Properties

Label 2-1008-63.58-c1-0-40
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: $\chi_{1008} (625, \cdot )$
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

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

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