Properties

Degree 2
Conductor $ 2^{4} \cdot 3^{2} \cdot 7 $
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

\( d \)  =  \(2\)
\( N \)  =  \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
\( \varepsilon \)  =  $-0.0633 - 0.997i$
motivic weight  =  \(1\)
character  :  $\chi_{1008} (271, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 1008,\ (\ :1/2),\ -0.0633 - 0.997i)\)
\(L(1)\)  \(\approx\)  \(0.4828254863\)
\(L(\frac12)\)  \(\approx\)  \(0.4828254863\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 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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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

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

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