↑

Properties

Label 2-1029-1029.554-c0-0-0
Degree $2$
Conductor $1029$
Sign $0.947 - 0.319i$
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  + (−0.0960 + 0.995i)3-s + (0.926 − 0.375i)4-s + (0.0320 − 0.999i)7-s + (−0.981 − 0.191i)9-s + (0.284 + 0.958i)12-s + (0.525 + 0.447i)13-s + (0.718 − 0.695i)16-s + 1.60·19-s + (0.991 + 0.127i)21-s + (−0.761 + 0.648i)25-s + (0.284 − 0.958i)27-s + (−0.345 − 0.938i)28-s + (−0.412 + 1.80i)31-s + (−0.981 + 0.191i)36-s + (−1.17 − 1.29i)37-s + ⋯
L(s)  = 1  + (−0.0960 + 0.995i)3-s + (0.926 − 0.375i)4-s + (0.0320 − 0.999i)7-s + (−0.981 − 0.191i)9-s + (0.284 + 0.958i)12-s + (0.525 + 0.447i)13-s + (0.718 − 0.695i)16-s + 1.60·19-s + (0.991 + 0.127i)21-s + (−0.761 + 0.648i)25-s + (0.284 − 0.958i)27-s + (−0.345 − 0.938i)28-s + (−0.412 + 1.80i)31-s + (−0.981 + 0.191i)36-s + (−1.17 − 1.29i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 - 0.319i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 - 0.319i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1029\)    =    \(3 \cdot 7^{3}\)
Sign: $0.947 - 0.319i$
Analytic conductor: \(0.513537\)
Root analytic conductor: \(0.716615\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1029} (554, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1029,\ (\ :0),\ 0.947 - 0.319i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.224913081\)
\(L(\frac12)\) \(\approx\) \(1.224913081\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.0960 - 0.995i)T \)
7 \( 1 + (-0.0320 + 0.999i)T \)
good2 \( 1 + (-0.926 + 0.375i)T^{2} \)
5 \( 1 + (0.761 - 0.648i)T^{2} \)
11 \( 1 + (0.672 - 0.740i)T^{2} \)
13 \( 1 + (-0.525 - 0.447i)T + (0.159 + 0.987i)T^{2} \)
17 \( 1 + (0.0960 - 0.995i)T^{2} \)
19 \( 1 - 1.60T + T^{2} \)
23 \( 1 + (-0.991 - 0.127i)T^{2} \)
29 \( 1 + (-0.991 + 0.127i)T^{2} \)
31 \( 1 + (0.412 - 1.80i)T + (-0.900 - 0.433i)T^{2} \)
37 \( 1 + (1.17 + 1.29i)T + (-0.0960 + 0.995i)T^{2} \)
41 \( 1 + (-0.801 - 0.598i)T^{2} \)
43 \( 1 + (0.678 + 0.441i)T + (0.404 + 0.914i)T^{2} \)
47 \( 1 + (-0.871 - 0.490i)T^{2} \)
53 \( 1 + (0.345 - 0.938i)T^{2} \)
59 \( 1 + (-0.801 + 0.598i)T^{2} \)
61 \( 1 + (0.182 - 1.12i)T + (-0.949 - 0.315i)T^{2} \)
67 \( 1 + (-0.0427 - 0.187i)T + (-0.900 + 0.433i)T^{2} \)
71 \( 1 + (-0.991 - 0.127i)T^{2} \)
73 \( 1 + (1.30 - 0.341i)T + (0.871 - 0.490i)T^{2} \)
79 \( 1 + (-0.199 + 0.249i)T + (-0.222 - 0.974i)T^{2} \)
83 \( 1 + (0.997 - 0.0640i)T^{2} \)
89 \( 1 + (0.572 - 0.820i)T^{2} \)
97 \( 1 + (-0.205 + 0.901i)T + (-0.900 - 0.433i)T^{2} \)
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

−10.30067647374913904229856664239, −9.582244220227095503371937413597, −8.688980204913786664236295860348, −7.48278695831543150571725601363, −6.89742040106331450222248165956, −5.76715560977525711629078962896, −5.08222243101213538916238564678, −3.83480544519443543569035726948, −3.13816071382768436241335707414, −1.49259123510526978117192485457, 1.58743958431776239616451492745, 2.60739856712265153749057251226, 3.44793800386913505409477199289, 5.27281273168771738940283579373, 6.00158733502969319692638965057, 6.66920186459932583613583509129, 7.75181119653801699186218906196, 8.082513668668444743656107036892, 9.115682996037534427144233153344, 10.16025047163511863628232395737

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