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Properties

Label 2-1008-63.16-c1-0-39
Degree $2$
Conductor $1008$
Sign $-0.296 + 0.954i$
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  + 1.73i·3-s + 2·5-s + (−2 − 1.73i)7-s − 2.99·9-s − 4·11-s + (−1.5 − 2.59i)13-s + 3.46i·15-s + (−3.5 − 6.06i)17-s + (2.5 − 4.33i)19-s + (2.99 − 3.46i)21-s − 4·23-s − 25-s − 5.19i·27-s + (0.5 − 0.866i)29-s + (−1.5 + 2.59i)31-s + ⋯
L(s)  = 1  + 0.999i·3-s + 0.894·5-s + (−0.755 − 0.654i)7-s − 0.999·9-s − 1.20·11-s + (−0.416 − 0.720i)13-s + 0.894i·15-s + (−0.848 − 1.47i)17-s + (0.573 − 0.993i)19-s + (0.654 − 0.755i)21-s − 0.834·23-s − 0.200·25-s − 0.999i·27-s + (0.0928 − 0.160i)29-s + (−0.269 + 0.466i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.296 + 0.954i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.296 + 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-0.296 + 0.954i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (961, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ -0.296 + 0.954i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5597161737\)
\(L(\frac12)\) \(\approx\) \(0.5597161737\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.73iT \)
7 \( 1 + (2 + 1.73i)T \)
good5 \( 1 - 2T + 5T^{2} \)
11 \( 1 + 4T + 11T^{2} \)
13 \( 1 + (1.5 + 2.59i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (3.5 + 6.06i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.5 + 4.33i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 4T + 23T^{2} \)
29 \( 1 + (-0.5 + 0.866i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.5 - 2.59i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (5.5 - 9.52i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.5 - 7.79i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.5 + 4.33i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.5 - 2.59i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.5 + 2.59i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.5 - 6.06i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.5 + 2.59i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.5 + 11.2i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 8T + 71T^{2} \)
73 \( 1 + (3.5 + 6.06i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.5 + 7.79i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.5 + 0.866i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (7.5 - 12.9i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-8.5 + 14.7i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.702991699231847102673901016457, −9.314973770740014031115097195289, −8.122186233885769782009433309624, −7.16862066147858595167628013561, −6.15420811065330038775964780036, −5.21562391543536164186758884384, −4.62818912346627854723909090341, −3.18336229079832367691854275084, −2.55012168007172695241117274551, −0.22675700639785588232923458412, 1.92241044667580427237582752730, 2.42094554376502999101240905399, 3.84218806776274006326149134720, 5.57825119060736193350311316766, 5.82362049077587346458997702789, 6.75065656430071195415206964635, 7.68413598091776155100365090809, 8.531585213248452425097026950193, 9.340461954375362709521909120861, 10.15329223174100279428251145849

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