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Properties

Label 2-429-13.4-c1-0-19
Degree $2$
Conductor $429$
Sign $0.999 + 0.0194i$
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  + (1.56 + 0.905i)2-s + (−0.5 + 0.866i)3-s + (0.638 + 1.10i)4-s − 2.99i·5-s + (−1.56 + 0.905i)6-s + (3.93 − 2.27i)7-s − 1.31i·8-s + (−0.499 − 0.866i)9-s + (2.71 − 4.69i)10-s + (−0.866 − 0.5i)11-s − 1.27·12-s + (−3.60 − 0.116i)13-s + 8.22·14-s + (2.59 + 1.49i)15-s + (2.46 − 4.26i)16-s + (2.79 + 4.84i)17-s + ⋯
L(s)  = 1  + (1.10 + 0.639i)2-s + (−0.288 + 0.499i)3-s + (0.319 + 0.552i)4-s − 1.34i·5-s + (−0.639 + 0.369i)6-s + (1.48 − 0.858i)7-s − 0.463i·8-s + (−0.166 − 0.288i)9-s + (0.857 − 1.48i)10-s + (−0.261 − 0.150i)11-s − 0.368·12-s + (−0.999 − 0.0322i)13-s + 2.19·14-s + (0.670 + 0.386i)15-s + (0.615 − 1.06i)16-s + (0.678 + 1.17i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0194i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0194i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $0.999 + 0.0194i$
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: \((2,\ 429,\ (\ :1/2),\ 0.999 + 0.0194i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.35821 - 0.0229527i\)
\(L(\frac12)\) \(\approx\) \(2.35821 - 0.0229527i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.60 + 0.116i)T \)
good2 \( 1 + (-1.56 - 0.905i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + 2.99iT - 5T^{2} \)
7 \( 1 + (-3.93 + 2.27i)T + (3.5 - 6.06i)T^{2} \)
17 \( 1 + (-2.79 - 4.84i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.372 - 0.214i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.86 - 6.68i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.28 - 3.95i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 0.104iT - 31T^{2} \)
37 \( 1 + (-7.18 - 4.14i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-9.17 - 5.29i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.14 - 1.98i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 9.86iT - 47T^{2} \)
53 \( 1 + 10.6T + 53T^{2} \)
59 \( 1 + (7.64 - 4.41i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.73 - 4.72i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.85 - 3.38i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-2.16 + 1.25i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 13.2iT - 73T^{2} \)
79 \( 1 - 7.11T + 79T^{2} \)
83 \( 1 - 0.999iT - 83T^{2} \)
89 \( 1 + (6.28 + 3.62i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-8.20 + 4.73i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.36817496263973613827616744383, −10.28039393484797356081301475341, −9.406569763093301576447936637494, −8.069552725060711980341957596617, −7.55651508223182090083983033878, −5.97209191582514839320019789807, −5.14109837289797919681020683167, −4.62082508979959569622844668520, −3.80341546928862903966428441835, −1.30074397968747324895419962461, 2.26884014514575268717009900385, 2.67295947744493840551087947683, 4.38476644396805250024366756534, 5.25730722793348663167246857188, 6.16260549962211154004752120002, 7.49871979802466976364043149134, 8.071790754541970057314913506625, 9.615876203221754039999191262679, 11.03944871795888315612423110814, 11.10669809364012540781812527680

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