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Properties

Label 2-1029-1029.995-c0-0-0
Degree $2$
Conductor $1029$
Sign $0.341 - 0.940i$
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  + (0.718 + 0.695i)3-s + (−0.997 − 0.0640i)4-s + (0.967 + 0.253i)7-s + (0.0320 + 0.999i)9-s + (−0.672 − 0.740i)12-s + (−1.52 + 1.13i)13-s + (0.991 + 0.127i)16-s + 0.809·19-s + (0.518 + 0.855i)21-s + (0.801 + 0.598i)25-s + (−0.672 + 0.740i)27-s + (−0.949 − 0.315i)28-s + (0.444 − 1.94i)31-s + (0.0320 − 0.999i)36-s + (−1.06 − 0.429i)37-s + ⋯
L(s)  = 1  + (0.718 + 0.695i)3-s + (−0.997 − 0.0640i)4-s + (0.967 + 0.253i)7-s + (0.0320 + 0.999i)9-s + (−0.672 − 0.740i)12-s + (−1.52 + 1.13i)13-s + (0.991 + 0.127i)16-s + 0.809·19-s + (0.518 + 0.855i)21-s + (0.801 + 0.598i)25-s + (−0.672 + 0.740i)27-s + (−0.949 − 0.315i)28-s + (0.444 − 1.94i)31-s + (0.0320 − 0.999i)36-s + (−1.06 − 0.429i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.341 - 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.341 - 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1029\)    =    \(3 \cdot 7^{3}\)
Sign: $0.341 - 0.940i$
Analytic conductor: \(0.513537\)
Root analytic conductor: \(0.716615\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1029} (995, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1029,\ (\ :0),\ 0.341 - 0.940i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.085402869\)
\(L(\frac12)\) \(\approx\) \(1.085402869\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.718 - 0.695i)T \)
7 \( 1 + (-0.967 - 0.253i)T \)
good2 \( 1 + (0.997 + 0.0640i)T^{2} \)
5 \( 1 + (-0.801 - 0.598i)T^{2} \)
11 \( 1 + (-0.926 + 0.375i)T^{2} \)
13 \( 1 + (1.52 - 1.13i)T + (0.284 - 0.958i)T^{2} \)
17 \( 1 + (-0.718 - 0.695i)T^{2} \)
19 \( 1 - 0.809T + T^{2} \)
23 \( 1 + (-0.518 - 0.855i)T^{2} \)
29 \( 1 + (-0.518 + 0.855i)T^{2} \)
31 \( 1 + (-0.444 + 1.94i)T + (-0.900 - 0.433i)T^{2} \)
37 \( 1 + (1.06 + 0.429i)T + (0.718 + 0.695i)T^{2} \)
41 \( 1 + (-0.404 + 0.914i)T^{2} \)
43 \( 1 + (-0.188 - 1.95i)T + (-0.981 + 0.191i)T^{2} \)
47 \( 1 + (0.572 + 0.820i)T^{2} \)
53 \( 1 + (0.949 - 0.315i)T^{2} \)
59 \( 1 + (-0.404 - 0.914i)T^{2} \)
61 \( 1 + (-0.0908 - 0.306i)T + (-0.838 + 0.545i)T^{2} \)
67 \( 1 + (0.319 + 1.40i)T + (-0.900 + 0.433i)T^{2} \)
71 \( 1 + (-0.518 - 0.855i)T^{2} \)
73 \( 1 + (0.857 + 1.64i)T + (-0.572 + 0.820i)T^{2} \)
79 \( 1 + (-0.354 + 0.444i)T + (-0.222 - 0.974i)T^{2} \)
83 \( 1 + (-0.871 + 0.490i)T^{2} \)
89 \( 1 + (-0.159 + 0.987i)T^{2} \)
97 \( 1 + (-0.338 + 1.48i)T + (-0.900 - 0.433i)T^{2} \)
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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

−9.948041318178741677861865086721, −9.429438326523779149611491383429, −8.854989737367791923201969867472, −7.919522307370716014345246834812, −7.33507170328466462760156264733, −5.68943696130306236393135925038, −4.68759940226286921383643703844, −4.48665805833223613561937736553, −3.13204816919304827085564811823, −1.89101818197616024987767248860, 1.07691294775288489017485958331, 2.59970577864019047390994363415, 3.61767498437639243202155277201, 4.86110961773939893971762559340, 5.38645267242630996260787513966, 6.97476874595565803295280122052, 7.57623510068054097587090459043, 8.397957215762952217242815329314, 8.860766741927397990199567491905, 9.987508643547061178453240270985

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