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Properties

Label 2-429-429.194-c1-0-46
Degree $2$
Conductor $429$
Sign $-0.760 + 0.648i$
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  + (1.65 − 2.28i)2-s + (−0.535 + 1.64i)3-s + (−1.84 − 5.67i)4-s + (2.37 − 1.72i)5-s + (2.87 + 3.95i)6-s + (−10.6 − 3.45i)8-s + (−2.42 − 1.76i)9-s − 8.29i·10-s + (1.47 + 2.96i)11-s + 10.3·12-s + (2.11 − 2.91i)13-s + (1.57 + 4.84i)15-s + (−15.8 + 11.5i)16-s + (−8.05 + 2.61i)18-s + (−14.1 − 10.3i)20-s + ⋯
L(s)  = 1  + (1.17 − 1.61i)2-s + (−0.309 + 0.951i)3-s + (−0.921 − 2.83i)4-s + (1.06 − 0.772i)5-s + (1.17 + 1.61i)6-s + (−3.76 − 1.22i)8-s + (−0.809 − 0.587i)9-s − 2.62i·10-s + (0.445 + 0.895i)11-s + 2.98·12-s + (0.587 − 0.809i)13-s + (0.406 + 1.24i)15-s + (−3.97 + 2.88i)16-s + (−1.89 + 0.616i)18-s + (−3.17 − 2.30i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.760 + 0.648i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.760 + 0.648i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $-0.760 + 0.648i$
Analytic conductor: \(3.42558\)
Root analytic conductor: \(1.85083\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{429} (194, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 429,\ (\ :1/2),\ -0.760 + 0.648i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.814981 - 2.21213i\)
\(L(\frac12)\) \(\approx\) \(0.814981 - 2.21213i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.535 - 1.64i)T \)
11 \( 1 + (-1.47 - 2.96i)T \)
13 \( 1 + (-2.11 + 2.91i)T \)
good2 \( 1 + (-1.65 + 2.28i)T + (-0.618 - 1.90i)T^{2} \)
5 \( 1 + (-2.37 + 1.72i)T + (1.54 - 4.75i)T^{2} \)
7 \( 1 + (-5.66 + 4.11i)T^{2} \)
17 \( 1 + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-9.57 - 29.4i)T^{2} \)
37 \( 1 + (29.9 - 21.7i)T^{2} \)
41 \( 1 + (-7.44 - 2.41i)T + (33.1 + 24.0i)T^{2} \)
43 \( 1 - 7.66iT - 43T^{2} \)
47 \( 1 + (0.508 - 1.56i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (-3.92 - 12.0i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (1.35 + 1.86i)T + (-18.8 + 58.0i)T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 + (3.98 - 2.89i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-10.4 + 14.3i)T + (-24.4 - 75.1i)T^{2} \)
83 \( 1 + (-5.06 - 6.96i)T + (-25.6 + 78.9i)T^{2} \)
89 \( 1 + 18.3T + 89T^{2} \)
97 \( 1 + (-29.9 - 92.2i)T^{2} \)
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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

−10.83997130702201984536601850818, −10.10657036188526758056022363612, −9.518022627216111929228456141679, −8.842680836490747725339885745816, −6.23662949389227937170360074064, −5.55462182419606995295455799498, −4.76637779148865449450913830149, −3.92250853414485477842193878236, −2.64707096153923871198020863511, −1.23058197322924602028789022355, 2.51648395049333028475951550414, 3.79601975320634885935706556635, 5.31829492685866923489682735624, 6.10037838599495201092621388887, 6.53752133104006734071227440308, 7.33712118567642876748416230207, 8.412613074297778212335268642240, 9.209044438303697285575447501534, 11.00102366065653787437304353186, 11.85279087656640103732039560115

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