↑

Properties

Label 2-429-11.9-c1-0-9
Degree $2$
Conductor $429$
Sign $-0.541 - 0.840i$
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  + (−0.488 + 1.50i)2-s + (0.809 − 0.587i)3-s + (−0.399 − 0.290i)4-s + (1.09 + 3.36i)5-s + (0.488 + 1.50i)6-s + (3.87 + 2.81i)7-s + (−1.92 + 1.39i)8-s + (0.309 − 0.951i)9-s − 5.58·10-s + (−0.697 − 3.24i)11-s − 0.494·12-s + (0.309 − 0.951i)13-s + (−6.12 + 4.44i)14-s + (2.86 + 2.07i)15-s + (−1.46 − 4.51i)16-s + (−1.69 − 5.20i)17-s + ⋯
L(s)  = 1  + (−0.345 + 1.06i)2-s + (0.467 − 0.339i)3-s + (−0.199 − 0.145i)4-s + (0.488 + 1.50i)5-s + (0.199 + 0.613i)6-s + (1.46 + 1.06i)7-s + (−0.680 + 0.494i)8-s + (0.103 − 0.317i)9-s − 1.76·10-s + (−0.210 − 0.977i)11-s − 0.142·12-s + (0.0857 − 0.263i)13-s + (−1.63 + 1.18i)14-s + (0.738 + 0.536i)15-s + (−0.366 − 1.12i)16-s + (−0.409 − 1.26i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.541 - 0.840i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.541 - 0.840i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $-0.541 - 0.840i$
Analytic conductor: \(3.42558\)
Root analytic conductor: \(1.85083\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{429} (196, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 429,\ (\ :1/2),\ -0.541 - 0.840i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.805267 + 1.47587i\)
\(L(\frac12)\) \(\approx\) \(0.805267 + 1.47587i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.809 + 0.587i)T \)
11 \( 1 + (0.697 + 3.24i)T \)
13 \( 1 + (-0.309 + 0.951i)T \)
good2 \( 1 + (0.488 - 1.50i)T + (-1.61 - 1.17i)T^{2} \)
5 \( 1 + (-1.09 - 3.36i)T + (-4.04 + 2.93i)T^{2} \)
7 \( 1 + (-3.87 - 2.81i)T + (2.16 + 6.65i)T^{2} \)
17 \( 1 + (1.69 + 5.20i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-4.58 + 3.33i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + 6.72T + 23T^{2} \)
29 \( 1 + (-3.06 - 2.22i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (1.31 - 4.03i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (6.25 + 4.54i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-1.48 + 1.07i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 5.88T + 43T^{2} \)
47 \( 1 + (-4.66 + 3.39i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-0.787 + 2.42i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-0.226 - 0.164i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (1.50 + 4.63i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + 15.5T + 67T^{2} \)
71 \( 1 + (-3.92 - 12.0i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (6.42 + 4.67i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-0.194 + 0.597i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-1.20 - 3.70i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 - 6.59T + 89T^{2} \)
97 \( 1 + (1.39 - 4.30i)T + (-78.4 - 57.0i)T^{2} \)
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

−11.45862980256736580047910313039, −10.64456903039267351540049809468, −9.234934705354865680006713811705, −8.559924909412565516688231907536, −7.65205789915540764924560502429, −7.03347594384573988216322265771, −5.96368684785241014239277202702, −5.26178144234336819685190037900, −3.03435365916116314643916741767, −2.30439690904503209598778653405, 1.35366370744324583662424006353, 1.95718313040387138178140621612, 4.01428924762300029564700947957, 4.58019720154407037358790034894, 5.85164198619694298842815342039, 7.60403082014908446240778607828, 8.343305930198493613254717739233, 9.249732896332174293068063154504, 10.10761431033216674559899685089, 10.59362348081207643904154320006

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