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Properties

Label 2-429-143.21-c1-0-24
Degree $2$
Conductor $429$
Sign $-0.892 - 0.450i$
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  + (0.735 − 0.735i)2-s − 3-s + 0.918i·4-s + (−2.75 − 2.75i)5-s + (−0.735 + 0.735i)6-s + (0.0552 + 0.0552i)7-s + (2.14 + 2.14i)8-s + 9-s − 4.05·10-s + (−3.21 + 0.833i)11-s − 0.918i·12-s + (−3.28 − 1.49i)13-s + 0.0812·14-s + (2.75 + 2.75i)15-s + 1.31·16-s − 4.44·17-s + ⋯
L(s)  = 1  + (0.519 − 0.519i)2-s − 0.577·3-s + 0.459i·4-s + (−1.23 − 1.23i)5-s + (−0.300 + 0.300i)6-s + (0.0208 + 0.0208i)7-s + (0.758 + 0.758i)8-s + 0.333·9-s − 1.28·10-s + (−0.967 + 0.251i)11-s − 0.265i·12-s + (−0.910 − 0.414i)13-s + 0.0217·14-s + (0.711 + 0.711i)15-s + 0.329·16-s − 1.07·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.892 - 0.450i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.892 - 0.450i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $-0.892 - 0.450i$
Analytic conductor: \(3.42558\)
Root analytic conductor: \(1.85083\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{429} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 429,\ (\ :1/2),\ -0.892 - 0.450i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0226679 + 0.0953015i\)
\(L(\frac12)\) \(\approx\) \(0.0226679 + 0.0953015i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.21 - 0.833i)T \)
13 \( 1 + (3.28 + 1.49i)T \)
good2 \( 1 + (-0.735 + 0.735i)T - 2iT^{2} \)
5 \( 1 + (2.75 + 2.75i)T + 5iT^{2} \)
7 \( 1 + (-0.0552 - 0.0552i)T + 7iT^{2} \)
17 \( 1 + 4.44T + 17T^{2} \)
19 \( 1 + (4.92 - 4.92i)T - 19iT^{2} \)
23 \( 1 + 2.35iT - 23T^{2} \)
29 \( 1 + 8.51iT - 29T^{2} \)
31 \( 1 + (-2.06 - 2.06i)T + 31iT^{2} \)
37 \( 1 + (1.50 - 1.50i)T - 37iT^{2} \)
41 \( 1 + (2.88 - 2.88i)T - 41iT^{2} \)
43 \( 1 + 0.467T + 43T^{2} \)
47 \( 1 + (-8.38 + 8.38i)T - 47iT^{2} \)
53 \( 1 - 7.45T + 53T^{2} \)
59 \( 1 + (1.39 - 1.39i)T - 59iT^{2} \)
61 \( 1 + 12.7iT - 61T^{2} \)
67 \( 1 + (-2.57 - 2.57i)T + 67iT^{2} \)
71 \( 1 + (2.77 + 2.77i)T + 71iT^{2} \)
73 \( 1 + (0.533 + 0.533i)T + 73iT^{2} \)
79 \( 1 - 15.5iT - 79T^{2} \)
83 \( 1 + (5.01 - 5.01i)T - 83iT^{2} \)
89 \( 1 + (10.3 - 10.3i)T - 89iT^{2} \)
97 \( 1 + (1.43 + 1.43i)T + 97iT^{2} \)
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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

−10.95045942268486353804164022281, −10.01757895496230792030423796120, −8.424585474831711946892891492751, −8.122814532809853076590103806102, −7.04850999861408650896917879504, −5.37897857741883757551011077869, −4.57316217779268794214151576082, −3.94755902848435707758913923506, −2.29205177695497370228406837521, −0.05294508616986108020867713659, 2.61055082419991567052406910738, 4.14888082951546044402707583200, 4.90277441227197227874939783862, 6.15000944408232324271252315339, 7.07513117530995154779702401348, 7.40693609061811539755418188838, 8.894581412180268928438008291517, 10.35744621677901135954999351213, 10.79240482220664682068901104362, 11.46654995281328858868915960490

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