Properties

Label 2-429-429.428-c1-0-42
Degree $2$
Conductor $429$
Sign $-0.990 - 0.138i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 1.79i·2-s + (−0.240 + 1.71i)3-s − 1.22·4-s + (3.08 + 0.431i)6-s − 5.23·7-s − 1.39i·8-s + (−2.88 − 0.824i)9-s − 3.31i·11-s + (0.294 − 2.10i)12-s + 3.60·13-s + 9.39i·14-s − 4.94·16-s + (−1.48 + 5.17i)18-s − 7.00·19-s + (1.25 − 8.97i)21-s − 5.95·22-s + ⋯
L(s)  = 1  − 1.26i·2-s + (−0.138 + 0.990i)3-s − 0.612·4-s + (1.25 + 0.176i)6-s − 1.97·7-s − 0.492i·8-s + (−0.961 − 0.274i)9-s − 1.00i·11-s + (0.0849 − 0.606i)12-s + 1.00·13-s + 2.50i·14-s − 1.23·16-s + (−0.349 + 1.22i)18-s − 1.60·19-s + (0.274 − 1.95i)21-s − 1.26·22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 - 0.138i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.990 - 0.138i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $-0.990 - 0.138i$
Analytic conductor: \(3.42558\)
Root analytic conductor: \(1.85083\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{429} (428, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 429,\ (\ :1/2),\ -0.990 - 0.138i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0324106 + 0.464615i\)
\(L(\frac12)\) \(\approx\) \(0.0324106 + 0.464615i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.240 - 1.71i)T \)
11 \( 1 + 3.31iT \)
13 \( 1 - 3.60T \)
good2 \( 1 + 1.79iT - 2T^{2} \)
5 \( 1 + 5T^{2} \)
7 \( 1 + 5.23T + 7T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + 7.00T + 19T^{2} \)
23 \( 1 + 6.19iT - 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 - 7.55iT - 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 2.65iT - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 6.10T + 73T^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 + 17.5iT - 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 - 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.61340225112606574396655280605, −10.09621800988871643857728715912, −9.220443836702836402785948865080, −8.532357644977074963617994121070, −6.44641851954641807363361700480, −6.08817641109469990781126677222, −4.21015840115354118315708847823, −3.50633770236643508773230653860, −2.67950142751477985719572847679, −0.27692121190235620558100467852, 2.23272531935447420161228503637, 3.78671255189883856605112902232, 5.55670109304934954819462031661, 6.30965862367012349154723168965, 6.80843757629515273292576083519, 7.63032242191580243520027241901, 8.665653243040392346104393825594, 9.523849311081488977472079998799, 10.70636671702002045099883386433, 11.87220434245156212890859279002

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