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Properties

Label 2-1008-63.59-c1-0-40
Degree $2$
Conductor $1008$
Sign $-0.0145 + 0.999i$
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  + (1.03 − 1.39i)3-s + 2.20·5-s + (−2.16 − 1.52i)7-s + (−0.876 − 2.86i)9-s + 0.181i·11-s + (2.50 + 1.44i)13-s + (2.27 − 3.07i)15-s + (1.98 − 3.43i)17-s + (−0.867 + 0.500i)19-s + (−4.34 + 1.44i)21-s − 5.61i·23-s − 0.121·25-s + (−4.89 − 1.73i)27-s + (0.703 − 0.406i)29-s + (6.89 − 3.98i)31-s + ⋯
L(s)  = 1  + (0.594 − 0.803i)3-s + 0.987·5-s + (−0.818 − 0.574i)7-s + (−0.292 − 0.956i)9-s + 0.0548i·11-s + (0.694 + 0.400i)13-s + (0.587 − 0.793i)15-s + (0.481 − 0.833i)17-s + (−0.198 + 0.114i)19-s + (−0.948 + 0.315i)21-s − 1.17i·23-s − 0.0242·25-s + (−0.942 − 0.334i)27-s + (0.130 − 0.0754i)29-s + (1.23 − 0.715i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0145 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0145 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-0.0145 + 0.999i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (689, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ -0.0145 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.079118339\)
\(L(\frac12)\) \(\approx\) \(2.079118339\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.03 + 1.39i)T \)
7 \( 1 + (2.16 + 1.52i)T \)
good5 \( 1 - 2.20T + 5T^{2} \)
11 \( 1 - 0.181iT - 11T^{2} \)
13 \( 1 + (-2.50 - 1.44i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.98 + 3.43i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.867 - 0.500i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 5.61iT - 23T^{2} \)
29 \( 1 + (-0.703 + 0.406i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-6.89 + 3.98i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.25 - 2.17i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.612 - 1.06i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.47 + 9.48i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (3.57 - 6.19i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.75 + 1.01i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.27 - 5.67i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.97 - 4.02i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.44 - 5.96i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 11.4iT - 71T^{2} \)
73 \( 1 + (10.1 + 5.88i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-6.35 + 11.0i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-7.19 - 12.4i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (7.11 + 12.3i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (3.01 - 1.73i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.779574300608307723798573914104, −8.929506548213947951748237852196, −8.139800191546703794098111070365, −7.06990267129933442500123645257, −6.46837516496408239825278946121, −5.77472642113523323284397020651, −4.30652431628614938856487763957, −3.16090933069750632378805534226, −2.22888546906438732265520594899, −0.912044712277997847336927135324, 1.79173921396007854478820004136, 2.98105195979013068845608657249, 3.70574101533098211429929365437, 5.07219603128711491415333662593, 5.82946732973482498735413968311, 6.55845347216455667597190804376, 7.996860941188587740441959344811, 8.628673632927356901092227660971, 9.562667849863762094366612379135, 9.913062812574755441042868714911

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