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Properties

Label 2-429-143.18-c1-0-16
Degree $2$
Conductor $429$
Sign $0.473 + 0.880i$
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  + (−0.114 − 0.724i)2-s + (0.309 − 0.951i)3-s + (1.39 − 0.451i)4-s + (−0.292 + 1.84i)5-s + (−0.724 − 0.114i)6-s + (2.11 + 1.08i)7-s + (−1.15 − 2.26i)8-s + (−0.809 − 0.587i)9-s + 1.36·10-s + (−0.393 − 3.29i)11-s − 1.46i·12-s + (2.53 + 2.56i)13-s + (0.538 − 1.65i)14-s + (1.66 + 0.847i)15-s + (0.860 − 0.625i)16-s + (2.17 − 1.58i)17-s + ⋯
L(s)  = 1  + (−0.0810 − 0.512i)2-s + (0.178 − 0.549i)3-s + (0.695 − 0.225i)4-s + (−0.130 + 0.824i)5-s + (−0.295 − 0.0468i)6-s + (0.801 + 0.408i)7-s + (−0.407 − 0.799i)8-s + (−0.269 − 0.195i)9-s + 0.433·10-s + (−0.118 − 0.992i)11-s − 0.422i·12-s + (0.702 + 0.712i)13-s + (0.144 − 0.443i)14-s + (0.429 + 0.218i)15-s + (0.215 − 0.156i)16-s + (0.528 − 0.383i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.473 + 0.880i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.473 + 0.880i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $0.473 + 0.880i$
Analytic conductor: \(3.42558\)
Root analytic conductor: \(1.85083\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{429} (304, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 429,\ (\ :1/2),\ 0.473 + 0.880i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.51803 - 0.907380i\)
\(L(\frac12)\) \(\approx\) \(1.51803 - 0.907380i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.309 + 0.951i)T \)
11 \( 1 + (0.393 + 3.29i)T \)
13 \( 1 + (-2.53 - 2.56i)T \)
good2 \( 1 + (0.114 + 0.724i)T + (-1.90 + 0.618i)T^{2} \)
5 \( 1 + (0.292 - 1.84i)T + (-4.75 - 1.54i)T^{2} \)
7 \( 1 + (-2.11 - 1.08i)T + (4.11 + 5.66i)T^{2} \)
17 \( 1 + (-2.17 + 1.58i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (0.0384 - 0.0195i)T + (11.1 - 15.3i)T^{2} \)
23 \( 1 - 0.141iT - 23T^{2} \)
29 \( 1 + (-2.55 + 0.829i)T + (23.4 - 17.0i)T^{2} \)
31 \( 1 + (8.20 - 1.29i)T + (29.4 - 9.57i)T^{2} \)
37 \( 1 + (-1.78 + 3.49i)T + (-21.7 - 29.9i)T^{2} \)
41 \( 1 + (1.76 - 0.897i)T + (24.0 - 33.1i)T^{2} \)
43 \( 1 + 4.73T + 43T^{2} \)
47 \( 1 + (4.62 - 2.35i)T + (27.6 - 38.0i)T^{2} \)
53 \( 1 + (1.29 + 0.943i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (5.25 - 10.3i)T + (-34.6 - 47.7i)T^{2} \)
61 \( 1 + (-2.71 - 3.73i)T + (-18.8 + 58.0i)T^{2} \)
67 \( 1 + (0.334 - 0.334i)T - 67iT^{2} \)
71 \( 1 + (1.91 - 12.1i)T + (-67.5 - 21.9i)T^{2} \)
73 \( 1 + (-10.8 - 5.50i)T + (42.9 + 59.0i)T^{2} \)
79 \( 1 + (-4.53 + 6.24i)T + (-24.4 - 75.1i)T^{2} \)
83 \( 1 + (-1.28 - 0.203i)T + (78.9 + 25.6i)T^{2} \)
89 \( 1 + (4.41 + 4.41i)T + 89iT^{2} \)
97 \( 1 + (18.5 - 2.93i)T + (92.2 - 29.9i)T^{2} \)
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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

−11.26865379305836043745489695709, −10.43618187386110752078096637712, −9.200311474874931661673880362333, −8.238907935776151573346799366099, −7.22540775087718634292422483576, −6.42524431520128749914843674794, −5.46982777407110851670594832361, −3.59902102958333240423495571266, −2.63907765886010326234592872672, −1.41511469646755968058168898465, 1.70945280562254136976465007920, 3.38392927727601752613033203498, 4.67786240099547939022030878276, 5.47194473507041147153389796796, 6.74576024397675747579971465664, 7.934342399874681348156294849792, 8.237509748356580024326997511605, 9.403281490440266845846820439301, 10.53093468127408382359627023962, 11.16783973103963741832216915623

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