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Properties

Label 2-1029-1029.323-c0-0-0
Degree $2$
Conductor $1029$
Sign $0.994 - 0.105i$
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  + (0.967 − 0.253i)3-s + (0.518 + 0.855i)4-s + (−0.572 − 0.820i)7-s + (0.871 − 0.490i)9-s + (0.718 + 0.695i)12-s + (−0.544 + 0.599i)13-s + (−0.462 + 0.886i)16-s + 1.85·19-s + (−0.761 − 0.648i)21-s + (−0.672 − 0.740i)25-s + (0.718 − 0.695i)27-s + (0.404 − 0.914i)28-s + (−0.934 − 0.449i)31-s + (0.871 + 0.490i)36-s + (−1.88 + 0.242i)37-s + ⋯
L(s)  = 1  + (0.967 − 0.253i)3-s + (0.518 + 0.855i)4-s + (−0.572 − 0.820i)7-s + (0.871 − 0.490i)9-s + (0.718 + 0.695i)12-s + (−0.544 + 0.599i)13-s + (−0.462 + 0.886i)16-s + 1.85·19-s + (−0.761 − 0.648i)21-s + (−0.672 − 0.740i)25-s + (0.718 − 0.695i)27-s + (0.404 − 0.914i)28-s + (−0.934 − 0.449i)31-s + (0.871 + 0.490i)36-s + (−1.88 + 0.242i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.105i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.105i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1029\)    =    \(3 \cdot 7^{3}\)
Sign: $0.994 - 0.105i$
Analytic conductor: \(0.513537\)
Root analytic conductor: \(0.716615\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1029} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1029,\ (\ :0),\ 0.994 - 0.105i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.482025688\)
\(L(\frac12)\) \(\approx\) \(1.482025688\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.967 + 0.253i)T \)
7 \( 1 + (0.572 + 0.820i)T \)
good2 \( 1 + (-0.518 - 0.855i)T^{2} \)
5 \( 1 + (0.672 + 0.740i)T^{2} \)
11 \( 1 + (-0.991 - 0.127i)T^{2} \)
13 \( 1 + (0.544 - 0.599i)T + (-0.0960 - 0.995i)T^{2} \)
17 \( 1 + (-0.967 + 0.253i)T^{2} \)
19 \( 1 - 1.85T + T^{2} \)
23 \( 1 + (0.761 + 0.648i)T^{2} \)
29 \( 1 + (0.761 - 0.648i)T^{2} \)
31 \( 1 + (0.934 + 0.449i)T + (0.623 + 0.781i)T^{2} \)
37 \( 1 + (1.88 - 0.242i)T + (0.967 - 0.253i)T^{2} \)
41 \( 1 + (-0.926 - 0.375i)T^{2} \)
43 \( 1 + (0.0639 - 1.99i)T + (-0.997 - 0.0640i)T^{2} \)
47 \( 1 + (0.949 - 0.315i)T^{2} \)
53 \( 1 + (-0.404 - 0.914i)T^{2} \)
59 \( 1 + (-0.926 + 0.375i)T^{2} \)
61 \( 1 + (-0.160 + 1.66i)T + (-0.981 - 0.191i)T^{2} \)
67 \( 1 + (1.74 - 0.839i)T + (0.623 - 0.781i)T^{2} \)
71 \( 1 + (0.761 + 0.648i)T^{2} \)
73 \( 1 + (-0.316 + 1.95i)T + (-0.949 - 0.315i)T^{2} \)
79 \( 1 + (-0.0427 - 0.187i)T + (-0.900 + 0.433i)T^{2} \)
83 \( 1 + (0.345 + 0.938i)T^{2} \)
89 \( 1 + (0.838 - 0.545i)T^{2} \)
97 \( 1 + (0.512 + 0.246i)T + (0.623 + 0.781i)T^{2} \)
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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

−9.876959618371047260811176642719, −9.395680020499616783961192188944, −8.350470217128051951467399352644, −7.49111596639367456190517923127, −7.17645092699455287741478562734, −6.23764227134476373744167586132, −4.61589252270033542553056800160, −3.60529956642607784340197160849, −3.01007122518934638029185365219, −1.75220445406271491633037391634, 1.71266771300399751140007406378, 2.77767950935599804250941492797, 3.58899413989859371472823823517, 5.24329727036921451647048436732, 5.56470218657466335610070142527, 7.00888222686581291420173808985, 7.46202467798250222114241570398, 8.719083452342693150699569246742, 9.362746777709912137817766707874, 9.998484497318298298392291827258

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