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Properties

Label 2-1008-48.11-c1-0-9
Degree $2$
Conductor $1008$
Sign $0.942 - 0.333i$
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  + (−0.579 − 1.29i)2-s + (−1.32 + 1.49i)4-s + (−0.667 + 0.667i)5-s + 7-s + (2.69 + 0.848i)8-s + (1.24 + 0.474i)10-s + (−1.57 − 1.57i)11-s + (−1.83 + 1.83i)13-s + (−0.579 − 1.29i)14-s + (−0.468 − 3.97i)16-s − 3.40i·17-s + (3.18 + 3.18i)19-s + (−0.110 − 1.88i)20-s + (−1.11 + 2.94i)22-s + 0.793i·23-s + ⋯
L(s)  = 1  + (−0.409 − 0.912i)2-s + (−0.664 + 0.747i)4-s + (−0.298 + 0.298i)5-s + 0.377·7-s + (0.953 + 0.300i)8-s + (0.394 + 0.150i)10-s + (−0.475 − 0.475i)11-s + (−0.507 + 0.507i)13-s + (−0.154 − 0.344i)14-s + (−0.117 − 0.993i)16-s − 0.826i·17-s + (0.730 + 0.730i)19-s + (−0.0247 − 0.421i)20-s + (−0.238 + 0.627i)22-s + 0.165i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.942 - 0.333i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.942 - 0.333i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.942 - 0.333i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (827, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ 0.942 - 0.333i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9549296147\)
\(L(\frac12)\) \(\approx\) \(0.9549296147\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.579 + 1.29i)T \)
3 \( 1 \)
7 \( 1 - T \)
good5 \( 1 + (0.667 - 0.667i)T - 5iT^{2} \)
11 \( 1 + (1.57 + 1.57i)T + 11iT^{2} \)
13 \( 1 + (1.83 - 1.83i)T - 13iT^{2} \)
17 \( 1 + 3.40iT - 17T^{2} \)
19 \( 1 + (-3.18 - 3.18i)T + 19iT^{2} \)
23 \( 1 - 0.793iT - 23T^{2} \)
29 \( 1 + (-1.73 - 1.73i)T + 29iT^{2} \)
31 \( 1 - 3.28iT - 31T^{2} \)
37 \( 1 + (-7.72 - 7.72i)T + 37iT^{2} \)
41 \( 1 + 7.19T + 41T^{2} \)
43 \( 1 + (5.84 - 5.84i)T - 43iT^{2} \)
47 \( 1 - 13.0T + 47T^{2} \)
53 \( 1 + (3.34 - 3.34i)T - 53iT^{2} \)
59 \( 1 + (-7.41 - 7.41i)T + 59iT^{2} \)
61 \( 1 + (-1.93 + 1.93i)T - 61iT^{2} \)
67 \( 1 + (-6.38 - 6.38i)T + 67iT^{2} \)
71 \( 1 - 3.41iT - 71T^{2} \)
73 \( 1 + 8.13iT - 73T^{2} \)
79 \( 1 + 0.0502iT - 79T^{2} \)
83 \( 1 + (2.29 - 2.29i)T - 83iT^{2} \)
89 \( 1 - 7.18T + 89T^{2} \)
97 \( 1 - 1.49T + 97T^{2} \)
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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

−10.04491216141457659971334306132, −9.351618738085236325747481399692, −8.421183121538242965635507097726, −7.69303682087109596091452072685, −6.93890941414166557769793654665, −5.43834068384576489972004094789, −4.60910828574393469678102679346, −3.45576014920110813037360681216, −2.62793679583447978563769564110, −1.23142184678451130886764932363, 0.57598124881722018551228975978, 2.24864222238046414133817869599, 3.97735108797729954192704783728, 4.88951531579061873004580432256, 5.59555348566003660440679634478, 6.67111227917877000707064566331, 7.56804262628321533580242639782, 8.111993834953212354078396818219, 8.908207706084070436832862601352, 9.851946985083954166036185362971

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