Properties

Label 2-1008-7.3-c0-0-0
Degree $2$
Conductor $1008$
Sign $0.605 + 0.795i$
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  + (−0.5 − 0.866i)7-s − 1.73i·13-s + (1.5 − 0.866i)19-s + (−0.5 + 0.866i)25-s + (1.5 + 0.866i)31-s + (−0.5 − 0.866i)37-s − 43-s + (−0.499 + 0.866i)49-s + (−0.5 + 0.866i)67-s + (1.5 + 0.866i)73-s + (−0.5 − 0.866i)79-s + (−1.49 + 0.866i)91-s + (−1.5 + 0.866i)103-s + (−0.5 + 0.866i)109-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)7-s − 1.73i·13-s + (1.5 − 0.866i)19-s + (−0.5 + 0.866i)25-s + (1.5 + 0.866i)31-s + (−0.5 − 0.866i)37-s − 43-s + (−0.499 + 0.866i)49-s + (−0.5 + 0.866i)67-s + (1.5 + 0.866i)73-s + (−0.5 − 0.866i)79-s + (−1.49 + 0.866i)91-s + (−1.5 + 0.866i)103-s + (−0.5 + 0.866i)109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.605 + 0.795i$
Analytic conductor: \(0.503057\)
Root analytic conductor: \(0.709265\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (577, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :0),\ 0.605 + 0.795i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9581719678\)
\(L(\frac12)\) \(\approx\) \(0.9581719678\)
\(L(1)\) 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 \( 1 \)
7 \( 1 + (0.5 + 0.866i)T \)
good5 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + 1.73iT - T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + T + T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (-0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 - T^{2} \)
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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

−10.08491808314350664402309233785, −9.382805276666471909970367477289, −8.271244122775923874585854312013, −7.50096675308085746290180006517, −6.81402807316948403893814708305, −5.66661645191246669430452462208, −4.90511058277431890498961637896, −3.58277185294013142610900405903, −2.89861079002382372345450620010, −0.986056299792670216456693117002, 1.77648099154799379502969259784, 2.95292994725343177894182080733, 4.07641828237881044156690447671, 5.11497174044819521420915479736, 6.14868134254294200480782182368, 6.74044434232873756386108573808, 7.889106721122961081874377707394, 8.672773592887718594803035420928, 9.636972443884801630147498142305, 9.919545298429506787495383923076

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