Properties

Degree 2
Conductor $ 2^{4} \cdot 3^{2} \cdot 7 $
Sign $0.895 - 0.444i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.5 + 0.866i)5-s + (−2 − 1.73i)7-s + (−1.5 − 0.866i)11-s + (4.5 + 2.59i)17-s + (3.5 + 6.06i)19-s + (7.5 − 4.33i)23-s + (−1 + 1.73i)25-s + 6·29-s + (2.5 − 4.33i)31-s + (4.5 + 0.866i)35-s + (2.5 + 4.33i)37-s + 6.92i·41-s − 3.46i·43-s + (−1.5 − 2.59i)47-s + (1.00 + 6.92i)49-s + ⋯
L(s)  = 1  + (−0.670 + 0.387i)5-s + (−0.755 − 0.654i)7-s + (−0.452 − 0.261i)11-s + (1.09 + 0.630i)17-s + (0.802 + 1.39i)19-s + (1.56 − 0.902i)23-s + (−0.200 + 0.346i)25-s + 1.11·29-s + (0.449 − 0.777i)31-s + (0.760 + 0.146i)35-s + (0.410 + 0.711i)37-s + 1.08i·41-s − 0.528i·43-s + (−0.218 − 0.378i)47-s + (0.142 + 0.989i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.895 - 0.444i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.895 - 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
\( \varepsilon \)  =  $0.895 - 0.444i$
motivic weight  =  \(1\)
character  :  $\chi_{1008} (703, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 1008,\ (\ :1/2),\ 0.895 - 0.444i)\)
\(L(1)\)  \(\approx\)  \(1.247521705\)
\(L(\frac12)\)  \(\approx\)  \(1.247521705\)
\(L(\frac{3}{2})\)   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 \( 1 + (2 + 1.73i)T \)
good5 \( 1 + (1.5 - 0.866i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.5 + 0.866i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 + (-4.5 - 2.59i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.5 - 6.06i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-7.5 + 4.33i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + (-2.5 + 4.33i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.5 - 4.33i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 6.92iT - 41T^{2} \)
43 \( 1 + 3.46iT - 43T^{2} \)
47 \( 1 + (1.5 + 2.59i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.5 - 7.79i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.5 - 7.79i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-7.5 + 4.33i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.5 + 2.59i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 3.46iT - 71T^{2} \)
73 \( 1 + (-1.5 - 0.866i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.5 + 2.59i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 12T + 83T^{2} \)
89 \( 1 + (-10.5 + 6.06i)T + (44.5 - 77.0i)T^{2} \)
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

−10.20772464581508182193139810827, −9.311231069464597711469604589178, −8.079951732229647135050464620021, −7.66668604542390609621200542253, −6.66638701623869078768142136065, −5.86876215308068170833862885488, −4.67308370474788940286620195938, −3.55597129459332455583221588646, −2.98321202324996564331742231004, −1.01797583092597064386126394471, 0.76715783948987930153868730182, 2.68758930852754300653482093215, 3.41647627513991490218230436430, 4.84179636730765277610529246676, 5.37141986948431102430541794778, 6.64567947275818139807415883882, 7.38804932730689304189785873614, 8.253093769894076859742591889342, 9.196984923626832527721402007778, 9.678633000191916942296011043060

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