↑

Properties

Label 2-429-11.5-c1-0-17
Degree $2$
Conductor $429$
Sign $0.331 - 0.943i$
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  + (0.589 + 1.81i)2-s + (0.809 + 0.587i)3-s + (−1.33 + 0.966i)4-s + (0.697 − 2.14i)5-s + (−0.589 + 1.81i)6-s + (2.71 − 1.96i)7-s + (0.549 + 0.398i)8-s + (0.309 + 0.951i)9-s + 4.30·10-s + (−0.0739 − 3.31i)11-s − 1.64·12-s + (0.309 + 0.951i)13-s + (5.17 + 3.75i)14-s + (1.82 − 1.32i)15-s + (−1.41 + 4.36i)16-s + (−0.677 + 2.08i)17-s + ⋯
L(s)  = 1  + (0.417 + 1.28i)2-s + (0.467 + 0.339i)3-s + (−0.665 + 0.483i)4-s + (0.311 − 0.959i)5-s + (−0.240 + 0.741i)6-s + (1.02 − 0.744i)7-s + (0.194 + 0.141i)8-s + (0.103 + 0.317i)9-s + 1.36·10-s + (−0.0223 − 0.999i)11-s − 0.474·12-s + (0.0857 + 0.263i)13-s + (1.38 + 1.00i)14-s + (0.471 − 0.342i)15-s + (−0.354 + 1.09i)16-s + (−0.164 + 0.505i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.331 - 0.943i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.331 - 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $0.331 - 0.943i$
Analytic conductor: \(3.42558\)
Root analytic conductor: \(1.85083\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{429} (313, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 429,\ (\ :1/2),\ 0.331 - 0.943i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.87431 + 1.32847i\)
\(L(\frac12)\) \(\approx\) \(1.87431 + 1.32847i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.0739 + 3.31i)T \)
13 \( 1 + (-0.309 - 0.951i)T \)
good2 \( 1 + (-0.589 - 1.81i)T + (-1.61 + 1.17i)T^{2} \)
5 \( 1 + (-0.697 + 2.14i)T + (-4.04 - 2.93i)T^{2} \)
7 \( 1 + (-2.71 + 1.96i)T + (2.16 - 6.65i)T^{2} \)
17 \( 1 + (0.677 - 2.08i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (4.13 + 3.00i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + 3.28T + 23T^{2} \)
29 \( 1 + (7.15 - 5.19i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-1.91 - 5.87i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (0.639 - 0.464i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (8.59 + 6.24i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 4.63T + 43T^{2} \)
47 \( 1 + (1.35 + 0.983i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-2.06 - 6.34i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (6.30 - 4.58i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (0.173 - 0.534i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 - 10.9T + 67T^{2} \)
71 \( 1 + (-2.27 + 6.99i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-5.38 + 3.91i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (0.600 + 1.84i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (5.05 - 15.5i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 - 13.3T + 89T^{2} \)
97 \( 1 + (0.678 + 2.08i)T + (-78.4 + 57.0i)T^{2} \)
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

−11.09363714787362364004731676618, −10.56964562787220813064331592334, −8.964251566811329086332188472369, −8.536586163409356054750352029550, −7.70221697675078703814900485428, −6.65853419541734391403337747879, −5.48942020137566151874341161706, −4.77464609735207939097272977737, −3.87288686342834018867072456415, −1.67879880664882857585164152176, 1.97307931446638623760848265480, 2.36571325408497336175756440724, 3.73198408148486428418052192767, 4.84307837233242304395545050339, 6.21762057946484907556510817930, 7.39416041207114651758949648059, 8.245581544437598353979089680161, 9.586611668782806118378185484924, 10.20555651891643959114250222839, 11.21474276055514323010131404607

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