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Properties

Label 2-429-13.9-c1-0-15
Degree $2$
Conductor $429$
Sign $-0.100 + 0.994i$
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  + (0.900 − 1.55i)2-s + (0.5 − 0.866i)3-s + (−0.621 − 1.07i)4-s + 1.40·5-s + (−0.900 − 1.55i)6-s + (0.888 + 1.53i)7-s + 1.36·8-s + (−0.499 − 0.866i)9-s + (1.26 − 2.19i)10-s + (−0.5 + 0.866i)11-s − 1.24·12-s + (−0.600 − 3.55i)13-s + 3.20·14-s + (0.703 − 1.21i)15-s + (2.47 − 4.27i)16-s + (−0.374 − 0.648i)17-s + ⋯
L(s)  = 1  + (0.636 − 1.10i)2-s + (0.288 − 0.499i)3-s + (−0.310 − 0.538i)4-s + 0.629·5-s + (−0.367 − 0.636i)6-s + (0.335 + 0.581i)7-s + 0.481·8-s + (−0.166 − 0.288i)9-s + (0.400 − 0.693i)10-s + (−0.150 + 0.261i)11-s − 0.358·12-s + (−0.166 − 0.986i)13-s + 0.855·14-s + (0.181 − 0.314i)15-s + (0.617 − 1.06i)16-s + (−0.0908 − 0.157i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.100 + 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.100 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $-0.100 + 0.994i$
Analytic conductor: \(3.42558\)
Root analytic conductor: \(1.85083\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{429} (100, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 429,\ (\ :1/2),\ -0.100 + 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.62175 - 1.79433i\)
\(L(\frac12)\) \(\approx\) \(1.62175 - 1.79433i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 + (0.600 + 3.55i)T \)
good2 \( 1 + (-0.900 + 1.55i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 - 1.40T + 5T^{2} \)
7 \( 1 + (-0.888 - 1.53i)T + (-3.5 + 6.06i)T^{2} \)
17 \( 1 + (0.374 + 0.648i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.0118 + 0.0205i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.44 - 5.97i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.92 - 3.34i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 0.519T + 31T^{2} \)
37 \( 1 + (0.906 - 1.56i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.08 - 1.87i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.0405 - 0.0701i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 5.08T + 47T^{2} \)
53 \( 1 + 10.1T + 53T^{2} \)
59 \( 1 + (-4.06 - 7.03i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.186 - 0.323i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.77 - 3.06i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (5.55 + 9.62i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 11.9T + 73T^{2} \)
79 \( 1 - 9.80T + 79T^{2} \)
83 \( 1 + 4.95T + 83T^{2} \)
89 \( 1 + (7.62 - 13.2i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (3.75 + 6.50i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.12879727046779108455305519629, −10.16962272116144357688764283795, −9.396478885769927443300542631363, −8.143024952965178843298517320409, −7.33216509475003494930825589061, −5.85363248518242696178486703477, −5.06111172386865361872562085683, −3.62055741840304991349278984814, −2.54040920218137109961267009164, −1.62115901827461814017290553257, 2.04582898837512281972278789896, 3.93662448889372933901379986625, 4.67335982412951151894742568728, 5.75944016111729944524863268756, 6.56853867440268672624612830210, 7.58296588979990444677458802006, 8.460223046825383038420457185608, 9.582561405303559999274860275163, 10.42676839451344620688793592528, 11.25102947493297415276354053666

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