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Properties

Label 2-1008-63.4-c1-0-27
Degree $2$
Conductor $1008$
Sign $-0.595 + 0.803i$
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  + (−1.09 − 1.33i)3-s − 0.146·5-s + (−0.0802 + 2.64i)7-s + (−0.580 + 2.94i)9-s − 1.66·11-s + (0.0999 − 0.173i)13-s + (0.160 + 0.195i)15-s + (3.13 − 5.43i)17-s + (−3.45 − 5.99i)19-s + (3.62 − 2.80i)21-s + 6.18·23-s − 4.97·25-s + (4.57 − 2.46i)27-s + (−2.46 − 4.27i)29-s + (−1.25 − 2.18i)31-s + ⋯
L(s)  = 1  + (−0.635 − 0.772i)3-s − 0.0654·5-s + (−0.0303 + 0.999i)7-s + (−0.193 + 0.981i)9-s − 0.501·11-s + (0.0277 − 0.0480i)13-s + (0.0415 + 0.0505i)15-s + (0.760 − 1.31i)17-s + (−0.793 − 1.37i)19-s + (0.791 − 0.611i)21-s + 1.28·23-s − 0.995·25-s + (0.880 − 0.473i)27-s + (−0.458 − 0.793i)29-s + (−0.226 − 0.391i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.595 + 0.803i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.595 + 0.803i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-0.595 + 0.803i$
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: \((2,\ 1008,\ (\ :1/2),\ -0.595 + 0.803i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7373886170\)
\(L(\frac12)\) \(\approx\) \(0.7373886170\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.09 + 1.33i)T \)
7 \( 1 + (0.0802 - 2.64i)T \)
good5 \( 1 + 0.146T + 5T^{2} \)
11 \( 1 + 1.66T + 11T^{2} \)
13 \( 1 + (-0.0999 + 0.173i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-3.13 + 5.43i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.45 + 5.99i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 6.18T + 23T^{2} \)
29 \( 1 + (2.46 + 4.27i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.25 + 2.18i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.50 + 6.06i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.15 + 2.00i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.940 - 1.62i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.905 - 1.56i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.67 - 4.62i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.28 + 3.95i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.339 + 0.587i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.09 + 5.35i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 1.27T + 71T^{2} \)
73 \( 1 + (0.778 - 1.34i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.39 + 11.0i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.75 + 6.50i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-4.53 - 7.85i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (3.98 + 6.90i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.545079393886598016496688236906, −8.877259064900346796609650626974, −7.80322685905414786979769694279, −7.20486572672365607928375717615, −6.19262000882139860148903982434, −5.42352993331072958894777310396, −4.71210951524755410478581433298, −2.96449379114014703104196657470, −2.09990517912321776552669176690, −0.37670006015039905100427127169, 1.40988635140530493584469491231, 3.39044422689488737308105870919, 3.99254552602934164349236763065, 5.03938565246227374633953736297, 5.90090608065334896756615509680, 6.76443510195015765350034754091, 7.79340914110394224510546716617, 8.604559936279045666338398690837, 9.716819172926486683368413256387, 10.43738643999672482937179651349

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