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Properties

Label 2-429-33.17-c1-0-11
Degree $2$
Conductor $429$
Sign $-0.830 + 0.556i$
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  + (−0.867 − 2.66i)2-s + (−1.64 + 0.556i)3-s + (−4.75 + 3.45i)4-s + (1.12 + 0.366i)5-s + (2.90 + 3.89i)6-s + (−1.16 − 1.60i)7-s + (8.79 + 6.39i)8-s + (2.38 − 1.82i)9-s − 3.32i·10-s + (3.07 + 1.25i)11-s + (5.87 − 8.30i)12-s + (0.951 − 0.309i)13-s + (−3.27 + 4.50i)14-s + (−2.05 + 0.0264i)15-s + (5.79 − 17.8i)16-s + (0.208 − 0.640i)17-s + ⋯
L(s)  = 1  + (−0.613 − 1.88i)2-s + (−0.946 + 0.321i)3-s + (−2.37 + 1.72i)4-s + (0.503 + 0.163i)5-s + (1.18 + 1.59i)6-s + (−0.440 − 0.606i)7-s + (3.10 + 2.25i)8-s + (0.793 − 0.608i)9-s − 1.05i·10-s + (0.926 + 0.377i)11-s + (1.69 − 2.39i)12-s + (0.263 − 0.0857i)13-s + (−0.874 + 1.20i)14-s + (−0.529 + 0.00684i)15-s + (1.44 − 4.46i)16-s + (0.0505 − 0.155i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.830 + 0.556i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.830 + 0.556i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $-0.830 + 0.556i$
Analytic conductor: \(3.42558\)
Root analytic conductor: \(1.85083\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{429} (248, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 429,\ (\ :1/2),\ -0.830 + 0.556i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.185268 - 0.609098i\)
\(L(\frac12)\) \(\approx\) \(0.185268 - 0.609098i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.64 - 0.556i)T \)
11 \( 1 + (-3.07 - 1.25i)T \)
13 \( 1 + (-0.951 + 0.309i)T \)
good2 \( 1 + (0.867 + 2.66i)T + (-1.61 + 1.17i)T^{2} \)
5 \( 1 + (-1.12 - 0.366i)T + (4.04 + 2.93i)T^{2} \)
7 \( 1 + (1.16 + 1.60i)T + (-2.16 + 6.65i)T^{2} \)
17 \( 1 + (-0.208 + 0.640i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (2.70 - 3.71i)T + (-5.87 - 18.0i)T^{2} \)
23 \( 1 + 3.03iT - 23T^{2} \)
29 \( 1 + (-3.53 + 2.56i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (1.02 + 3.14i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-9.07 + 6.59i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (1.37 + 0.998i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 0.610iT - 43T^{2} \)
47 \( 1 + (-7.17 + 9.87i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (-6.81 + 2.21i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (-4.01 - 5.51i)T + (-18.2 + 56.1i)T^{2} \)
61 \( 1 + (-4.63 - 1.50i)T + (49.3 + 35.8i)T^{2} \)
67 \( 1 + 8.07T + 67T^{2} \)
71 \( 1 + (-2.24 - 0.730i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (1.81 + 2.49i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (5.52 - 1.79i)T + (63.9 - 46.4i)T^{2} \)
83 \( 1 + (-1.50 + 4.64i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 - 6.98iT - 89T^{2} \)
97 \( 1 + (2.07 + 6.38i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.64269591327096028031083780178, −10.12688783339754801920403524785, −9.569570200013163947819887087674, −8.555363494850076653860967423657, −7.19538128868725774715423448365, −5.85154723596418611954263310131, −4.31569609125378697519137454983, −3.82216987604846608999321787430, −2.15301293339049517162890935511, −0.71108136621704736934124364438, 1.18913175585960356947455192286, 4.32696454564819287855288806893, 5.38667519296868194900414916757, 6.15323809000710020491230956036, 6.60945049121502621684468199727, 7.63943881705518188244818809815, 8.769784299877228490858624885486, 9.380540231286327692384476199644, 10.26407941602077691017006177239, 11.38028881128880929272943671023

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