.31·14-s + (−2.46 − 2.46i)15-s + 3.72·16-s − 1.84·17-s + ⋯
L(s)  = 1  + (−1.01 + 1.01i)2-s + 0.577·3-s − 1.06i·4-s + (−1.10 − 1.10i)5-s + (−0.586 + 0.586i)6-s + (0.304 + 0.304i)7-s + (0.0653 + 0.0653i)8-s + 0.333·9-s + 2.24·10-s + (0.150 + 0.988i)11-s − 0.614i·12-s + (−0.275 + 0.961i)13-s − 0.619·14-s + (−0.637 − 0.637i)15-s + 0.931·16-s − 0.448·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.912 - 0.409i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.912 - 0.409i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $-0.912 - 0.409i$
Analytic conductor: \(3.42558\)
Root analytic conductor: \(1.85083\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{429} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 429,\ (\ :1/2),\ -0.912 - 0.409i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.112236 + 0.523818i\)
\(L(\frac12)\) \(\approx\) \(0.112236 + 0.523818i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T \)
11 \( 1 + (-0.500 - 3.27i)T \)
13 \( 1 + (0.994 - 3.46i)T \)
good2 \( 1 + (1.43 - 1.43i)T - 2iT^{2} \)
5 \( 1 + (2.46 + 2.46i)T + 5iT^{2} \)
7 \( 1 + (-0.806 - 0.806i)T + 7iT^{2} \)
17 \( 1 + 1.84T + 17T^{2} \)
19 \( 1 + (5.23 - 5.23i)T - 19iT^{2} \)
23 \( 1 - 8.45iT - 23T^{2} \)
29 \( 1 + 5.73iT - 29T^{2} \)
31 \( 1 + (-4.19 - 4.19i)T + 31iT^{2} \)
37 \( 1 + (-1.19 + 1.19i)T - 37iT^{2} \)
41 \( 1 + (1.04 - 1.04i)T - 41iT^{2} \)
43 \( 1 - 2.53T + 43T^{2} \)
47 \( 1 + (2.58 - 2.58i)T - 47iT^{2} \)
53 \( 1 - 3.52T + 53T^{2} \)
59 \( 1 + (8.26 - 8.26i)T - 59iT^{2} \)
61 \( 1 - 2.26iT - 61T^{2} \)
67 \( 1 + (10.7 + 10.7i)T + 67iT^{2} \)
71 \( 1 + (-3.91 - 3.91i)T + 71iT^{2} \)
73 \( 1 + (7.89 + 7.89i)T + 73iT^{2} \)
79 \( 1 - 3.25iT - 79T^{2} \)
83 \( 1 + (2.19 - 2.19i)T - 83iT^{2} \)
89 \( 1 + (-11.1 + 11.1i)T - 89iT^{2} \)
97 \( 1 + (-2.64 - 2.64i)T + 97iT^{2} \)
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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

−11.76539805488492606882778100253, −10.19691471402025156091603115016, −9.252714074951383452192727747126, −8.751224905701945189427489400961, −7.892337368515399478288446133222, −7.41887730990144878845409774427, −6.22827301365270681485925467269, −4.76011665042640400492935395948, −3.89378697672065690426350121497, −1.67025369268776095936640513437, 0.44777317666177679440991133030, 2.53441584342265592327351078754, 3.19096215832144063333422116590, 4.41933726340357605964124851141, 6.39299084796769127301921753507, 7.45825605152807336976467017434, 8.346380396814733065601812058615, 8.801476649409060898402383668727, 10.17141033491802326740655898756, 10.86398024213903798046638763971

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