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Properties

Label 2-1008-21.17-c1-0-9
Degree $2$
Conductor $1008$
Sign $0.569 + 0.821i$
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  + (0.358 + 0.621i)5-s + (−2.62 − 0.358i)7-s + (−2.59 − 1.5i)11-s + 2.44i·13-s + (2.95 − 5.12i)17-s + (5.12 − 2.95i)19-s + (3.67 − 2.12i)23-s + (2.24 − 3.88i)25-s − 7.24i·29-s + (7.86 + 4.54i)31-s + (−0.717 − 1.75i)35-s + (0.121 + 0.210i)37-s − 11.8·41-s + 0.242·43-s + (−2.95 − 5.12i)47-s + ⋯
L(s)  = 1  + (0.160 + 0.277i)5-s + (−0.990 − 0.135i)7-s + (−0.783 − 0.452i)11-s + 0.679i·13-s + (0.717 − 1.24i)17-s + (1.17 − 0.678i)19-s + (0.766 − 0.442i)23-s + (0.448 − 0.776i)25-s − 1.34i·29-s + (1.41 + 0.815i)31-s + (−0.121 − 0.297i)35-s + (0.0199 + 0.0345i)37-s − 1.84·41-s + 0.0370·43-s + (−0.431 − 0.747i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.569 + 0.821i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.569 + 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.569 + 0.821i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ 0.569 + 0.821i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.299688999\)
\(L(\frac12)\) \(\approx\) \(1.299688999\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.62 + 0.358i)T \)
good5 \( 1 + (-0.358 - 0.621i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (2.59 + 1.5i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 2.44iT - 13T^{2} \)
17 \( 1 + (-2.95 + 5.12i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-5.12 + 2.95i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.67 + 2.12i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 7.24iT - 29T^{2} \)
31 \( 1 + (-7.86 - 4.54i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.121 - 0.210i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 11.8T + 41T^{2} \)
43 \( 1 - 0.242T + 43T^{2} \)
47 \( 1 + (2.95 + 5.12i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6.27 - 3.62i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.03 + 6.98i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.878 + 0.507i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (5 - 8.66i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 1.75iT - 71T^{2} \)
73 \( 1 + (1.24 + 0.717i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.37 + 2.38i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 6.63T + 83T^{2} \)
89 \( 1 + (5.19 + 9i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 13.5iT - 97T^{2} \)
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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

−9.945805877273669358971008143687, −9.119111671613819022564667264310, −8.202779150542064406707809213307, −7.10552672608846909139922697171, −6.62494896688516899573821128442, −5.50986346895755284759860691398, −4.65132182712114367377631836444, −3.22094859148805480719699496387, −2.66058102394086865155140391569, −0.66141477857497147414947989932, 1.29254779314192831606474194132, 2.90159877467659169842477299386, 3.61999160606434422281047187433, 5.11548656240469998714712935123, 5.65836314562095092308160082860, 6.71912462315741829556255410214, 7.62755818056633122005631190932, 8.403615509853653038205338132444, 9.404657366504397757617580713422, 10.09373291952711533538108570780

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