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Properties

Label 2-1008-7.3-c2-0-5
Degree $2$
Conductor $1008$
Sign $-0.605 - 0.795i$
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  + (−4.24 + 2.44i)5-s + (−3.5 − 6.06i)7-s + (8.48 − 14.6i)11-s − 1.73i·13-s + (4.24 + 2.44i)17-s + (−25.5 + 14.7i)19-s + (4.24 + 7.34i)23-s + (−0.500 + 0.866i)25-s + 33.9·29-s + (10.5 + 6.06i)31-s + (29.6 + 17.1i)35-s + (23.5 + 40.7i)37-s − 68.5i·41-s − 31·43-s + (−72.1 + 41.6i)47-s + ⋯
L(s)  = 1  + (−0.848 + 0.489i)5-s + (−0.5 − 0.866i)7-s + (0.771 − 1.33i)11-s − 0.133i·13-s + (0.249 + 0.144i)17-s + (−1.34 + 0.774i)19-s + (0.184 + 0.319i)23-s + (−0.0200 + 0.0346i)25-s + 1.17·29-s + (0.338 + 0.195i)31-s + (0.848 + 0.489i)35-s + (0.635 + 1.10i)37-s − 1.67i·41-s − 0.720·43-s + (−1.53 + 0.885i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-0.605 - 0.795i$
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: \((2,\ 1008,\ (\ :1),\ -0.605 - 0.795i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.4381903747\)
\(L(\frac12)\) \(\approx\) \(0.4381903747\)
\(L(2)\) 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 \( 1 + (4.24 - 2.44i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (-8.48 + 14.6i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + 1.73iT - 169T^{2} \)
17 \( 1 + (-4.24 - 2.44i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (25.5 - 14.7i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (-4.24 - 7.34i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 - 33.9T + 841T^{2} \)
31 \( 1 + (-10.5 - 6.06i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (-23.5 - 40.7i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + 68.5iT - 1.68e3T^{2} \)
43 \( 1 + 31T + 1.84e3T^{2} \)
47 \( 1 + (72.1 - 41.6i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (38.1 - 66.1i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (72.1 + 41.6i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (72 - 41.5i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (15.5 - 26.8i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 59.3T + 5.04e3T^{2} \)
73 \( 1 + (70.5 + 40.7i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-20.5 - 35.5i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 4.89iT - 6.88e3T^{2} \)
89 \( 1 + (50.9 - 29.3i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 - 41.5iT - 9.40e3T^{2} \)
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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

−10.29113380501131355743018562742, −9.185981075726626329325956623185, −8.266775279643038540144942374374, −7.65140945561369725843514107971, −6.54890581661314507492359240779, −6.13725237290941942040324846395, −4.58050375182865119633453760012, −3.67625075955971181367613876724, −3.09879517689966041174213812141, −1.19766621852964604344450572197, 0.15041729896340897976458726390, 1.85670891843390927474068906161, 3.05231368985149313972449235984, 4.36771897613729171727138306468, 4.78883499375013648112029746138, 6.26725989500938214032815727469, 6.80955053275422904521453906477, 7.958264879722002654235620873769, 8.646021827349351490729585807663, 9.428672635424859209058846597235

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