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Properties

Label 2-429-429.233-c1-0-26
Degree $2$
Conductor $429$
Sign $-0.124 - 0.992i$
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  + (2.60 + 0.847i)2-s + (−1.40 + 1.01i)3-s + (4.46 + 3.24i)4-s + (0.309 + 0.951i)5-s + (−4.51 + 1.46i)6-s + (5.67 + 7.81i)8-s + (0.927 − 2.85i)9-s + 2.74i·10-s + (−0.486 − 3.28i)11-s − 9.56·12-s + (−3.42 − 1.11i)13-s + (−1.40 − 1.01i)15-s + (4.76 + 14.6i)16-s + (4.83 − 6.65i)18-s + (−1.70 + 5.25i)20-s + ⋯
L(s)  = 1  + (1.84 + 0.599i)2-s + (−0.809 + 0.587i)3-s + (2.23 + 1.62i)4-s + (0.138 + 0.425i)5-s + (−1.84 + 0.599i)6-s + (2.00 + 2.76i)8-s + (0.309 − 0.951i)9-s + 0.867i·10-s + (−0.146 − 0.989i)11-s − 2.76·12-s + (−0.951 − 0.309i)13-s + (−0.361 − 0.262i)15-s + (1.19 + 3.66i)16-s + (1.13 − 1.56i)18-s + (−0.381 + 1.17i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.124 - 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.124 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $-0.124 - 0.992i$
Analytic conductor: \(3.42558\)
Root analytic conductor: \(1.85083\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{429} (233, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 429,\ (\ :1/2),\ -0.124 - 0.992i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.09280 + 2.37270i\)
\(L(\frac12)\) \(\approx\) \(2.09280 + 2.37270i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.40 - 1.01i)T \)
11 \( 1 + (0.486 + 3.28i)T \)
13 \( 1 + (3.42 + 1.11i)T \)
good2 \( 1 + (-2.60 - 0.847i)T + (1.61 + 1.17i)T^{2} \)
5 \( 1 + (-0.309 - 0.951i)T + (-4.04 + 2.93i)T^{2} \)
7 \( 1 + (2.16 + 6.65i)T^{2} \)
17 \( 1 + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (5.87 - 18.0i)T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (25.0 + 18.2i)T^{2} \)
37 \( 1 + (-11.4 - 35.1i)T^{2} \)
41 \( 1 + (-5.95 - 8.20i)T + (-12.6 + 38.9i)T^{2} \)
43 \( 1 + 12.4iT - 43T^{2} \)
47 \( 1 + (4.66 - 3.39i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (11.9 + 8.66i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-14.6 + 4.76i)T + (49.3 - 35.8i)T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 + (-4.98 - 15.3i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (5.16 + 1.67i)T + (63.9 + 46.4i)T^{2} \)
83 \( 1 + (17.1 - 5.57i)T + (67.1 - 48.7i)T^{2} \)
89 \( 1 + 4.31T + 89T^{2} \)
97 \( 1 + (78.4 + 57.0i)T^{2} \)
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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

−11.51212564695320406271127279739, −10.98125559561339116660936187240, −9.987730266025667735121249722587, −8.333434153635997528889002410118, −7.13500176061975558139685049418, −6.38671216791864074273726679388, −5.53975078383314828917883558937, −4.82671362343557881900127191268, −3.72683708626968074520241848359, −2.72979690147814493218853119433, 1.54331317747369352452983461269, 2.65356326253729855791822650575, 4.37769722799813670564364595964, 4.95602061236998316822538046987, 5.83796439614497312710861538189, 6.84323500711657796351972026528, 7.53826286989842282412048981481, 9.590257332563293585864932314512, 10.48800029873171864212047783413, 11.32513217181732800435880261862

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