Properties

Degree 2
Conductor $ 2^{4} \cdot 3^{2} \cdot 7 $
Sign $0.549 + 0.835i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.64 − 0.545i)3-s + (0.849 − 1.47i)5-s + (−0.5 − 0.866i)7-s + (2.40 + 1.79i)9-s + (1.23 + 2.14i)11-s + (−0.388 + 0.673i)13-s + (−2.19 + 1.95i)15-s + 2.81·17-s + 4.98·19-s + (0.349 + 1.69i)21-s + (0.356 − 0.616i)23-s + (1.05 + 1.82i)25-s + (−2.97 − 4.25i)27-s + (−2.25 − 3.90i)29-s + (2.54 − 4.41i)31-s + ⋯
L(s)  = 1  + (−0.949 − 0.314i)3-s + (0.380 − 0.658i)5-s + (−0.188 − 0.327i)7-s + (0.801 + 0.597i)9-s + (0.373 + 0.646i)11-s + (−0.107 + 0.186i)13-s + (−0.567 + 0.505i)15-s + 0.681·17-s + 1.14·19-s + (0.0763 + 0.370i)21-s + (0.0742 − 0.128i)23-s + (0.211 + 0.365i)25-s + (−0.572 − 0.819i)27-s + (−0.418 − 0.725i)29-s + (0.457 − 0.793i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.549 + 0.835i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.549 + 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
\( \varepsilon \)  =  $0.549 + 0.835i$
motivic weight  =  \(1\)
character  :  $\chi_{1008} (337, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 1008,\ (\ :1/2),\ 0.549 + 0.835i)\)
\(L(1)\)  \(\approx\)  \(1.262253944\)
\(L(\frac12)\)  \(\approx\)  \(1.262253944\)
\(L(\frac{3}{2})\)   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 \( 1 + (1.64 + 0.545i)T \)
7 \( 1 + (0.5 + 0.866i)T \)
good5 \( 1 + (-0.849 + 1.47i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.23 - 2.14i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.388 - 0.673i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 2.81T + 17T^{2} \)
19 \( 1 - 4.98T + 19T^{2} \)
23 \( 1 + (-0.356 + 0.616i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.25 + 3.90i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.54 + 4.41i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 6.87T + 37T^{2} \)
41 \( 1 + (-2.93 + 5.08i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.32 + 4.03i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-6.49 - 11.2i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 1.88T + 53T^{2} \)
59 \( 1 + (-7.14 + 12.3i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (7.15 + 12.3i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.99 + 6.91i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 10.2T + 71T^{2} \)
73 \( 1 - 4.98T + 73T^{2} \)
79 \( 1 + (4.60 + 7.97i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.40 - 7.63i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 9.65T + 89T^{2} \)
97 \( 1 + (4.32 + 7.48i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.727606728195061880533178283630, −9.339247431094825479802977225965, −7.949760629287195292743727464085, −7.25833466029076660569719569585, −6.39073599063810628837218932047, −5.45774096732587870982354702020, −4.81480360222961662271719978010, −3.70296531609430440258744601911, −1.96787806254207747472817339877, −0.823823687012042183320912864673, 1.14243048187873955093498399281, 2.88025037662652357120487895643, 3.79818926427333372453326641029, 5.17993430191430297266470912020, 5.71661652153089687008437548779, 6.63632664627882867643723871696, 7.29183646429192302290699294480, 8.571651360434897471103270042831, 9.477315614088366329034262968126, 10.24544183595851746002391435879

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