Properties

Degree 2
Conductor $ 2^{4} \cdot 3^{2} \cdot 7 $
Sign $0.998 + 0.0529i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.71 + 0.255i)3-s + (1.81 − 3.13i)5-s + (0.5 + 0.866i)7-s + (2.86 − 0.875i)9-s + (1.95 + 3.39i)11-s + (−2.53 + 4.39i)13-s + (−2.30 + 5.83i)15-s + 1.03·17-s + 2.50·19-s + (−1.07 − 1.35i)21-s + (2.47 − 4.29i)23-s + (−4.06 − 7.04i)25-s + (−4.69 + 2.23i)27-s + (4.60 + 7.97i)29-s + (−0.422 + 0.731i)31-s + ⋯
L(s)  = 1  + (−0.989 + 0.147i)3-s + (0.810 − 1.40i)5-s + (0.188 + 0.327i)7-s + (0.956 − 0.291i)9-s + (0.590 + 1.02i)11-s + (−0.703 + 1.21i)13-s + (−0.594 + 1.50i)15-s + 0.250·17-s + 0.575·19-s + (−0.235 − 0.295i)21-s + (0.516 − 0.895i)23-s + (−0.813 − 1.40i)25-s + (−0.902 + 0.429i)27-s + (0.854 + 1.48i)29-s + (−0.0758 + 0.131i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0529i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0529i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
\( \varepsilon \)  =  $0.998 + 0.0529i$
motivic weight  =  \(1\)
character  :  $\chi_{1008} (337, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 1008,\ (\ :1/2),\ 0.998 + 0.0529i)\)
\(L(1)\)  \(\approx\)  \(1.430235451\)
\(L(\frac12)\)  \(\approx\)  \(1.430235451\)
\(L(\frac{3}{2})\)   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 \( 1 + (1.71 - 0.255i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
good5 \( 1 + (-1.81 + 3.13i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.95 - 3.39i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.53 - 4.39i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 1.03T + 17T^{2} \)
19 \( 1 - 2.50T + 19T^{2} \)
23 \( 1 + (-2.47 + 4.29i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.60 - 7.97i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.422 - 0.731i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 4.84T + 37T^{2} \)
41 \( 1 + (-2.07 + 3.59i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.20 + 3.81i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (3.93 + 6.82i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 12.2T + 53T^{2} \)
59 \( 1 + (5.60 - 9.70i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.208 + 0.360i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.02 + 8.70i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 5.05T + 71T^{2} \)
73 \( 1 - 7.20T + 73T^{2} \)
79 \( 1 + (-7.56 - 13.1i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.932 - 1.61i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 0.669T + 89T^{2} \)
97 \( 1 + (7.63 + 13.2i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.767109919040719141391788275727, −9.323513544162589419084711708276, −8.578337201984243619366910815328, −7.16039884782235399277522929786, −6.54105891187425255837441999525, −5.36412096112139987816640096344, −4.91749343545374917000943027908, −4.16844378365029645867305905813, −2.08586393242414142155157691636, −1.11070930205010049140711775876, 0.977220995333953795482071952560, 2.57238193044226460596354933740, 3.56041807731883871826669307090, 4.99341134149126926130745149952, 5.90887080692705977292652798543, 6.36977546219676106848934342645, 7.33605400977824105954321826829, 7.992193523011680517008559261638, 9.658260321200064995863145891672, 9.974463188783365509474824841174

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