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Properties

Label 2-1008-252.187-c1-0-43
Degree $2$
Conductor $1008$
Sign $-0.299 + 0.954i$
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  + (1.14 + 1.30i)3-s − 4.29i·5-s + (−0.334 − 2.62i)7-s + (−0.386 + 2.97i)9-s − 3.60i·11-s + (−4.45 + 2.57i)13-s + (5.58 − 4.91i)15-s + (−0.886 + 0.511i)17-s + (−0.662 + 1.14i)19-s + (3.03 − 3.43i)21-s − 3.14i·23-s − 13.4·25-s + (−4.31 + 2.89i)27-s + (−0.373 + 0.646i)29-s + (4.64 − 8.05i)31-s + ⋯
L(s)  = 1  + (0.660 + 0.751i)3-s − 1.92i·5-s + (−0.126 − 0.991i)7-s + (−0.128 + 0.991i)9-s − 1.08i·11-s + (−1.23 + 0.713i)13-s + (1.44 − 1.26i)15-s + (−0.215 + 0.124i)17-s + (−0.152 + 0.263i)19-s + (0.661 − 0.749i)21-s − 0.655i·23-s − 2.69·25-s + (−0.830 + 0.557i)27-s + (−0.0692 + 0.119i)29-s + (0.834 − 1.44i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.299 + 0.954i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.299 + 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-0.299 + 0.954i$
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: \((2,\ 1008,\ (\ :1/2),\ -0.299 + 0.954i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.439312718\)
\(L(\frac12)\) \(\approx\) \(1.439312718\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.14 - 1.30i)T \)
7 \( 1 + (0.334 + 2.62i)T \)
good5 \( 1 + 4.29iT - 5T^{2} \)
11 \( 1 + 3.60iT - 11T^{2} \)
13 \( 1 + (4.45 - 2.57i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.886 - 0.511i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.662 - 1.14i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 3.14iT - 23T^{2} \)
29 \( 1 + (0.373 - 0.646i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-4.64 + 8.05i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.761 - 1.31i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.22 + 2.43i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (6.14 + 3.54i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.70 - 6.41i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.0746 + 0.129i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.996 - 1.72i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.05 + 2.91i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-7.86 - 4.53i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 11.3iT - 71T^{2} \)
73 \( 1 + (-4.04 + 2.33i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-9.68 + 5.58i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-6.90 + 11.9i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-14.0 - 8.12i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (6.25 + 3.60i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.557464476204838702174427634930, −8.965808746401390194425174162893, −8.209481633740163083425717441656, −7.58397238723670756039893736597, −6.13590635375121496141821886101, −4.97566531434682945899519855147, −4.46532030610841818488668840275, −3.67124872427778674675475186702, −2.11770137577225495583047923179, −0.57041704956460320937423232095, 2.16933005277411679865374955199, 2.64697660046486344225295036754, 3.52734715234420551674410398101, 5.13325705316566388310866347555, 6.30827301513817567644963888872, 6.95784035768456240396610245219, 7.50364759694467750028264823844, 8.377342749663866464219750204032, 9.626254865283536428441772049926, 9.954845908237661952832653415814

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