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Properties

Label 2-1008-252.187-c1-0-15
Degree $2$
Conductor $1008$
Sign $0.471 - 0.882i$
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  + (0.232 − 1.71i)3-s + 3.40i·5-s + (2.63 − 0.211i)7-s + (−2.89 − 0.798i)9-s + 5.50i·11-s + (1.96 − 1.13i)13-s + (5.84 + 0.791i)15-s + (−6.64 + 3.83i)17-s + (0.850 − 1.47i)19-s + (0.250 − 4.57i)21-s − 1.26i·23-s − 6.58·25-s + (−2.04 + 4.77i)27-s + (−3.04 + 5.26i)29-s + (−3.66 + 6.34i)31-s + ⋯
L(s)  = 1  + (0.134 − 0.990i)3-s + 1.52i·5-s + (0.996 − 0.0799i)7-s + (−0.963 − 0.266i)9-s + 1.66i·11-s + (0.545 − 0.314i)13-s + (1.50 + 0.204i)15-s + (−1.61 + 0.931i)17-s + (0.195 − 0.338i)19-s + (0.0546 − 0.998i)21-s − 0.264i·23-s − 1.31·25-s + (−0.393 + 0.919i)27-s + (−0.564 + 0.978i)29-s + (−0.657 + 1.13i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.471 - 0.882i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.471 - 0.882i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.471 - 0.882i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (943, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ 0.471 - 0.882i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.549472663\)
\(L(\frac12)\) \(\approx\) \(1.549472663\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.232 + 1.71i)T \)
7 \( 1 + (-2.63 + 0.211i)T \)
good5 \( 1 - 3.40iT - 5T^{2} \)
11 \( 1 - 5.50iT - 11T^{2} \)
13 \( 1 + (-1.96 + 1.13i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (6.64 - 3.83i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.850 + 1.47i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 1.26iT - 23T^{2} \)
29 \( 1 + (3.04 - 5.26i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.66 - 6.34i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.928 - 1.60i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.99 + 2.88i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-6.68 - 3.85i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.822 - 1.42i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.90 + 5.02i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.48 + 6.04i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-9.64 + 5.56i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-7.13 - 4.12i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 5.47iT - 71T^{2} \)
73 \( 1 + (12.5 - 7.25i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.96 + 1.71i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-5.67 + 9.82i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-4.67 - 2.69i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.903 + 0.521i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.39885801492801843161815535194, −9.129499146654063398492726234303, −8.282543031749874971531887744683, −7.30445569282647725541438345686, −6.98520208866097276178159169884, −6.15259358403294445847894066162, −4.91217792462537627816693131224, −3.70136759835118206824699747270, −2.42056240382354998355621456220, −1.74862324058594211006931790973, 0.69616525995507710402483337648, 2.31034269603533811801106592491, 3.88824849024461437394817934996, 4.46105130662487178249373749129, 5.42261807689510934648857602977, 5.93919959595872591189285999577, 7.68235363051066224934076918477, 8.519446159537543186481878939625, 8.922840362629065399372529198228, 9.491805766070882517921617214505

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