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Properties

Label 2-1008-252.67-c0-0-0
Degree $2$
Conductor $1008$
Sign $-0.110 - 0.993i$
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  + (−0.866 + 0.5i)3-s − 5-s + (0.866 + 0.5i)7-s + (0.499 − 0.866i)9-s + i·11-s + (0.5 − 0.866i)13-s + (0.866 − 0.5i)15-s + (−0.5 + 0.866i)17-s + (−0.866 + 0.5i)19-s − 0.999·21-s + i·23-s + 0.999i·27-s + (0.5 + 0.866i)29-s + (−0.5 − 0.866i)33-s + (−0.866 − 0.5i)35-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)3-s − 5-s + (0.866 + 0.5i)7-s + (0.499 − 0.866i)9-s + i·11-s + (0.5 − 0.866i)13-s + (0.866 − 0.5i)15-s + (−0.5 + 0.866i)17-s + (−0.866 + 0.5i)19-s − 0.999·21-s + i·23-s + 0.999i·27-s + (0.5 + 0.866i)29-s + (−0.5 − 0.866i)33-s + (−0.866 − 0.5i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.110 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.110 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-0.110 - 0.993i$
Analytic conductor: \(0.503057\)
Root analytic conductor: \(0.709265\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (319, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :0),\ -0.110 - 0.993i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6171817191\)
\(L(\frac12)\) \(\approx\) \(0.6171817191\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.866 - 0.5i)T \)
7 \( 1 + (-0.866 - 0.5i)T \)
good5 \( 1 + T + T^{2} \)
11 \( 1 - iT - T^{2} \)
13 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
17 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
23 \( 1 - iT - T^{2} \)
29 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
41 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
59 \( 1 + (-1.73 + i)T + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 + 2iT - T^{2} \)
73 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
89 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
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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

−10.47798594653419237999896184923, −9.782453491328016472892917300071, −8.530066936067179422337228040283, −8.047784938718644521691283578883, −6.98124885064642614002063132719, −6.03184782441006453297710064859, −5.06557388815782573449609739789, −4.35816064857474253969256452712, −3.45445309405558840351209183118, −1.64476090390049158851296281161, 0.68565665448265552785458784317, 2.28449149183528066827763165953, 4.02858587833784317557035126449, 4.54750616976340650115264709035, 5.68302501438110682787747212629, 6.70940938097809815351325426679, 7.29562196791550169204664183095, 8.277124194738644185558325920446, 8.783411887066597328959371939042, 10.31415663385545271075124963517

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