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Properties

Label 2-429-11.3-c1-0-6
Degree $2$
Conductor $429$
Sign $0.505 - 0.862i$
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  + (1.19 − 0.865i)2-s + (0.309 + 0.951i)3-s + (0.0521 − 0.160i)4-s + (−0.917 − 0.666i)5-s + (1.19 + 0.865i)6-s + (−1.26 + 3.88i)7-s + (0.833 + 2.56i)8-s + (−0.809 + 0.587i)9-s − 1.66·10-s + (−1.60 + 2.90i)11-s + 0.168·12-s + (0.809 − 0.587i)13-s + (1.85 + 5.71i)14-s + (0.350 − 1.07i)15-s + (3.48 + 2.53i)16-s + (2.56 + 1.86i)17-s + ⋯
L(s)  = 1  + (0.842 − 0.612i)2-s + (0.178 + 0.549i)3-s + (0.0260 − 0.0802i)4-s + (−0.410 − 0.297i)5-s + (0.486 + 0.353i)6-s + (−0.476 + 1.46i)7-s + (0.294 + 0.906i)8-s + (−0.269 + 0.195i)9-s − 0.527·10-s + (−0.484 + 0.874i)11-s + 0.0487·12-s + (0.224 − 0.163i)13-s + (0.496 + 1.52i)14-s + (0.0904 − 0.278i)15-s + (0.871 + 0.633i)16-s + (0.622 + 0.452i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.505 - 0.862i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.505 - 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $0.505 - 0.862i$
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: \((2,\ 429,\ (\ :1/2),\ 0.505 - 0.862i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.60703 + 0.920896i\)
\(L(\frac12)\) \(\approx\) \(1.60703 + 0.920896i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.60 - 2.90i)T \)
13 \( 1 + (-0.809 + 0.587i)T \)
good2 \( 1 + (-1.19 + 0.865i)T + (0.618 - 1.90i)T^{2} \)
5 \( 1 + (0.917 + 0.666i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (1.26 - 3.88i)T + (-5.66 - 4.11i)T^{2} \)
17 \( 1 + (-2.56 - 1.86i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (2.63 + 8.10i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 7.96T + 23T^{2} \)
29 \( 1 + (-0.151 + 0.465i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-3.62 + 2.63i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (2.89 - 8.91i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-2.30 - 7.09i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 7.71T + 43T^{2} \)
47 \( 1 + (0.00750 + 0.0231i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (1.60 - 1.16i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-1.41 + 4.35i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (10.3 + 7.48i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 1.26T + 67T^{2} \)
71 \( 1 + (0.793 + 0.576i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (0.843 - 2.59i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-11.6 + 8.48i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-3.10 - 2.25i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 17.8T + 89T^{2} \)
97 \( 1 + (-0.293 + 0.213i)T + (29.9 - 92.2i)T^{2} \)
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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

−11.49916193496909140212949985967, −10.66417463616524120193934582865, −9.466483860466205941134198772687, −8.715411878149814322013644195910, −7.86961558207472675080098041961, −6.32840722229460163791934511303, −5.05898053512173916394897952801, −4.57985226601865290308730119020, −3.13914880592576130766592350710, −2.45070009991835878113803076549, 0.943849441341204632174659142573, 3.33236595077257391597523443918, 3.98187987799753006799651563268, 5.40698262428236815191910452621, 6.32138771532619510697113576199, 7.26267850245386688987838279609, 7.72250583756798713505526586429, 9.127086292783782003867787413625, 10.40726179417370996240819123463, 10.83573550669320413992335886869

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