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Properties

Label 2-429-13.12-c1-0-20
Degree $2$
Conductor $429$
Sign $-0.981 + 0.189i$
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  − 1.40i·2-s − 3-s + 0.0273·4-s − 3.56i·5-s + 1.40i·6-s − 0.116i·7-s − 2.84i·8-s + 9-s − 5.01·10-s − i·11-s − 0.0273·12-s + (0.684 + 3.53i)13-s − 0.163·14-s + 3.56i·15-s − 3.94·16-s + 0.845·17-s + ⋯
L(s)  = 1  − 0.993i·2-s − 0.577·3-s + 0.0136·4-s − 1.59i·5-s + 0.573i·6-s − 0.0440i·7-s − 1.00i·8-s + 0.333·9-s − 1.58·10-s − 0.301i·11-s − 0.00790·12-s + (0.189 + 0.981i)13-s − 0.0437·14-s + 0.921i·15-s − 0.986·16-s + 0.204·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.981 + 0.189i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.981 + 0.189i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $-0.981 + 0.189i$
Analytic conductor: \(3.42558\)
Root analytic conductor: \(1.85083\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{429} (298, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 429,\ (\ :1/2),\ -0.981 + 0.189i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.111516 - 1.16395i\)
\(L(\frac12)\) \(\approx\) \(0.111516 - 1.16395i\)
\(L(\frac{3}{2})\) 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 \( 1 + iT \)
13 \( 1 + (-0.684 - 3.53i)T \)
good2 \( 1 + 1.40iT - 2T^{2} \)
5 \( 1 + 3.56iT - 5T^{2} \)
7 \( 1 + 0.116iT - 7T^{2} \)
17 \( 1 - 0.845T + 17T^{2} \)
19 \( 1 + 2.19iT - 19T^{2} \)
23 \( 1 + 1.73T + 23T^{2} \)
29 \( 1 + 4.43T + 29T^{2} \)
31 \( 1 + 6.18iT - 31T^{2} \)
37 \( 1 - 5.66iT - 37T^{2} \)
41 \( 1 - 4.86iT - 41T^{2} \)
43 \( 1 + 3.65T + 43T^{2} \)
47 \( 1 - 0.856iT - 47T^{2} \)
53 \( 1 - 12.0T + 53T^{2} \)
59 \( 1 + 4.63iT - 59T^{2} \)
61 \( 1 - 12.8T + 61T^{2} \)
67 \( 1 + 12.5iT - 67T^{2} \)
71 \( 1 + 3.93iT - 71T^{2} \)
73 \( 1 - 5.81iT - 73T^{2} \)
79 \( 1 - 7.74T + 79T^{2} \)
83 \( 1 - 6.83iT - 83T^{2} \)
89 \( 1 + 8.05iT - 89T^{2} \)
97 \( 1 - 15.3iT - 97T^{2} \)
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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

−11.01896340898424124826235350824, −9.843551969691673562317113748561, −9.247166140280498429838924955228, −8.209426243755523727851109031629, −6.91395496957978842939267068514, −5.80687872675075152697665317121, −4.68651305307591643970995942737, −3.81125064783163944484539463647, −2.00010608265168253683870131816, −0.806019447076460519592398588456, 2.34075540908433229104065293642, 3.66677203457351890118148996783, 5.41090987355153007514531040463, 6.00685282264917001312725571678, 7.04567254165236931937484489430, 7.41423749050349525521795099310, 8.553429010654994849654650304531, 10.11494224415136560800119752962, 10.62668698993157515198717762269, 11.43011726302844659968252344857

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