Properties

Label 2-1008-7.3-c2-0-11
Degree $2$
Conductor $1008$
Sign $-0.540 - 0.841i$
Analytic cond. $27.4660$
Root an. cond. $5.24080$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.18 + 1.25i)5-s + (3.00 + 6.31i)7-s + (2.78 − 4.81i)11-s + 6.50i·13-s + (17.6 + 10.1i)17-s + (−9.98 + 5.76i)19-s + (−16.3 − 28.2i)23-s + (−9.32 + 16.1i)25-s + 19.7·29-s + (5.23 + 3.02i)31-s + (−14.5 − 9.99i)35-s + (−5.71 − 9.89i)37-s + 69.9i·41-s − 40.6·43-s + (33.5 − 19.3i)47-s + ⋯
L(s)  = 1  + (−0.436 + 0.251i)5-s + (0.429 + 0.902i)7-s + (0.252 − 0.437i)11-s + 0.500i·13-s + (1.03 + 0.597i)17-s + (−0.525 + 0.303i)19-s + (−0.709 − 1.22i)23-s + (−0.373 + 0.646i)25-s + 0.679·29-s + (0.168 + 0.0974i)31-s + (−0.415 − 0.285i)35-s + (−0.154 − 0.267i)37-s + 1.70i·41-s − 0.945·43-s + (0.713 − 0.411i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.540 - 0.841i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.540 - 0.841i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-0.540 - 0.841i$
Analytic conductor: \(27.4660\)
Root analytic conductor: \(5.24080\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (577, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1),\ -0.540 - 0.841i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.263308053\)
\(L(\frac12)\) \(\approx\) \(1.263308053\)
\(L(2)\) 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 \( 1 + (-3.00 - 6.31i)T \)
good5 \( 1 + (2.18 - 1.25i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (-2.78 + 4.81i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 - 6.50iT - 169T^{2} \)
17 \( 1 + (-17.6 - 10.1i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (9.98 - 5.76i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (16.3 + 28.2i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 - 19.7T + 841T^{2} \)
31 \( 1 + (-5.23 - 3.02i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (5.71 + 9.89i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 - 69.9iT - 1.68e3T^{2} \)
43 \( 1 + 40.6T + 1.84e3T^{2} \)
47 \( 1 + (-33.5 + 19.3i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (16.4 - 28.5i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (61.4 + 35.4i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (35.8 - 20.7i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (47.9 - 82.9i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 57.6T + 5.04e3T^{2} \)
73 \( 1 + (-28.2 - 16.3i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-31.1 - 53.9i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 35.3iT - 6.88e3T^{2} \)
89 \( 1 + (-6.60 + 3.81i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 - 130. iT - 9.40e3T^{2} \)
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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

−10.11042472413291088790601232883, −9.104399305695014761248140512155, −8.331694787227227203230584199260, −7.77089266752801509165043034532, −6.51123328949789715520084909443, −5.89526371583201835575031646693, −4.78923849400104517245338930335, −3.80168722584035941748963250542, −2.71319153703638514845193479347, −1.46827874232569200613348128202, 0.40695423232611017038036296031, 1.68146899684116583924509556192, 3.23392684216337832332601139150, 4.17996140115946368928214960846, 4.97850125663866701226983494911, 6.04683415748670662398859077463, 7.20988069042870267699236813120, 7.73324946768779795054290235497, 8.517434306196631604620488308980, 9.627672733020159020451924486793

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