Properties

Degree $2$
Conductor $1008$
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

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.895 + 0.444i$
Motivic weight: \(1\)
Character: $\chi_{1008} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ 0.895 + 0.444i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.247521705\)
\(L(\frac12)\) \(\approx\) \(1.247521705\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$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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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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