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Properties

Label 2-1008-48.11-c1-0-8
Degree $2$
Conductor $1008$
Sign $-0.595 - 0.803i$
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  + (−0.890 + 1.09i)2-s + (−0.412 − 1.95i)4-s + (−1.65 + 1.65i)5-s + 7-s + (2.51 + 1.29i)8-s + (−0.343 − 3.29i)10-s + (−0.993 − 0.993i)11-s + (2.62 − 2.62i)13-s + (−0.890 + 1.09i)14-s + (−3.65 + 1.61i)16-s + 2.77i·17-s + (1.56 + 1.56i)19-s + (3.92 + 2.55i)20-s + (1.97 − 0.206i)22-s + 1.05i·23-s + ⋯
L(s)  = 1  + (−0.629 + 0.776i)2-s + (−0.206 − 0.978i)4-s + (−0.741 + 0.741i)5-s + 0.377·7-s + (0.889 + 0.456i)8-s + (−0.108 − 1.04i)10-s + (−0.299 − 0.299i)11-s + (0.728 − 0.728i)13-s + (−0.238 + 0.293i)14-s + (−0.914 + 0.403i)16-s + 0.673i·17-s + (0.358 + 0.358i)19-s + (0.877 + 0.572i)20-s + (0.421 − 0.0439i)22-s + 0.219i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.595 - 0.803i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.595 - 0.803i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-0.595 - 0.803i$
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: \((2,\ 1008,\ (\ :1/2),\ -0.595 - 0.803i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8529349660\)
\(L(\frac12)\) \(\approx\) \(0.8529349660\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.890 - 1.09i)T \)
3 \( 1 \)
7 \( 1 - T \)
good5 \( 1 + (1.65 - 1.65i)T - 5iT^{2} \)
11 \( 1 + (0.993 + 0.993i)T + 11iT^{2} \)
13 \( 1 + (-2.62 + 2.62i)T - 13iT^{2} \)
17 \( 1 - 2.77iT - 17T^{2} \)
19 \( 1 + (-1.56 - 1.56i)T + 19iT^{2} \)
23 \( 1 - 1.05iT - 23T^{2} \)
29 \( 1 + (-3.47 - 3.47i)T + 29iT^{2} \)
31 \( 1 - 1.06iT - 31T^{2} \)
37 \( 1 + (0.0657 + 0.0657i)T + 37iT^{2} \)
41 \( 1 - 6.31T + 41T^{2} \)
43 \( 1 + (2.38 - 2.38i)T - 43iT^{2} \)
47 \( 1 + 1.47T + 47T^{2} \)
53 \( 1 + (7.63 - 7.63i)T - 53iT^{2} \)
59 \( 1 + (4.15 + 4.15i)T + 59iT^{2} \)
61 \( 1 + (7.78 - 7.78i)T - 61iT^{2} \)
67 \( 1 + (-1.98 - 1.98i)T + 67iT^{2} \)
71 \( 1 - 13.0iT - 71T^{2} \)
73 \( 1 - 9.50iT - 73T^{2} \)
79 \( 1 - 9.85iT - 79T^{2} \)
83 \( 1 + (-1.13 + 1.13i)T - 83iT^{2} \)
89 \( 1 - 7.04T + 89T^{2} \)
97 \( 1 - 5.35T + 97T^{2} \)
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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

−10.36220552719575520119258753358, −9.278452695924930551201643174114, −8.291100398282323115792075607393, −7.88269958604631842003540618219, −7.04441762651207215452702211749, −6.12675514293716372844109650856, −5.33341839024165692482963623339, −4.12376948420052529265856306957, −2.99308414873982985923916142266, −1.26406871769995271924907050413, 0.55527772851868302963038710741, 1.88628489571001293647714046093, 3.20060214288433759736124719655, 4.33262666619886847467727339943, 4.86829356248067980738721354500, 6.45314789323338045783550766967, 7.57460187393229135462226660265, 8.111414173691711112123000383180, 8.952068760733789217705882500608, 9.531846359126752807134791381414

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