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Properties

Label 2-1008-63.4-c1-0-10
Degree $2$
Conductor $1008$
Sign $-0.932 - 0.360i$
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  + (−0.619 + 1.61i)3-s + 1.76·5-s + (1.85 + 1.88i)7-s + (−2.23 − 2.00i)9-s − 6.12·11-s + (−0.380 + 0.658i)13-s + (−1.09 + 2.84i)15-s + (−3.42 + 5.92i)17-s + (−0.971 − 1.68i)19-s + (−4.20 + 1.82i)21-s + 0.421·23-s − 1.89·25-s + (4.62 − 2.36i)27-s + (0.732 + 1.26i)29-s + (3.85 + 6.67i)31-s + ⋯
L(s)  = 1  + (−0.357 + 0.933i)3-s + 0.787·5-s + (0.699 + 0.714i)7-s + (−0.744 − 0.668i)9-s − 1.84·11-s + (−0.105 + 0.182i)13-s + (−0.281 + 0.735i)15-s + (−0.829 + 1.43i)17-s + (−0.222 − 0.385i)19-s + (−0.917 + 0.398i)21-s + 0.0877·23-s − 0.379·25-s + (0.890 − 0.455i)27-s + (0.135 + 0.235i)29-s + (0.691 + 1.19i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.932 - 0.360i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.932 - 0.360i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-0.932 - 0.360i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ -0.932 - 0.360i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9802226948\)
\(L(\frac12)\) \(\approx\) \(0.9802226948\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.619 - 1.61i)T \)
7 \( 1 + (-1.85 - 1.88i)T \)
good5 \( 1 - 1.76T + 5T^{2} \)
11 \( 1 + 6.12T + 11T^{2} \)
13 \( 1 + (0.380 - 0.658i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (3.42 - 5.92i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.971 + 1.68i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 0.421T + 23T^{2} \)
29 \( 1 + (-0.732 - 1.26i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.85 - 6.67i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.44 - 2.49i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (3.47 - 6.01i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.33 + 7.49i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.830 + 1.43i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.112 - 0.195i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.993 - 1.72i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.17 + 8.96i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.39 - 5.87i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 10.7T + 71T^{2} \)
73 \( 1 + (-0.153 + 0.265i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (6.72 - 11.6i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-1.56 - 2.70i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-1.30 - 2.25i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.81 + 3.14i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.38189225547734825905116619440, −9.686130829719009485487802486135, −8.560343719429158833740328540803, −8.273182243629200814748915806201, −6.73725624446179996464127806201, −5.76788724466432163558940045793, −5.19607774219170389925403083351, −4.42936745977076228009183857697, −2.96648913038438996129268562835, −1.98406000552015288990483591268, 0.42949306127341024110573346703, 1.98521449318396799475856611423, 2.74724866109442533872490762208, 4.60723857661338138857017007134, 5.33180840133492559648434901668, 6.13767777638052871905468649640, 7.24267969715650438086165033458, 7.74018835764880491132403204420, 8.535820929234162913085156819224, 9.792007018649337092551344414719

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