Properties

Label 2-429-13.4-c1-0-11
Degree $2$
Conductor $429$
Sign $-0.412 + 0.911i$
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  + (−1.70 − 0.986i)2-s + (0.5 − 0.866i)3-s + (0.946 + 1.63i)4-s + 1.03i·5-s + (−1.70 + 0.986i)6-s + (−0.142 + 0.0822i)7-s + 0.210i·8-s + (−0.499 − 0.866i)9-s + (1.01 − 1.76i)10-s + (0.866 + 0.5i)11-s + 1.89·12-s + (1.52 − 3.26i)13-s + 0.324·14-s + (0.893 + 0.515i)15-s + (2.10 − 3.63i)16-s + (−2.92 − 5.06i)17-s + ⋯
L(s)  = 1  + (−1.20 − 0.697i)2-s + (0.288 − 0.499i)3-s + (0.473 + 0.819i)4-s + 0.461i·5-s + (−0.697 + 0.402i)6-s + (−0.0538 + 0.0310i)7-s + 0.0744i·8-s + (−0.166 − 0.288i)9-s + (0.321 − 0.557i)10-s + (0.261 + 0.150i)11-s + 0.546·12-s + (0.423 − 0.905i)13-s + 0.0867·14-s + (0.230 + 0.133i)15-s + (0.525 − 0.909i)16-s + (−0.709 − 1.22i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.412 + 0.911i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.412 + 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $-0.412 + 0.911i$
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: \((2,\ 429,\ (\ :1/2),\ -0.412 + 0.911i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.409461 - 0.634580i\)
\(L(\frac12)\) \(\approx\) \(0.409461 - 0.634580i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.52 + 3.26i)T \)
good2 \( 1 + (1.70 + 0.986i)T + (1 + 1.73i)T^{2} \)
5 \( 1 - 1.03iT - 5T^{2} \)
7 \( 1 + (0.142 - 0.0822i)T + (3.5 - 6.06i)T^{2} \)
17 \( 1 + (2.92 + 5.06i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.05 + 2.34i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.41 - 2.45i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.72 + 4.72i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 4.77iT - 31T^{2} \)
37 \( 1 + (-3.32 - 1.92i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.98 + 1.14i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.13 + 7.15i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 0.407iT - 47T^{2} \)
53 \( 1 + 3.70T + 53T^{2} \)
59 \( 1 + (-1.08 + 0.626i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (7.30 + 12.6i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.97 - 1.71i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-13.5 + 7.84i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 13.0iT - 73T^{2} \)
79 \( 1 - 7.46T + 79T^{2} \)
83 \( 1 + 1.16iT - 83T^{2} \)
89 \( 1 + (-11.3 - 6.56i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (15.4 - 8.94i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.90764897786464126023191802601, −9.788597044552925242294528352861, −9.273294003398377638928167601532, −8.220574163703256314180081903877, −7.51266167063998655570295791102, −6.50217605471290524204190947835, −5.10818929921613543760984297140, −3.25108343343586626059906518017, −2.32936741345973300302651314743, −0.76491824925335162644458412155, 1.46342278365236148564032626260, 3.53972469616552047713544124711, 4.68173042267224444103506072613, 6.13210606031092365888960132981, 6.92440253336962165514861967017, 8.166163058558546909544033046476, 8.685484420002832537324634414735, 9.372299235728733180018268973727, 10.25598464688816182810969199341, 11.05160677656308882897937523831

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