↑

Properties

Label 2-1029-147.74-c0-0-0
Degree $2$
Conductor $1029$
Sign $0.768 - 0.639i$
Analytic cond. $0.513537$
Root an. cond. $0.716615$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.365 + 0.930i)3-s + (0.826 − 0.563i)4-s + (−0.733 + 0.680i)9-s + (0.826 + 0.563i)12-s + (−0.277 + 1.21i)13-s + (0.365 − 0.930i)16-s + (0.222 + 0.385i)19-s + (0.955 + 0.294i)25-s + (−0.900 − 0.433i)27-s + (0.900 − 1.56i)31-s + (−0.222 + 0.974i)36-s + (−1.48 − 1.01i)37-s + (−1.23 + 0.185i)39-s + (−0.277 − 0.347i)43-s + 48-s + ⋯
L(s)  = 1  + (0.365 + 0.930i)3-s + (0.826 − 0.563i)4-s + (−0.733 + 0.680i)9-s + (0.826 + 0.563i)12-s + (−0.277 + 1.21i)13-s + (0.365 − 0.930i)16-s + (0.222 + 0.385i)19-s + (0.955 + 0.294i)25-s + (−0.900 − 0.433i)27-s + (0.900 − 1.56i)31-s + (−0.222 + 0.974i)36-s + (−1.48 − 1.01i)37-s + (−1.23 + 0.185i)39-s + (−0.277 − 0.347i)43-s + 48-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.768 - 0.639i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.768 - 0.639i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1029\)    =    \(3 \cdot 7^{3}\)
Sign: $0.768 - 0.639i$
Analytic conductor: \(0.513537\)
Root analytic conductor: \(0.716615\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1029} (410, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1029,\ (\ :0),\ 0.768 - 0.639i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.336730374\)
\(L(\frac12)\) \(\approx\) \(1.336730374\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.365 - 0.930i)T \)
7 \( 1 \)
good2 \( 1 + (-0.826 + 0.563i)T^{2} \)
5 \( 1 + (-0.955 - 0.294i)T^{2} \)
11 \( 1 + (-0.0747 - 0.997i)T^{2} \)
13 \( 1 + (0.277 - 1.21i)T + (-0.900 - 0.433i)T^{2} \)
17 \( 1 + (0.988 + 0.149i)T^{2} \)
19 \( 1 + (-0.222 - 0.385i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.988 - 0.149i)T^{2} \)
29 \( 1 + (-0.623 + 0.781i)T^{2} \)
31 \( 1 + (-0.900 + 1.56i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (1.48 + 1.01i)T + (0.365 + 0.930i)T^{2} \)
41 \( 1 + (0.222 + 0.974i)T^{2} \)
43 \( 1 + (0.277 + 0.347i)T + (-0.222 + 0.974i)T^{2} \)
47 \( 1 + (-0.826 + 0.563i)T^{2} \)
53 \( 1 + (-0.365 + 0.930i)T^{2} \)
59 \( 1 + (-0.955 + 0.294i)T^{2} \)
61 \( 1 + (1.48 + 1.01i)T + (0.365 + 0.930i)T^{2} \)
67 \( 1 + (0.623 - 1.07i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (-0.623 - 0.781i)T^{2} \)
73 \( 1 + (1.72 + 0.531i)T + (0.826 + 0.563i)T^{2} \)
79 \( 1 + (-0.900 - 1.56i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.900 - 0.433i)T^{2} \)
89 \( 1 + (-0.0747 + 0.997i)T^{2} \)
97 \( 1 + 0.445T + T^{2} \)
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

−10.20710376919109477716640513618, −9.508106653764010139215765614659, −8.804366042057570354495823372854, −7.71274846129437512764886484861, −6.84995186729388416831572003099, −5.90005767484849658497949193909, −5.01189244360378503093097421457, −4.05691577635481705112300503810, −2.89665001648663424329088321831, −1.86424179801745720666101686960, 1.44156436727123709278304354038, 2.82284748713415947946660735945, 3.24104016250552048607108915112, 4.91398167198924808698141188190, 6.07449808351966883201723777079, 6.84605151417874653697720469758, 7.49193156658918105871702353413, 8.287525088631155541720592949751, 8.872329463121189047472127036719, 10.23653253813698368337140656671

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