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Properties

Label 2-1008-28.11-c2-0-7
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  + (−2.13 − 3.70i)5-s + (0.774 + 6.95i)7-s + (2.68 + 1.55i)11-s + 2.72·13-s + (2.54 − 4.41i)17-s + (−22.2 + 12.8i)19-s + (10.5 − 6.09i)23-s + (3.36 − 5.82i)25-s − 41.0·29-s + (−0.149 − 0.0863i)31-s + (24.0 − 17.7i)35-s + (8.46 + 14.6i)37-s − 36.7·41-s + 53.3i·43-s + (21 − 12.1i)47-s + ⋯
L(s)  = 1  + (−0.427 − 0.740i)5-s + (0.110 + 0.993i)7-s + (0.244 + 0.141i)11-s + 0.209·13-s + (0.149 − 0.259i)17-s + (−1.17 + 0.677i)19-s + (0.458 − 0.264i)23-s + (0.134 − 0.232i)25-s − 1.41·29-s + (−0.00482 − 0.00278i)31-s + (0.688 − 0.506i)35-s + (0.228 + 0.396i)37-s − 0.896·41-s + 1.24i·43-s + (0.446 − 0.257i)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: $\chi_{1008} (991, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1),\ -0.540 - 0.841i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.7939460576\)
\(L(\frac12)\) \(\approx\) \(0.7939460576\)
\(L(2)\) 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 \( 1 + (-0.774 - 6.95i)T \)
good5 \( 1 + (2.13 + 3.70i)T + (-12.5 + 21.6i)T^{2} \)
11 \( 1 + (-2.68 - 1.55i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 - 2.72T + 169T^{2} \)
17 \( 1 + (-2.54 + 4.41i)T + (-144.5 - 250. i)T^{2} \)
19 \( 1 + (22.2 - 12.8i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (-10.5 + 6.09i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + 41.0T + 841T^{2} \)
31 \( 1 + (0.149 + 0.0863i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (-8.46 - 14.6i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + 36.7T + 1.68e3T^{2} \)
43 \( 1 - 53.3iT - 1.84e3T^{2} \)
47 \( 1 + (-21 + 12.1i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (51.6 - 89.5i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (25.3 + 14.6i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-16.4 - 28.4i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-14.7 - 8.50i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 76.7iT - 5.04e3T^{2} \)
73 \( 1 + (49.6 - 85.9i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-75.9 + 43.8i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 151. iT - 6.88e3T^{2} \)
89 \( 1 + (34.2 + 59.3i)T + (-3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 104.T + 9.40e3T^{2} \)
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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

−9.916648594330257193213857597928, −9.056772644257832346893851619347, −8.492216923119975272098699416476, −7.75747671542888482390391787052, −6.56997319626101362057041811617, −5.72625933002332810102751692463, −4.81039814959306023414259282314, −3.94009589128434047444189579023, −2.63888156590024498564335513079, −1.40229817049541041228572021831, 0.25094212220752666850694508973, 1.82398091332492710099866451333, 3.29643618104553436662377544526, 3.95386778719397762241858413277, 5.03088973886365797937512035688, 6.28376292714440206306255172660, 7.02274979173870723588255494725, 7.65127326667543005483997013148, 8.626606571021796453220292848224, 9.519594400871933795024592397801

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