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Properties

Label 2-1029-1029.449-c0-0-0
Degree $2$
Conductor $1029$
Sign $0.978 + 0.204i$
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  + (−0.572 + 0.820i)3-s + (−0.761 − 0.648i)4-s + (−0.949 + 0.315i)7-s + (−0.345 − 0.938i)9-s + (0.967 − 0.253i)12-s + (1.33 − 1.28i)13-s + (0.159 + 0.987i)16-s + 1.98·19-s + (0.284 − 0.958i)21-s + (0.718 + 0.695i)25-s + (0.967 + 0.253i)27-s + (0.926 + 0.375i)28-s + (−0.949 − 1.19i)31-s + (−0.345 + 0.938i)36-s + (−0.374 − 0.717i)37-s + ⋯
L(s)  = 1  + (−0.572 + 0.820i)3-s + (−0.761 − 0.648i)4-s + (−0.949 + 0.315i)7-s + (−0.345 − 0.938i)9-s + (0.967 − 0.253i)12-s + (1.33 − 1.28i)13-s + (0.159 + 0.987i)16-s + 1.98·19-s + (0.284 − 0.958i)21-s + (0.718 + 0.695i)25-s + (0.967 + 0.253i)27-s + (0.926 + 0.375i)28-s + (−0.949 − 1.19i)31-s + (−0.345 + 0.938i)36-s + (−0.374 − 0.717i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.978 + 0.204i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.978 + 0.204i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1029\)    =    \(3 \cdot 7^{3}\)
Sign: $0.978 + 0.204i$
Analytic conductor: \(0.513537\)
Root analytic conductor: \(0.716615\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1029} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1029,\ (\ :0),\ 0.978 + 0.204i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6639613232\)
\(L(\frac12)\) \(\approx\) \(0.6639613232\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.572 - 0.820i)T \)
7 \( 1 + (0.949 - 0.315i)T \)
good2 \( 1 + (0.761 + 0.648i)T^{2} \)
5 \( 1 + (-0.718 - 0.695i)T^{2} \)
11 \( 1 + (0.462 - 0.886i)T^{2} \)
13 \( 1 + (-1.33 + 1.28i)T + (0.0320 - 0.999i)T^{2} \)
17 \( 1 + (0.572 - 0.820i)T^{2} \)
19 \( 1 - 1.98T + T^{2} \)
23 \( 1 + (-0.284 + 0.958i)T^{2} \)
29 \( 1 + (-0.284 - 0.958i)T^{2} \)
31 \( 1 + (0.949 + 1.19i)T + (-0.222 + 0.974i)T^{2} \)
37 \( 1 + (0.374 + 0.717i)T + (-0.572 + 0.820i)T^{2} \)
41 \( 1 + (-0.991 + 0.127i)T^{2} \)
43 \( 1 + (-0.903 - 0.508i)T + (0.518 + 0.855i)T^{2} \)
47 \( 1 + (-0.404 - 0.914i)T^{2} \)
53 \( 1 + (-0.926 + 0.375i)T^{2} \)
59 \( 1 + (-0.991 - 0.127i)T^{2} \)
61 \( 1 + (0.0629 + 1.96i)T + (-0.997 + 0.0640i)T^{2} \)
67 \( 1 + (0.713 - 0.894i)T + (-0.222 - 0.974i)T^{2} \)
71 \( 1 + (-0.284 + 0.958i)T^{2} \)
73 \( 1 + (-0.775 + 0.504i)T + (0.404 - 0.914i)T^{2} \)
79 \( 1 + (0.0577 - 0.0278i)T + (0.623 - 0.781i)T^{2} \)
83 \( 1 + (-0.801 - 0.598i)T^{2} \)
89 \( 1 + (0.981 + 0.191i)T^{2} \)
97 \( 1 + (0.119 + 0.150i)T + (-0.222 + 0.974i)T^{2} \)
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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

−10.04398064104510637679528923492, −9.355550433295708703276277798166, −8.900963588685437401942540581776, −7.67197272929745039534703404314, −6.33267054688992662243150528222, −5.64116585802455617253353374442, −5.18486164765434642192283521278, −3.81538540998561155455722490025, −3.21511205696453292371160755514, −0.873750276845986285491539638644, 1.19154475847330643527207413431, 2.98218480885725959111917260408, 3.89308355207852598902437751750, 5.01633211959098711757708566010, 6.01166508879982560105341813400, 6.92299843483940933518601904036, 7.46110022972303192584822376344, 8.613471987749345754299536750691, 9.142432881058828662049750178708, 10.15129639942211134432109954123

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