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Properties

Label 2-1008-48.35-c1-0-22
Degree $2$
Conductor $1008$
Sign $0.574 + 0.818i$
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  + (−1.40 − 0.139i)2-s + (1.96 + 0.393i)4-s + (−1.17 − 1.17i)5-s + 7-s + (−2.70 − 0.827i)8-s + (1.49 + 1.82i)10-s + (4.54 − 4.54i)11-s + (2.56 + 2.56i)13-s + (−1.40 − 0.139i)14-s + (3.69 + 1.54i)16-s + 2.05i·17-s + (−3.64 + 3.64i)19-s + (−1.84 − 2.77i)20-s + (−7.03 + 5.76i)22-s + 2.27i·23-s + ⋯
L(s)  = 1  + (−0.995 − 0.0988i)2-s + (0.980 + 0.196i)4-s + (−0.527 − 0.527i)5-s + 0.377·7-s + (−0.956 − 0.292i)8-s + (0.472 + 0.576i)10-s + (1.37 − 1.37i)11-s + (0.710 + 0.710i)13-s + (−0.376 − 0.0373i)14-s + (0.922 + 0.385i)16-s + 0.497i·17-s + (−0.836 + 0.836i)19-s + (−0.413 − 0.620i)20-s + (−1.50 + 1.22i)22-s + 0.473i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.574 + 0.818i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.574 + 0.818i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.574 + 0.818i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ 0.574 + 0.818i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.026037199\)
\(L(\frac12)\) \(\approx\) \(1.026037199\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.40 + 0.139i)T \)
3 \( 1 \)
7 \( 1 - T \)
good5 \( 1 + (1.17 + 1.17i)T + 5iT^{2} \)
11 \( 1 + (-4.54 + 4.54i)T - 11iT^{2} \)
13 \( 1 + (-2.56 - 2.56i)T + 13iT^{2} \)
17 \( 1 - 2.05iT - 17T^{2} \)
19 \( 1 + (3.64 - 3.64i)T - 19iT^{2} \)
23 \( 1 - 2.27iT - 23T^{2} \)
29 \( 1 + (0.544 - 0.544i)T - 29iT^{2} \)
31 \( 1 + 10.1iT - 31T^{2} \)
37 \( 1 + (-4.71 + 4.71i)T - 37iT^{2} \)
41 \( 1 - 0.487T + 41T^{2} \)
43 \( 1 + (-7.56 - 7.56i)T + 43iT^{2} \)
47 \( 1 + 0.768T + 47T^{2} \)
53 \( 1 + (-0.269 - 0.269i)T + 53iT^{2} \)
59 \( 1 + (-0.0979 + 0.0979i)T - 59iT^{2} \)
61 \( 1 + (7.41 + 7.41i)T + 61iT^{2} \)
67 \( 1 + (-6.83 + 6.83i)T - 67iT^{2} \)
71 \( 1 + 8.66iT - 71T^{2} \)
73 \( 1 + 13.6iT - 73T^{2} \)
79 \( 1 - 9.29iT - 79T^{2} \)
83 \( 1 + (-9.76 - 9.76i)T + 83iT^{2} \)
89 \( 1 - 7.27T + 89T^{2} \)
97 \( 1 - 10.4T + 97T^{2} \)
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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

−9.531246764543359608231707658043, −9.001664655787970695612370911166, −8.240868763311339853062136757409, −7.73936570798680902872417278327, −6.26238944932518587451745723254, −6.08676824479267604425699550692, −4.22333526419942160458666926184, −3.58188948094006048695761331224, −1.91462847798055176843778669518, −0.794806933020235063409606263711, 1.17986569641434803778329040011, 2.47273759632287435287940189936, 3.69906848087199183515779870151, 4.85151562934297749158110737435, 6.21223628420707504715423699714, 6.98951015888518915839444002938, 7.47709836729456970462286692673, 8.604553711777575496862503510914, 9.095379963946407254277742311766, 10.12528068409594777428281429830

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