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Properties

Label 2-1029-1029.953-c0-0-0
Degree $2$
Conductor $1029$
Sign $-0.323 + 0.946i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.462 + 0.886i)3-s + (−0.345 − 0.938i)4-s + (0.159 − 0.987i)7-s + (−0.572 − 0.820i)9-s + (0.991 + 0.127i)12-s + (−1.81 − 0.736i)13-s + (−0.761 + 0.648i)16-s − 1.99·19-s + (0.801 + 0.598i)21-s + (0.926 − 0.375i)25-s + (0.991 − 0.127i)27-s + (−0.981 + 0.191i)28-s + (0.622 − 0.299i)31-s + (−0.572 + 0.820i)36-s + (−0.868 − 1.43i)37-s + ⋯
L(s)  = 1  + (−0.462 + 0.886i)3-s + (−0.345 − 0.938i)4-s + (0.159 − 0.987i)7-s + (−0.572 − 0.820i)9-s + (0.991 + 0.127i)12-s + (−1.81 − 0.736i)13-s + (−0.761 + 0.648i)16-s − 1.99·19-s + (0.801 + 0.598i)21-s + (0.926 − 0.375i)25-s + (0.991 − 0.127i)27-s + (−0.981 + 0.191i)28-s + (0.622 − 0.299i)31-s + (−0.572 + 0.820i)36-s + (−0.868 − 1.43i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.323 + 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.323 + 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1029\)    =    \(3 \cdot 7^{3}\)
Sign: $-0.323 + 0.946i$
Analytic conductor: \(0.513537\)
Root analytic conductor: \(0.716615\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1029} (953, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1029,\ (\ :0),\ -0.323 + 0.946i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4914183539\)
\(L(\frac12)\) \(\approx\) \(0.4914183539\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.462 - 0.886i)T \)
7 \( 1 + (-0.159 + 0.987i)T \)
good2 \( 1 + (0.345 + 0.938i)T^{2} \)
5 \( 1 + (-0.926 + 0.375i)T^{2} \)
11 \( 1 + (-0.518 + 0.855i)T^{2} \)
13 \( 1 + (1.81 + 0.736i)T + (0.718 + 0.695i)T^{2} \)
17 \( 1 + (0.462 - 0.886i)T^{2} \)
19 \( 1 + 1.99T + T^{2} \)
23 \( 1 + (-0.801 - 0.598i)T^{2} \)
29 \( 1 + (-0.801 + 0.598i)T^{2} \)
31 \( 1 + (-0.622 + 0.299i)T + (0.623 - 0.781i)T^{2} \)
37 \( 1 + (0.868 + 1.43i)T + (-0.462 + 0.886i)T^{2} \)
41 \( 1 + (0.997 + 0.0640i)T^{2} \)
43 \( 1 + (-1.68 + 0.442i)T + (0.871 - 0.490i)T^{2} \)
47 \( 1 + (0.838 - 0.545i)T^{2} \)
53 \( 1 + (0.981 + 0.191i)T^{2} \)
59 \( 1 + (0.997 - 0.0640i)T^{2} \)
61 \( 1 + (0.137 - 0.133i)T + (0.0320 - 0.999i)T^{2} \)
67 \( 1 + (-0.833 - 0.401i)T + (0.623 + 0.781i)T^{2} \)
71 \( 1 + (-0.801 - 0.598i)T^{2} \)
73 \( 1 + (-0.294 + 0.993i)T + (-0.838 - 0.545i)T^{2} \)
79 \( 1 + (0.319 - 1.40i)T + (-0.900 - 0.433i)T^{2} \)
83 \( 1 + (0.949 - 0.315i)T^{2} \)
89 \( 1 + (0.0960 + 0.995i)T^{2} \)
97 \( 1 + (-1.21 + 0.583i)T + (0.623 - 0.781i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.10013748112417921698783751544, −9.322276962739672284629403569470, −8.435675065456786105620370501675, −7.23283224218591564377541552609, −6.36821160335302542895104402474, −5.38456217435947415347838017266, −4.64539361396648239617469866084, −4.03068563726279817198489508108, −2.42403937510176380081130947005, −0.46012331568026876273757532577, 2.10924207765236475227202061682, 2.81085484786300717728270864808, 4.49675163676471470598071056685, 5.09254377998088316207526677010, 6.38120781146620607824445769721, 7.01124486633731669475558053003, 7.921462819455055958140181358699, 8.613534080049536370751270792680, 9.303040929307854108806630254450, 10.49682440352868802343492806331

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