Properties

Degree $2$
Conductor $1008$
Sign $-0.0633 + 0.997i$
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 − 1.73i·13-s + (−2.5 − 4.33i)19-s + (6 − 3.46i)23-s + (3.5 − 6.06i)25-s + (2.5 − 4.33i)31-s + (−3 − 8.66i)35-s + (5.5 + 9.52i)37-s + 3.46i·41-s − 8.66i·43-s + (−3 − 5.19i)47-s + (−6.5 − 2.59i)49-s + (6 − 10.3i)53-s + ⋯
L(s)  = 1  + (−1.34 + 0.774i)5-s + (−0.188 + 0.981i)7-s + (−0.904 − 0.522i)11-s − 0.480i·13-s + (−0.573 − 0.993i)19-s + (1.25 − 0.722i)23-s + (0.700 − 1.21i)25-s + (0.449 − 0.777i)31-s + (−0.507 − 1.46i)35-s + (0.904 + 1.56i)37-s + 0.541i·41-s − 1.32i·43-s + (−0.437 − 0.757i)47-s + (−0.928 − 0.371i)49-s + (0.824 − 1.42i)53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0633 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0633 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-0.0633 + 0.997i$
Motivic weight: \(1\)
Character: $\chi_{1008} (703, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ -0.0633 + 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4828254863\)
\(L(\frac12)\) \(\approx\) \(0.4828254863\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.73iT - 13T^{2} \)
17 \( 1 + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.5 + 4.33i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-6 + 3.46i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + (-2.5 + 4.33i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5.5 - 9.52i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 3.46iT - 41T^{2} \)
43 \( 1 + 8.66iT - 43T^{2} \)
47 \( 1 + (3 + 5.19i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6 + 10.3i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-6 + 10.3i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (12 - 6.92i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (7.5 + 4.33i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 3.46iT - 71T^{2} \)
73 \( 1 + (4.5 + 2.59i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (10.5 - 6.06i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 18T + 83T^{2} \)
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

−9.844461022451727615447584396806, −8.565486820617685523328977339289, −8.274319603405284888248795870631, −7.22015101156579925092374443938, −6.48619652650809141664367753604, −5.38534997325518752022333756957, −4.43074147001037008747509306031, −3.12930014457128871505268877591, −2.65175136502634313387120931747, −0.24207554647875397757989957833, 1.24864156146184720534655393394, 3.05764118467688180907451683891, 4.20207987542667339646694479725, 4.59179730223089507811351871447, 5.83900768137407154594884121638, 7.24211190464527637064942741586, 7.53251725893981925389476069566, 8.425441559292425038998015201861, 9.270469766163606916296520725264, 10.28287541406076884592324924230

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