Properties

Degree 2
Conductor $ 2^{4} \cdot 3^{2} \cdot 7 $
Sign $0.5 + 0.866i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.11 − 1.32i)3-s + (0.439 + 0.761i)5-s + (0.5 − 0.866i)7-s + (−0.520 − 2.95i)9-s + (1.93 − 3.35i)11-s + (2.72 + 4.72i)13-s + (1.5 + 0.264i)15-s + 1.65·17-s − 2.41·19-s + (−0.592 − 1.62i)21-s + (1.58 + 2.73i)23-s + (2.11 − 3.66i)25-s + (−4.5 − 2.59i)27-s + (3.02 − 5.23i)29-s + (−2.27 − 3.94i)31-s + ⋯
L(s)  = 1  + (0.642 − 0.766i)3-s + (0.196 + 0.340i)5-s + (0.188 − 0.327i)7-s + (−0.173 − 0.984i)9-s + (0.584 − 1.01i)11-s + (0.756 + 1.30i)13-s + (0.387 + 0.0682i)15-s + 0.400·17-s − 0.553·19-s + (−0.129 − 0.355i)21-s + (0.329 + 0.571i)23-s + (0.422 − 0.732i)25-s + (−0.866 − 0.499i)27-s + (0.561 − 0.972i)29-s + (−0.408 − 0.708i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.5 + 0.866i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.5 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
\( \varepsilon \)  =  $0.5 + 0.866i$
motivic weight  =  \(1\)
character  :  $\chi_{1008} (673, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 1008,\ (\ :1/2),\ 0.5 + 0.866i)\)
\(L(1)\)  \(\approx\)  \(2.201575050\)
\(L(\frac12)\)  \(\approx\)  \(2.201575050\)
\(L(\frac{3}{2})\)   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 \( 1 + (-1.11 + 1.32i)T \)
7 \( 1 + (-0.5 + 0.866i)T \)
good5 \( 1 + (-0.439 - 0.761i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.93 + 3.35i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.72 - 4.72i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 1.65T + 17T^{2} \)
19 \( 1 + 2.41T + 19T^{2} \)
23 \( 1 + (-1.58 - 2.73i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.02 + 5.23i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.27 + 3.94i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 4.55T + 37T^{2} \)
41 \( 1 + (-0.592 - 1.02i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.0923 + 0.160i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.511 - 0.885i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 7.29T + 53T^{2} \)
59 \( 1 + (-3.33 - 5.76i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.29 + 2.24i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.47 + 2.56i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 3.68T + 71T^{2} \)
73 \( 1 + 12.7T + 73T^{2} \)
79 \( 1 + (2.97 - 5.15i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (0.109 - 0.189i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 11.0T + 89T^{2} \)
97 \( 1 + (6.25 - 10.8i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.647581851045754290252241054099, −8.824546491935362081186182693333, −8.290703276922955241009827101882, −7.24099846412641456310032281346, −6.50419620413243778186133385311, −5.88166780282746217921209615962, −4.24306421836378867503753801782, −3.45148047845483548542918218645, −2.24280527903032127388952740519, −1.07557105603813125082445035523, 1.54739200221319459854699463644, 2.87379612677785301862097154924, 3.81649840934088093443857162008, 4.89601658221015665136827576233, 5.51106899598835049743640217072, 6.79934270521761139760556170017, 7.79635229867851623251652596321, 8.750660061213648832307443604644, 9.029739825518235027177698170882, 10.25143137163148994206572070459

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