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Properties

Label 2-1008-7.4-c1-0-11
Degree $2$
Conductor $1008$
Sign $0.968 + 0.250i$
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  + (0.5 + 0.866i)5-s + (2.5 − 0.866i)7-s + (2.5 − 4.33i)11-s + 2·13-s + (−3 + 5.19i)17-s + (1 + 1.73i)19-s + (−3 − 5.19i)23-s + (2 − 3.46i)25-s − 3·29-s + (2.5 − 4.33i)31-s + (2 + 1.73i)35-s + (1 + 1.73i)37-s + 8·41-s + 4·43-s + (2 + 3.46i)47-s + ⋯
L(s)  = 1  + (0.223 + 0.387i)5-s + (0.944 − 0.327i)7-s + (0.753 − 1.30i)11-s + 0.554·13-s + (−0.727 + 1.26i)17-s + (0.229 + 0.397i)19-s + (−0.625 − 1.08i)23-s + (0.400 − 0.692i)25-s − 0.557·29-s + (0.449 − 0.777i)31-s + (0.338 + 0.292i)35-s + (0.164 + 0.284i)37-s + 1.24·41-s + 0.609·43-s + (0.291 + 0.505i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 + 0.250i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.968 + 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.968 + 0.250i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (865, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ 0.968 + 0.250i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.942486779\)
\(L(\frac12)\) \(\approx\) \(1.942486779\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 + (-2.5 + 0.866i)T \)
good5 \( 1 + (-0.5 - 0.866i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.5 + 4.33i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 + (3 - 5.19i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1 - 1.73i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3 + 5.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 3T + 29T^{2} \)
31 \( 1 + (-2.5 + 4.33i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1 - 1.73i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 8T + 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 + (-2 - 3.46i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.5 - 7.79i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.5 - 2.59i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6 - 10.3i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1 + 1.73i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 8T + 71T^{2} \)
73 \( 1 + (-7 + 12.1i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.5 - 0.866i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 17T + 83T^{2} \)
89 \( 1 + (9 + 15.5i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 3T + 97T^{2} \)
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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

−10.10051905916083327559476162561, −8.857126283461275325925549495738, −8.396202720501077644362789395296, −7.52725885682715943320558593367, −6.18510876913338181325759321643, −6.03549181075314301035101084618, −4.45597551617817180174755871541, −3.79488237679497876746623222698, −2.41200872600283332440526837655, −1.10121837595018112655442827978, 1.33905286141008585789474465930, 2.35038030609312245103349140896, 3.90126648109492614393109521578, 4.83810623470018991429148729167, 5.48092315071384705466987905613, 6.75777853345412627581954342757, 7.43385166399178688060561871033, 8.419191577310011041877260697941, 9.328792234842016858583479085048, 9.656186245915957256870375452217

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