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Properties

Label 2-1008-21.2-c2-0-21
Degree $2$
Conductor $1008$
Sign $0.851 + 0.524i$
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  + (3.67 + 2.12i)5-s + (3.5 − 6.06i)7-s + (−11.0 + 6.36i)11-s − 13-s + (14.6 − 8.48i)17-s + (11.5 − 19.9i)19-s + (14.6 + 8.48i)23-s + (−3.5 − 6.06i)25-s − 33.9i·29-s + (23.5 + 40.7i)31-s + (25.7 − 14.8i)35-s + (27.5 − 47.6i)37-s + 46.6i·41-s − 23·43-s + (3.67 + 2.12i)47-s + ⋯
L(s)  = 1  + (0.734 + 0.424i)5-s + (0.5 − 0.866i)7-s + (−1.00 + 0.578i)11-s − 0.0769·13-s + (0.864 − 0.499i)17-s + (0.605 − 1.04i)19-s + (0.638 + 0.368i)23-s + (−0.140 − 0.242i)25-s − 1.17i·29-s + (0.758 + 1.31i)31-s + (0.734 − 0.424i)35-s + (0.743 − 1.28i)37-s + 1.13i·41-s − 0.534·43-s + (0.0781 + 0.0451i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.851 + 0.524i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.851 + 0.524i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.851 + 0.524i$
Analytic conductor: \(27.4660\)
Root analytic conductor: \(5.24080\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (737, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1),\ 0.851 + 0.524i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.258567034\)
\(L(\frac12)\) \(\approx\) \(2.258567034\)
\(L(2)\) 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 \( 1 + (-3.5 + 6.06i)T \)
good5 \( 1 + (-3.67 - 2.12i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (11.0 - 6.36i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + T + 169T^{2} \)
17 \( 1 + (-14.6 + 8.48i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-11.5 + 19.9i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (-14.6 - 8.48i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + 33.9iT - 841T^{2} \)
31 \( 1 + (-23.5 - 40.7i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (-27.5 + 47.6i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 - 46.6iT - 1.68e3T^{2} \)
43 \( 1 + 23T + 1.84e3T^{2} \)
47 \( 1 + (-3.67 - 2.12i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-44.0 + 25.4i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-73.4 + 42.4i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (52 - 90.0i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (48.5 + 84.0i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 97.5iT - 5.04e3T^{2} \)
73 \( 1 + (32.5 + 56.2i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-56.5 + 97.8i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 29.6iT - 6.88e3T^{2} \)
89 \( 1 + (-117. - 67.8i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 104T + 9.40e3T^{2} \)
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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

−10.02401094076641622539397064706, −8.970768080526516620584486326826, −7.74568042812519908297760965897, −7.35003490064250274005198128758, −6.33969121326458103733786129829, −5.24212375375952715274864843038, −4.60281246362804469973087956494, −3.18911329434959911988336000570, −2.23168019773635448125308335648, −0.818293108166654686252154648320, 1.17607177420588161840035516127, 2.32808092776805742781258798862, 3.39058594129012499683810750698, 4.88756085715340661261181725415, 5.55362827281221382404826825682, 6.10575069936303329912572467751, 7.54439243720486689681278893374, 8.248820285175666898876705236212, 8.956462035313800011818338696660, 9.877041926144705611382862725719

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