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Properties

Label 2-1008-63.58-c1-0-44
Degree $2$
Conductor $1008$
Sign $-0.927 - 0.373i$
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  + (0.704 − 1.58i)3-s + (−1.05 − 1.82i)5-s + (−2.58 + 0.569i)7-s + (−2.00 − 2.22i)9-s + (0.199 − 0.345i)11-s + (1.44 − 2.49i)13-s + (−3.62 + 0.381i)15-s + (−0.176 − 0.305i)17-s + (−2.84 + 4.93i)19-s + (−0.918 + 4.48i)21-s + (−0.438 − 0.759i)23-s + (0.285 − 0.494i)25-s + (−4.94 + 1.60i)27-s + (0.874 + 1.51i)29-s − 9.13·31-s + ⋯
L(s)  = 1  + (0.406 − 0.913i)3-s + (−0.470 − 0.815i)5-s + (−0.976 + 0.215i)7-s + (−0.669 − 0.742i)9-s + (0.0601 − 0.104i)11-s + (0.400 − 0.693i)13-s + (−0.935 + 0.0985i)15-s + (−0.0428 − 0.0741i)17-s + (−0.653 + 1.13i)19-s + (−0.200 + 0.979i)21-s + (−0.0914 − 0.158i)23-s + (0.0571 − 0.0989i)25-s + (−0.950 + 0.309i)27-s + (0.162 + 0.281i)29-s − 1.64·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.927 - 0.373i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.927 - 0.373i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-0.927 - 0.373i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (625, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ -0.927 - 0.373i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6167162657\)
\(L(\frac12)\) \(\approx\) \(0.6167162657\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.704 + 1.58i)T \)
7 \( 1 + (2.58 - 0.569i)T \)
good5 \( 1 + (1.05 + 1.82i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.199 + 0.345i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.44 + 2.49i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.176 + 0.305i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.84 - 4.93i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.438 + 0.759i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.874 - 1.51i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 9.13T + 31T^{2} \)
37 \( 1 + (3.39 - 5.88i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.20 + 2.08i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.276 + 0.479i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 11.7T + 47T^{2} \)
53 \( 1 + (2.07 + 3.59i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 9.32T + 59T^{2} \)
61 \( 1 + 10.0T + 61T^{2} \)
67 \( 1 + 1.20T + 67T^{2} \)
71 \( 1 - 14.6T + 71T^{2} \)
73 \( 1 + (-0.315 - 0.546i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 - 2.48T + 79T^{2} \)
83 \( 1 + (-4.59 - 7.95i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-7.29 + 12.6i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (7.84 + 13.5i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.294822428788772587969393163400, −8.480050113726350260499760956659, −8.054132083238236573602516510864, −6.95608712146122050131442732601, −6.19092824677803883877935481280, −5.31577763825595002538717590161, −3.87822362310211004096073492081, −3.09748607083678835276526795878, −1.69090296929324341778660351310, −0.25543023703259421981866222316, 2.34075540472428052126085977828, 3.41759786685576675243145653016, 3.95801866057823662546789119006, 5.09194875250134669648192343611, 6.33633396467426384811717225289, 7.01353981130191612862406229615, 7.960243708444787018167775337702, 9.131631310336560008226002801261, 9.399495788055982235365481805383, 10.58497621351078065111468163511

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