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Properties

Label 2-1008-63.25-c1-0-0
Degree $2$
Conductor $1008$
Sign $-0.580 - 0.814i$
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  − 1.73i·3-s + (−1.5 + 2.59i)5-s + (2 + 1.73i)7-s − 2.99·9-s + (−1.5 − 2.59i)11-s + (0.5 + 0.866i)13-s + (4.5 + 2.59i)15-s + (−1.5 + 2.59i)17-s + (−3.5 − 6.06i)19-s + (2.99 − 3.46i)21-s + (−4.5 + 7.79i)23-s + (−2 − 3.46i)25-s + 5.19i·27-s + (−1.5 + 2.59i)29-s − 8·31-s + ⋯
L(s)  = 1  − 0.999i·3-s + (−0.670 + 1.16i)5-s + (0.755 + 0.654i)7-s − 0.999·9-s + (−0.452 − 0.783i)11-s + (0.138 + 0.240i)13-s + (1.16 + 0.670i)15-s + (−0.363 + 0.630i)17-s + (−0.802 − 1.39i)19-s + (0.654 − 0.755i)21-s + (−0.938 + 1.62i)23-s + (−0.400 − 0.692i)25-s + 0.999i·27-s + (−0.278 + 0.482i)29-s − 1.43·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.580 - 0.814i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.580 - 0.814i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-0.580 - 0.814i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (529, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ -0.580 - 0.814i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5206056605\)
\(L(\frac12)\) \(\approx\) \(0.5206056605\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.73iT \)
7 \( 1 + (-2 - 1.73i)T \)
good5 \( 1 + (1.5 - 2.59i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.5 + 2.59i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.5 - 0.866i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.5 - 2.59i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.5 + 6.06i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.5 - 7.79i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.5 - 2.59i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 8T + 31T^{2} \)
37 \( 1 + (-0.5 - 0.866i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.5 + 2.59i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + (1.5 - 2.59i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 - 4T + 67T^{2} \)
71 \( 1 + 12T + 71T^{2} \)
73 \( 1 + (5.5 - 9.52i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 16T + 79T^{2} \)
83 \( 1 + (4.5 - 7.79i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (1.5 + 2.59i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-0.5 + 0.866i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.66254984973676347377975244596, −9.135959489140984409522019366506, −8.428826621037116129879141103598, −7.66592794011883750422174849471, −7.01786376633221569298974729601, −6.11132563404614580886402593925, −5.30968695071870815697396813611, −3.80353503678282861715344759396, −2.79433002171681351406492356928, −1.81813297776412226049428693039, 0.22440494053290833674281439569, 2.04048886502950537780268404559, 3.75210440072080894988617971878, 4.44349045357828938087357071859, 4.93586228874475039371567334231, 6.01449311511046754430397384730, 7.45777023735805085764870625157, 8.223596465920531972536540562352, 8.696300147409355328495147833772, 9.776821956213305281573722737937

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