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Properties

Label 2-429-143.21-c1-0-0
Degree $2$
Conductor $429$
Sign $-0.386 + 0.922i$
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  + (−1.12 + 1.12i)2-s − 3-s − 0.513i·4-s + (0.850 + 0.850i)5-s + (1.12 − 1.12i)6-s + (−0.675 − 0.675i)7-s + (−1.66 − 1.66i)8-s + 9-s − 1.90·10-s + (−0.299 + 3.30i)11-s + 0.513i·12-s + (−3.60 − 0.0438i)13-s + 1.51·14-s + (−0.850 − 0.850i)15-s + 4.76·16-s − 7.43·17-s + ⋯
L(s)  = 1  + (−0.792 + 0.792i)2-s − 0.577·3-s − 0.256i·4-s + (0.380 + 0.380i)5-s + (0.457 − 0.457i)6-s + (−0.255 − 0.255i)7-s + (−0.589 − 0.589i)8-s + 0.333·9-s − 0.603·10-s + (−0.0903 + 0.995i)11-s + 0.148i·12-s + (−0.999 − 0.0121i)13-s + 0.404·14-s + (−0.219 − 0.219i)15-s + 1.19·16-s − 1.80·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.386 + 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $-0.386 + 0.922i$
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: \((2,\ 429,\ (\ :1/2),\ -0.386 + 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0338110 - 0.0508161i\)
\(L(\frac12)\) \(\approx\) \(0.0338110 - 0.0508161i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.299 - 3.30i)T \)
13 \( 1 + (3.60 + 0.0438i)T \)
good2 \( 1 + (1.12 - 1.12i)T - 2iT^{2} \)
5 \( 1 + (-0.850 - 0.850i)T + 5iT^{2} \)
7 \( 1 + (0.675 + 0.675i)T + 7iT^{2} \)
17 \( 1 + 7.43T + 17T^{2} \)
19 \( 1 + (-2.00 + 2.00i)T - 19iT^{2} \)
23 \( 1 - 3.77iT - 23T^{2} \)
29 \( 1 - 0.246iT - 29T^{2} \)
31 \( 1 + (7.23 + 7.23i)T + 31iT^{2} \)
37 \( 1 + (-0.872 + 0.872i)T - 37iT^{2} \)
41 \( 1 + (-2.78 + 2.78i)T - 41iT^{2} \)
43 \( 1 + 4.52T + 43T^{2} \)
47 \( 1 + (-2.70 + 2.70i)T - 47iT^{2} \)
53 \( 1 + 12.4T + 53T^{2} \)
59 \( 1 + (6.27 - 6.27i)T - 59iT^{2} \)
61 \( 1 + 3.96iT - 61T^{2} \)
67 \( 1 + (-6.55 - 6.55i)T + 67iT^{2} \)
71 \( 1 + (-2.41 - 2.41i)T + 71iT^{2} \)
73 \( 1 + (-5.88 - 5.88i)T + 73iT^{2} \)
79 \( 1 - 11.3iT - 79T^{2} \)
83 \( 1 + (-2.61 + 2.61i)T - 83iT^{2} \)
89 \( 1 + (5.39 - 5.39i)T - 89iT^{2} \)
97 \( 1 + (10.7 + 10.7i)T + 97iT^{2} \)
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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

−11.67753855672125149083028352781, −10.68299344793547469273696808087, −9.668997727879412225528022690691, −9.299504647521827532576316712909, −7.899384607072026330815204749013, −7.03561317530340810302364554419, −6.60476790164540142314373320816, −5.35337872725553799370452199175, −4.10442160876735615179293240876, −2.34207543308044732240757886575, 0.05103336692113170403188932978, 1.70997238423968563283221756085, 3.01898067568695670411331230087, 4.80209337752547208464507978228, 5.72643832939148742959420275942, 6.68898895643600042153798974745, 8.109745119191153130907444842580, 9.139497312180301085636609616596, 9.540996234079558451972374752255, 10.79792055464769654038431359050

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