Properties

Degree $2$
Conductor $1008$
Sign $0.895 - 0.444i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3 − 1.73i)5-s + (−0.5 + 2.59i)7-s + (−3 − 1.73i)11-s + 5.19i·13-s + (6 + 3.46i)17-s + (3.5 + 6.06i)19-s + (3.5 − 6.06i)25-s + (2.5 − 4.33i)31-s + (3 + 8.66i)35-s + (−0.5 − 0.866i)37-s − 10.3i·41-s − 1.73i·43-s + (3 + 5.19i)47-s + (−6.5 − 2.59i)49-s − 12·55-s + ⋯
L(s)  = 1  + (1.34 − 0.774i)5-s + (−0.188 + 0.981i)7-s + (−0.904 − 0.522i)11-s + 1.44i·13-s + (1.45 + 0.840i)17-s + (0.802 + 1.39i)19-s + (0.700 − 1.21i)25-s + (0.449 − 0.777i)31-s + (0.507 + 1.46i)35-s + (−0.0821 − 0.142i)37-s − 1.62i·41-s − 0.264i·43-s + (0.437 + 0.757i)47-s + (−0.928 − 0.371i)49-s − 1.61·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.895 - 0.444i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.895 - 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.895 - 0.444i$
Motivic weight: \(1\)
Character: $\chi_{1008} (703, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ 0.895 - 0.444i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.966649279\)
\(L(\frac12)\) \(\approx\) \(1.966649279\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 + (0.5 - 2.59i)T \)
good5 \( 1 + (-3 + 1.73i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (3 + 1.73i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 5.19iT - 13T^{2} \)
17 \( 1 + (-6 - 3.46i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.5 - 6.06i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + (-2.5 + 4.33i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 10.3iT - 41T^{2} \)
43 \( 1 + 1.73iT - 43T^{2} \)
47 \( 1 + (-3 - 5.19i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.5 + 0.866i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 3.46iT - 71T^{2} \)
73 \( 1 + (-7.5 - 4.33i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-13.5 + 7.79i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 6T + 83T^{2} \)
89 \( 1 + (6 - 3.46i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 6.92iT - 97T^{2} \)
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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

−9.837652035559882592675033631905, −9.299750802981559027408933590020, −8.513528867911589940564655701781, −7.69510935076730065069043461413, −6.24230118974771331470176126924, −5.71716745923910514169809733608, −5.14740552790729556275239509453, −3.71641947114840228992975792657, −2.38012340947232393790553933578, −1.46824724639271500189539877841, 1.00678612609309744881587395639, 2.69059908666722207005943690410, 3.19548452408356376078755964195, 4.95335560551707424016632813275, 5.48892943691590873712993826794, 6.60243585751128372839360272086, 7.33547611781743160015597808384, 8.013045436342747691905712330580, 9.484821166444071539078997288989, 10.00618018602297004553007880282

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