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Properties

Label 2-429-143.21-c1-0-15
Degree $2$
Conductor $429$
Sign $0.998 - 0.0515i$
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  + (−1.20 + 1.20i)2-s + 3-s − 0.908i·4-s + (−0.200 − 0.200i)5-s + (−1.20 + 1.20i)6-s + (−3.33 − 3.33i)7-s + (−1.31 − 1.31i)8-s + 9-s + 0.484·10-s + (2.95 + 1.51i)11-s − 0.908i·12-s + (2.46 − 2.63i)13-s + 8.04·14-s + (−0.200 − 0.200i)15-s + 4.99·16-s + 7.25·17-s + ⋯
L(s)  = 1  + (−0.852 + 0.852i)2-s + 0.577·3-s − 0.454i·4-s + (−0.0898 − 0.0898i)5-s + (−0.492 + 0.492i)6-s + (−1.26 − 1.26i)7-s + (−0.465 − 0.465i)8-s + 0.333·9-s + 0.153·10-s + (0.889 + 0.456i)11-s − 0.262i·12-s + (0.682 − 0.730i)13-s + 2.14·14-s + (−0.0518 − 0.0518i)15-s + 1.24·16-s + 1.75·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0515i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0515i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $0.998 - 0.0515i$
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: \((2,\ 429,\ (\ :1/2),\ 0.998 - 0.0515i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.941343 + 0.0242756i\)
\(L(\frac12)\) \(\approx\) \(0.941343 + 0.0242756i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.95 - 1.51i)T \)
13 \( 1 + (-2.46 + 2.63i)T \)
good2 \( 1 + (1.20 - 1.20i)T - 2iT^{2} \)
5 \( 1 + (0.200 + 0.200i)T + 5iT^{2} \)
7 \( 1 + (3.33 + 3.33i)T + 7iT^{2} \)
17 \( 1 - 7.25T + 17T^{2} \)
19 \( 1 + (1.18 - 1.18i)T - 19iT^{2} \)
23 \( 1 + 3.26iT - 23T^{2} \)
29 \( 1 - 5.67iT - 29T^{2} \)
31 \( 1 + (3.86 + 3.86i)T + 31iT^{2} \)
37 \( 1 + (-2.95 + 2.95i)T - 37iT^{2} \)
41 \( 1 + (-4.75 + 4.75i)T - 41iT^{2} \)
43 \( 1 + 2.81T + 43T^{2} \)
47 \( 1 + (-3.39 + 3.39i)T - 47iT^{2} \)
53 \( 1 + 5.44T + 53T^{2} \)
59 \( 1 + (5.68 - 5.68i)T - 59iT^{2} \)
61 \( 1 + 14.3iT - 61T^{2} \)
67 \( 1 + (0.748 + 0.748i)T + 67iT^{2} \)
71 \( 1 + (-6.94 - 6.94i)T + 71iT^{2} \)
73 \( 1 + (-2.84 - 2.84i)T + 73iT^{2} \)
79 \( 1 - 3.04iT - 79T^{2} \)
83 \( 1 + (4.10 - 4.10i)T - 83iT^{2} \)
89 \( 1 + (-6.08 + 6.08i)T - 89iT^{2} \)
97 \( 1 + (1.76 + 1.76i)T + 97iT^{2} \)
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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

−10.68098910327570382495066697091, −9.928690633568052035687068083420, −9.348151476623626275960801501469, −8.301852027077552855489255470045, −7.53280567723834232871121470150, −6.79960860486083924600340212236, −5.94350845736984608890123334564, −3.96591220494671464782474008752, −3.30942561468632319960107573667, −0.841150339795851663629778336689, 1.46439171524546730250681973270, 2.89025142573511747913517664678, 3.59056139403764341798408909920, 5.64871017383319548094198933743, 6.40539877741587625632800238465, 7.86255786191669779892435874304, 8.931636649961374855575839507669, 9.321433698010960640193412357666, 9.933681216208297326645332888650, 11.17348190507738505102317704938

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