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Properties

Label 2-1008-63.16-c1-0-17
Degree $2$
Conductor $1008$
Sign $0.871 + 0.489i$
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  + (−0.134 + 1.72i)3-s − 3.43·5-s + (1.83 − 1.90i)7-s + (−2.96 − 0.465i)9-s − 4.40·11-s + (1.49 + 2.58i)13-s + (0.463 − 5.93i)15-s + (0.542 + 0.939i)17-s + (3.74 − 6.48i)19-s + (3.03 + 3.43i)21-s + 4.32·23-s + 6.80·25-s + (1.20 − 5.05i)27-s + (1.68 − 2.91i)29-s + (4.68 − 8.11i)31-s + ⋯
L(s)  = 1  + (−0.0778 + 0.996i)3-s − 1.53·5-s + (0.695 − 0.718i)7-s + (−0.987 − 0.155i)9-s − 1.32·11-s + (0.414 + 0.717i)13-s + (0.119 − 1.53i)15-s + (0.131 + 0.227i)17-s + (0.858 − 1.48i)19-s + (0.662 + 0.748i)21-s + 0.901·23-s + 1.36·25-s + (0.231 − 0.972i)27-s + (0.312 − 0.541i)29-s + (0.841 − 1.45i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.871 + 0.489i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.871 + 0.489i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.871 + 0.489i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (961, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ 0.871 + 0.489i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9041932069\)
\(L(\frac12)\) \(\approx\) \(0.9041932069\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.134 - 1.72i)T \)
7 \( 1 + (-1.83 + 1.90i)T \)
good5 \( 1 + 3.43T + 5T^{2} \)
11 \( 1 + 4.40T + 11T^{2} \)
13 \( 1 + (-1.49 - 2.58i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.542 - 0.939i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.74 + 6.48i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 4.32T + 23T^{2} \)
29 \( 1 + (-1.68 + 2.91i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-4.68 + 8.11i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.50 - 4.34i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.20 + 2.08i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.31 - 5.74i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.50 - 2.60i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.530 + 0.919i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-6.20 + 10.7i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.71 - 4.69i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.66 + 2.89i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 12.9T + 71T^{2} \)
73 \( 1 + (8.21 + 14.2i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.17 + 2.03i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.60 + 2.78i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-5.67 + 9.82i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (6.40 - 11.0i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.03757769433775319709474755482, −9.018537202411243707451738032457, −8.115310681219257005413047715612, −7.67734081559569122112741777840, −6.61977552609321413370574051400, −5.08995222113140885350607519244, −4.63292366850322527317443885193, −3.77490553693334013818105305509, −2.80757894917857796805424822032, −0.51075332369512180868114661153, 1.10443791582110071505984657370, 2.68391243086372887363283103096, 3.53296034496149410674278758192, 5.09558105575674461553613998954, 5.54032348333161985501664878679, 6.95424728299491614322372214304, 7.65650935148462760126407869552, 8.263618278074488486296444830662, 8.643883747715041806250055129076, 10.29370722729973284406130207293

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