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Properties

Label 2-1008-252.31-c1-0-8
Degree $2$
Conductor $1008$
Sign $-0.512 - 0.858i$
Analytic cond. $8.04892$
Root an. cond. $2.83706$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.33 + 1.10i)3-s + 1.09i·5-s + (1.10 + 2.40i)7-s + (0.576 − 2.94i)9-s + 0.100i·11-s + (4.27 + 2.46i)13-s + (−1.20 − 1.46i)15-s + (3.23 + 1.86i)17-s + (−2.54 − 4.41i)19-s + (−4.12 − 2.00i)21-s + 9.20i·23-s + 3.79·25-s + (2.46 + 4.57i)27-s + (−3.96 − 6.86i)29-s + (2.41 + 4.18i)31-s + ⋯
L(s)  = 1  + (−0.772 + 0.635i)3-s + 0.490i·5-s + (0.416 + 0.908i)7-s + (0.192 − 0.981i)9-s + 0.0303i·11-s + (1.18 + 0.683i)13-s + (−0.311 − 0.378i)15-s + (0.784 + 0.453i)17-s + (−0.584 − 1.01i)19-s + (−0.899 − 0.436i)21-s + 1.91i·23-s + 0.759·25-s + (0.475 + 0.879i)27-s + (−0.735 − 1.27i)29-s + (0.434 + 0.752i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.512 - 0.858i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.512 - 0.858i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-0.512 - 0.858i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ -0.512 - 0.858i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.207553699\)
\(L(\frac12)\) \(\approx\) \(1.207553699\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (1.33 - 1.10i)T \)
7 \( 1 + (-1.10 - 2.40i)T \)
good5 \( 1 - 1.09iT - 5T^{2} \)
11 \( 1 - 0.100iT - 11T^{2} \)
13 \( 1 + (-4.27 - 2.46i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-3.23 - 1.86i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.54 + 4.41i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 9.20iT - 23T^{2} \)
29 \( 1 + (3.96 + 6.86i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.41 - 4.18i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.77 + 4.80i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (3.91 + 2.25i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.73 - 1.00i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (3.17 - 5.49i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (6.53 - 11.3i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.44 + 2.50i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-8.21 - 4.74i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (10.8 - 6.28i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 14.5iT - 71T^{2} \)
73 \( 1 + (-5.92 - 3.41i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.31 - 1.33i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.112 + 0.195i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (4.90 - 2.83i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-6.47 + 3.73i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.39051833912586436012513685955, −9.343671551015344738450782293070, −8.868401041890021916366997248338, −7.72530540993336586635617580648, −6.60675437585304546033138602611, −5.92605702716210187285726196574, −5.15300532702557155601065444024, −4.08511656816808199700971225521, −3.11465284313523722852632980688, −1.54621856131966482297716637550, 0.67013906921263800885978198223, 1.66365571345954954663020420512, 3.40055227569781238490486897057, 4.58849904711146500785721511897, 5.34614504039244098985602836906, 6.34615967275693922348655903233, 7.04033957252983563881033135205, 8.224234451432649231122749530786, 8.381599422321842553207303437059, 10.00444235064244238458761609109

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