Properties

Label 2-1008-252.103-c1-0-17
Degree $2$
Conductor $1008$
Sign $0.775 - 0.631i$
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  + (1.69 − 0.348i)3-s + (−2.08 + 1.20i)5-s + (−2.53 − 0.763i)7-s + (2.75 − 1.18i)9-s + (2.81 + 1.62i)11-s + (4.35 + 2.51i)13-s + (−3.11 + 2.76i)15-s + (0.795 − 0.459i)17-s + (−3.22 + 5.57i)19-s + (−4.56 − 0.414i)21-s + (5.31 − 3.06i)23-s + (0.399 − 0.691i)25-s + (4.26 − 2.96i)27-s + (1.22 + 2.12i)29-s − 3.21·31-s + ⋯
L(s)  = 1  + (0.979 − 0.200i)3-s + (−0.932 + 0.538i)5-s + (−0.957 − 0.288i)7-s + (0.919 − 0.393i)9-s + (0.847 + 0.489i)11-s + (1.20 + 0.698i)13-s + (−0.805 + 0.714i)15-s + (0.192 − 0.111i)17-s + (−0.739 + 1.28i)19-s + (−0.995 − 0.0904i)21-s + (1.10 − 0.640i)23-s + (0.0798 − 0.138i)25-s + (0.821 − 0.570i)27-s + (0.228 + 0.394i)29-s − 0.578·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.775 - 0.631i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.775 - 0.631i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.775 - 0.631i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (607, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ 0.775 - 0.631i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.862127082\)
\(L(\frac12)\) \(\approx\) \(1.862127082\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.69 + 0.348i)T \)
7 \( 1 + (2.53 + 0.763i)T \)
good5 \( 1 + (2.08 - 1.20i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.81 - 1.62i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-4.35 - 2.51i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.795 + 0.459i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.22 - 5.57i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-5.31 + 3.06i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.22 - 2.12i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 3.21T + 31T^{2} \)
37 \( 1 + (4.08 - 7.07i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-10.3 - 5.99i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-10.2 + 5.89i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + 4.84T + 47T^{2} \)
53 \( 1 + (1.56 + 2.71i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 9.86T + 59T^{2} \)
61 \( 1 - 9.49iT - 61T^{2} \)
67 \( 1 - 0.359iT - 67T^{2} \)
71 \( 1 + 2.32iT - 71T^{2} \)
73 \( 1 + (4.01 - 2.31i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + 4.36iT - 79T^{2} \)
83 \( 1 + (8.68 + 15.0i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (9.68 + 5.59i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (9.79 - 5.65i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.973603222845290078689768654758, −9.068265381942012404088719240803, −8.501391595559111350531047197556, −7.44653610340663223144207484397, −6.85899229222483060722730432562, −6.15651569738043430580854792063, −4.21703587078706922385359772307, −3.79233639457073278621093560449, −2.91081888119993540413547030055, −1.39105119408901264967193631228, 0.880686406788536001308225073715, 2.68039485443220685290081993218, 3.65581128099730966451725004796, 4.16624250031995026621033048207, 5.54618357456670770158452204189, 6.61943272066562973107138923241, 7.50897505780545298932861142638, 8.447114237599144418766808616522, 8.971352483538404743640181094155, 9.504818046881072046712833960428

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