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Properties

Label 2-429-13.4-c1-0-22
Degree $2$
Conductor $429$
Sign $-0.294 + 0.955i$
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  + (0.219 + 0.126i)2-s + (0.5 − 0.866i)3-s + (−0.967 − 1.67i)4-s − 0.770i·5-s + (0.219 − 0.126i)6-s + (2.38 − 1.37i)7-s − 0.995i·8-s + (−0.499 − 0.866i)9-s + (0.0975 − 0.168i)10-s + (−0.866 − 0.5i)11-s − 1.93·12-s + (−1.10 + 3.43i)13-s + 0.697·14-s + (−0.667 − 0.385i)15-s + (−1.80 + 3.13i)16-s + (−3.27 − 5.68i)17-s + ⋯
L(s)  = 1  + (0.154 + 0.0894i)2-s + (0.288 − 0.499i)3-s + (−0.483 − 0.838i)4-s − 0.344i·5-s + (0.0894 − 0.0516i)6-s + (0.902 − 0.521i)7-s − 0.352i·8-s + (−0.166 − 0.288i)9-s + (0.0308 − 0.0534i)10-s + (−0.261 − 0.150i)11-s − 0.558·12-s + (−0.307 + 0.951i)13-s + 0.186·14-s + (−0.172 − 0.0995i)15-s + (−0.452 + 0.783i)16-s + (−0.795 − 1.37i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.294 + 0.955i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.294 + 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $-0.294 + 0.955i$
Analytic conductor: \(3.42558\)
Root analytic conductor: \(1.85083\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{429} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 429,\ (\ :1/2),\ -0.294 + 0.955i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.852438 - 1.15509i\)
\(L(\frac12)\) \(\approx\) \(0.852438 - 1.15509i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.5 + 0.866i)T \)
11 \( 1 + (0.866 + 0.5i)T \)
13 \( 1 + (1.10 - 3.43i)T \)
good2 \( 1 + (-0.219 - 0.126i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + 0.770iT - 5T^{2} \)
7 \( 1 + (-2.38 + 1.37i)T + (3.5 - 6.06i)T^{2} \)
17 \( 1 + (3.27 + 5.68i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.45 - 0.841i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.866 + 1.50i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.81 + 3.13i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 6.59iT - 31T^{2} \)
37 \( 1 + (-9.18 - 5.30i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.10 - 0.638i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.83 + 4.90i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 8.64iT - 47T^{2} \)
53 \( 1 + 3.28T + 53T^{2} \)
59 \( 1 + (-1.86 + 1.07i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.45 - 9.45i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-9.08 - 5.24i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-9.21 + 5.32i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 0.341iT - 73T^{2} \)
79 \( 1 - 3.17T + 79T^{2} \)
83 \( 1 + 8.70iT - 83T^{2} \)
89 \( 1 + (14.2 + 8.24i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-8.47 + 4.89i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.04073012952713948563714939919, −9.845452310347058610908970617242, −9.090382971003372734310465376927, −8.190566007380182211936125187442, −7.14103244516210842726057139046, −6.20283829647438279612202779935, −4.87051462096159120716464287804, −4.35380736113180530827606513878, −2.33244526416975849042840574321, −0.893194563316075108772099123983, 2.33687893779483103913822112528, 3.43908961157907815214195244798, 4.59208236145681922635707597414, 5.37849231569264530785723747031, 6.90832566794288118498993714765, 8.223950777190103400953722358603, 8.396320966413361384481659815287, 9.550936697000382983780828680917, 10.71592507472204531210808605642, 11.25021174554491205855763369320

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