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Properties

Label 2-1008-112.51-c0-0-0
Degree $2$
Conductor $1008$
Sign $0.997 + 0.0674i$
Analytic cond. $0.503057$
Root an. cond. $0.709265$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.258 − 0.965i)2-s + (−0.866 + 0.499i)4-s + (0.258 + 0.965i)5-s + (−0.5 − 0.866i)7-s + (0.707 + 0.707i)8-s + (0.866 − 0.499i)10-s + (−0.258 + 0.965i)11-s + (1 + i)13-s + (−0.707 + 0.707i)14-s + (0.500 − 0.866i)16-s + (−0.707 − 0.707i)20-s + 22-s + (−0.707 + 1.22i)23-s + (0.707 − 1.22i)26-s + (0.866 + 0.5i)28-s + (0.707 + 0.707i)29-s + ⋯
L(s)  = 1  + (−0.258 − 0.965i)2-s + (−0.866 + 0.499i)4-s + (0.258 + 0.965i)5-s + (−0.5 − 0.866i)7-s + (0.707 + 0.707i)8-s + (0.866 − 0.499i)10-s + (−0.258 + 0.965i)11-s + (1 + i)13-s + (−0.707 + 0.707i)14-s + (0.500 − 0.866i)16-s + (−0.707 − 0.707i)20-s + 22-s + (−0.707 + 1.22i)23-s + (0.707 − 1.22i)26-s + (0.866 + 0.5i)28-s + (0.707 + 0.707i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0674i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0674i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.997 + 0.0674i$
Analytic conductor: \(0.503057\)
Root analytic conductor: \(0.709265\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :0),\ 0.997 + 0.0674i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8169607936\)
\(L(\frac12)\) \(\approx\) \(0.8169607936\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.258 + 0.965i)T \)
3 \( 1 \)
7 \( 1 + (0.5 + 0.866i)T \)
good5 \( 1 + (-0.258 - 0.965i)T + (-0.866 + 0.5i)T^{2} \)
11 \( 1 + (0.258 - 0.965i)T + (-0.866 - 0.5i)T^{2} \)
13 \( 1 + (-1 - i)T + iT^{2} \)
17 \( 1 + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.866 + 0.5i)T^{2} \)
23 \( 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (-0.707 - 0.707i)T + iT^{2} \)
31 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (-0.866 + 0.5i)T^{2} \)
41 \( 1 + 1.41iT - T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (-0.965 - 0.258i)T + (0.866 + 0.5i)T^{2} \)
59 \( 1 + (-0.258 + 0.965i)T + (-0.866 - 0.5i)T^{2} \)
61 \( 1 + (1.36 - 0.366i)T + (0.866 - 0.5i)T^{2} \)
67 \( 1 + (0.866 + 0.5i)T^{2} \)
71 \( 1 + 1.41T + T^{2} \)
73 \( 1 + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (-0.707 + 0.707i)T - iT^{2} \)
89 \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \)
97 \( 1 - T + T^{2} \)
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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

−10.31574787716399159225925430880, −9.592291710762948539084345401205, −8.777271599427962693251754015010, −7.57522447909350276946575941304, −6.98302109783709852883535064907, −5.95424328369658528628034506061, −4.49102360438871762302448518864, −3.77052539717295612858146967316, −2.75896890021932332679562068305, −1.57687872548948320736883100339, 0.955069714973560626517743716396, 2.88818597623519621398257230510, 4.24526414141190827461049336074, 5.32068991521251058078115818134, 5.91262628224297500249301938436, 6.54186807312356518329068448531, 8.023418490753737577124241284730, 8.503914456539371251029047246842, 8.971041997394185773519971734316, 10.00540683453560069904050864325

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