Properties

Degree 2
Conductor $ 2^{4} \cdot 3^{2} \cdot 7 $
Sign $0.697 - 0.716i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.385 − 1.68i)3-s + (1.15 + 2.00i)5-s + (−0.5 + 0.866i)7-s + (−2.70 + 1.30i)9-s + (0.655 − 1.13i)11-s + (1.93 + 3.34i)13-s + (2.93 − 2.72i)15-s − 0.326·17-s − 3.08·19-s + (1.65 + 0.510i)21-s + (1.81 + 3.15i)23-s + (−0.169 + 0.293i)25-s + (3.24 + 4.05i)27-s + (−4.75 + 8.22i)29-s + (3.24 + 5.62i)31-s + ⋯
L(s)  = 1  + (−0.222 − 0.974i)3-s + (0.516 + 0.894i)5-s + (−0.188 + 0.327i)7-s + (−0.900 + 0.434i)9-s + (0.197 − 0.342i)11-s + (0.536 + 0.928i)13-s + (0.757 − 0.703i)15-s − 0.0792·17-s − 0.707·19-s + (0.361 + 0.111i)21-s + (0.379 + 0.656i)23-s + (−0.0338 + 0.0586i)25-s + (0.624 + 0.781i)27-s + (−0.882 + 1.52i)29-s + (0.583 + 1.01i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.697 - 0.716i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.697 - 0.716i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
\( \varepsilon \)  =  $0.697 - 0.716i$
motivic weight  =  \(1\)
character  :  $\chi_{1008} (673, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 1008,\ (\ :1/2),\ 0.697 - 0.716i)\)
\(L(1)\)  \(\approx\)  \(1.379795774\)
\(L(\frac12)\)  \(\approx\)  \(1.379795774\)
\(L(\frac{3}{2})\)   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 \( 1 + (0.385 + 1.68i)T \)
7 \( 1 + (0.5 - 0.866i)T \)
good5 \( 1 + (-1.15 - 2.00i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.655 + 1.13i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.93 - 3.34i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 0.326T + 17T^{2} \)
19 \( 1 + 3.08T + 19T^{2} \)
23 \( 1 + (-1.81 - 3.15i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.75 - 8.22i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.24 - 5.62i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 2.31T + 37T^{2} \)
41 \( 1 + (-4.74 - 8.22i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.0493 + 0.0855i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.108 + 0.187i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 13.7T + 53T^{2} \)
59 \( 1 + (1.22 + 2.12i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-7.68 + 13.3i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.93 + 5.08i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 1.77T + 71T^{2} \)
73 \( 1 + 5.99T + 73T^{2} \)
79 \( 1 + (7.29 - 12.6i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.04 - 5.26i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 5.52T + 89T^{2} \)
97 \( 1 + (-2.99 + 5.18i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.18898225517629911103656483987, −9.062055853826327389299100822160, −8.487634756155246330866684579669, −7.28132403833172272479586015128, −6.63511619856218424451507094305, −6.10862301605508995306014677555, −5.09791450443251215935423290310, −3.54494205123792899073369566456, −2.52434041632518813850684728423, −1.47307077394530054384548780923, 0.67698163222371326710688079041, 2.44180099203689080766932613397, 3.85083996976866560884685640932, 4.50209264408797911020956276671, 5.57513502041377734039527921512, 6.07542337613324953622643041757, 7.39959085492892927618080265475, 8.569445872025329596448125435209, 9.008476050614108434689920798104, 9.984412851887952790588327848854

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