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Properties

Label 2-1008-63.20-c1-0-42
Degree $2$
Conductor $1008$
Sign $-0.933 - 0.359i$
Analytic cond. $8.04892$
Root an. cond. $2.83706$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.410 − 1.68i)3-s + (−1.41 − 2.45i)5-s + (0.387 − 2.61i)7-s + (−2.66 + 1.38i)9-s + (−0.136 − 0.0789i)11-s + (3.41 − 1.97i)13-s + (−3.55 + 3.39i)15-s − 4.14·17-s − 6.33i·19-s + (−4.56 + 0.420i)21-s + (−0.472 + 0.273i)23-s + (−1.52 + 2.64i)25-s + (3.41 + 3.91i)27-s + (4.02 + 2.32i)29-s + (0.112 − 0.0647i)31-s + ⋯
L(s)  = 1  + (−0.236 − 0.971i)3-s + (−0.634 − 1.09i)5-s + (0.146 − 0.989i)7-s + (−0.887 + 0.460i)9-s + (−0.0412 − 0.0237i)11-s + (0.947 − 0.546i)13-s + (−0.917 + 0.876i)15-s − 1.00·17-s − 1.45i·19-s + (−0.995 + 0.0917i)21-s + (−0.0986 + 0.0569i)23-s + (−0.305 + 0.528i)25-s + (0.657 + 0.753i)27-s + (0.747 + 0.431i)29-s + (0.0201 − 0.0116i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.933 - 0.359i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.933 - 0.359i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-0.933 - 0.359i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (209, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ -0.933 - 0.359i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8955259486\)
\(L(\frac12)\) \(\approx\) \(0.8955259486\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.410 + 1.68i)T \)
7 \( 1 + (-0.387 + 2.61i)T \)
good5 \( 1 + (1.41 + 2.45i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.136 + 0.0789i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-3.41 + 1.97i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 4.14T + 17T^{2} \)
19 \( 1 + 6.33iT - 19T^{2} \)
23 \( 1 + (0.472 - 0.273i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.02 - 2.32i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.112 + 0.0647i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 2.46T + 37T^{2} \)
41 \( 1 + (-1.99 - 3.45i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.28 - 5.68i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (4.33 - 7.50i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 2.60iT - 53T^{2} \)
59 \( 1 + (-1.80 - 3.12i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.91 - 1.68i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.663 - 1.14i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 0.409iT - 71T^{2} \)
73 \( 1 + 15.0iT - 73T^{2} \)
79 \( 1 + (-2.16 + 3.74i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.22 + 5.58i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 5.05T + 89T^{2} \)
97 \( 1 + (-2.18 - 1.26i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.229666634835943776109972982511, −8.460711534314942116079200262999, −7.918082883265252984562154651035, −7.00003751391597489721477669722, −6.25894723825032673147504205906, −4.99988769146477234238745472409, −4.36079804724567198433566287980, −3.01442883682643728549295075749, −1.38163503860270969026424301446, −0.44689886937359850437693710635, 2.23038447067708229759323729107, 3.41940521607627375503376657868, 4.06289174054851839152620713791, 5.26136416129126436663980106234, 6.17355097792371496584497032840, 6.85123093199788685223705773682, 8.237578753535065835224954277807, 8.684691102067599024629858704102, 9.735076637764962516771327439022, 10.48433018159054431721814230150

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