↑

Properties

Label 2-1029-1029.218-c0-0-0
Degree $2$
Conductor $1029$
Sign $0.537 + 0.843i$
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  + (0.801 − 0.598i)3-s + (−0.838 + 0.545i)4-s + (−0.672 − 0.740i)7-s + (0.284 − 0.958i)9-s + (−0.345 + 0.938i)12-s + (1.68 − 0.949i)13-s + (0.404 − 0.914i)16-s − 1.14·19-s + (−0.981 − 0.191i)21-s + (0.871 + 0.490i)25-s + (−0.345 − 0.938i)27-s + (0.967 + 0.253i)28-s + (1.51 − 0.727i)31-s + (0.284 + 0.958i)36-s + (−1.36 + 0.452i)37-s + ⋯
L(s)  = 1  + (0.801 − 0.598i)3-s + (−0.838 + 0.545i)4-s + (−0.672 − 0.740i)7-s + (0.284 − 0.958i)9-s + (−0.345 + 0.938i)12-s + (1.68 − 0.949i)13-s + (0.404 − 0.914i)16-s − 1.14·19-s + (−0.981 − 0.191i)21-s + (0.871 + 0.490i)25-s + (−0.345 − 0.938i)27-s + (0.967 + 0.253i)28-s + (1.51 − 0.727i)31-s + (0.284 + 0.958i)36-s + (−1.36 + 0.452i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.537 + 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.537 + 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1029\)    =    \(3 \cdot 7^{3}\)
Sign: $0.537 + 0.843i$
Analytic conductor: \(0.513537\)
Root analytic conductor: \(0.716615\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1029} (218, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1029,\ (\ :0),\ 0.537 + 0.843i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.065654891\)
\(L(\frac12)\) \(\approx\) \(1.065654891\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.801 + 0.598i)T \)
7 \( 1 + (0.672 + 0.740i)T \)
good2 \( 1 + (0.838 - 0.545i)T^{2} \)
5 \( 1 + (-0.871 - 0.490i)T^{2} \)
11 \( 1 + (0.949 + 0.315i)T^{2} \)
13 \( 1 + (-1.68 + 0.949i)T + (0.518 - 0.855i)T^{2} \)
17 \( 1 + (-0.801 + 0.598i)T^{2} \)
19 \( 1 + 1.14T + T^{2} \)
23 \( 1 + (0.981 + 0.191i)T^{2} \)
29 \( 1 + (0.981 - 0.191i)T^{2} \)
31 \( 1 + (-1.51 + 0.727i)T + (0.623 - 0.781i)T^{2} \)
37 \( 1 + (1.36 - 0.452i)T + (0.801 - 0.598i)T^{2} \)
41 \( 1 + (0.572 + 0.820i)T^{2} \)
43 \( 1 + (0.243 - 0.206i)T + (0.159 - 0.987i)T^{2} \)
47 \( 1 + (-0.718 - 0.695i)T^{2} \)
53 \( 1 + (-0.967 + 0.253i)T^{2} \)
59 \( 1 + (0.572 - 0.820i)T^{2} \)
61 \( 1 + (-1.02 - 1.69i)T + (-0.462 + 0.886i)T^{2} \)
67 \( 1 + (1.44 + 0.695i)T + (0.623 + 0.781i)T^{2} \)
71 \( 1 + (0.981 + 0.191i)T^{2} \)
73 \( 1 + (1.75 - 0.712i)T + (0.718 - 0.695i)T^{2} \)
79 \( 1 + (0.230 - 1.01i)T + (-0.900 - 0.433i)T^{2} \)
83 \( 1 + (0.0960 + 0.995i)T^{2} \)
89 \( 1 + (-0.991 - 0.127i)T^{2} \)
97 \( 1 + (-1.79 + 0.865i)T + (0.623 - 0.781i)T^{2} \)
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

−9.956647806299045787597456699813, −8.797265404189481883463883348126, −8.532730487731391321945335636710, −7.66484095222097669938268557169, −6.76491654932868143004204987910, −5.92450270590607908651273985134, −4.41962458970531844909605310563, −3.61141393545803571164271686466, −2.91923947399779184239221808051, −1.05116699118176382698121341963, 1.78866021787280097920063433411, 3.17934778350601868003432891611, 4.08226018751526744544132966361, 4.88364583291608551689428784798, 6.01132655673954983237948205666, 6.71246083377741569094525899359, 8.367710831825748797980749881693, 8.729919065231547042276507306962, 9.210575729466267004357519673573, 10.25575485694670074415063040209

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