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Properties

Label 2-1008-63.20-c1-0-10
Degree $2$
Conductor $1008$
Sign $0.389 - 0.920i$
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  + (−1.40 + 1.00i)3-s + (1.17 + 2.03i)5-s + (−1.55 − 2.14i)7-s + (0.971 − 2.83i)9-s + (4.91 + 2.83i)11-s + (1.48 − 0.859i)13-s + (−3.70 − 1.68i)15-s − 1.76·17-s + 1.13i·19-s + (4.34 + 1.45i)21-s + (3.18 − 1.83i)23-s + (−0.259 + 0.449i)25-s + (1.48 + 4.97i)27-s + (3.59 + 2.07i)29-s + (7.24 − 4.18i)31-s + ⋯
L(s)  = 1  + (−0.813 + 0.581i)3-s + (0.525 + 0.909i)5-s + (−0.587 − 0.809i)7-s + (0.323 − 0.946i)9-s + (1.48 + 0.855i)11-s + (0.413 − 0.238i)13-s + (−0.956 − 0.434i)15-s − 0.429·17-s + 0.261i·19-s + (0.948 + 0.317i)21-s + (0.663 − 0.383i)23-s + (−0.0519 + 0.0899i)25-s + (0.286 + 0.958i)27-s + (0.668 + 0.385i)29-s + (1.30 − 0.751i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.389 - 0.920i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.389 - 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.389 - 0.920i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (209, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ 0.389 - 0.920i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.351304170\)
\(L(\frac12)\) \(\approx\) \(1.351304170\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.40 - 1.00i)T \)
7 \( 1 + (1.55 + 2.14i)T \)
good5 \( 1 + (-1.17 - 2.03i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-4.91 - 2.83i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.48 + 0.859i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 1.76T + 17T^{2} \)
19 \( 1 - 1.13iT - 19T^{2} \)
23 \( 1 + (-3.18 + 1.83i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.59 - 2.07i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-7.24 + 4.18i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 9.19T + 37T^{2} \)
41 \( 1 + (-3.99 - 6.92i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.76 - 3.04i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (5.90 - 10.2i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + (1.11 + 1.93i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-7.79 - 4.49i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.43 - 9.41i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 4.52iT - 71T^{2} \)
73 \( 1 - 5.34iT - 73T^{2} \)
79 \( 1 + (6.51 - 11.2i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-6.27 + 10.8i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 1.16T + 89T^{2} \)
97 \( 1 + (3.97 + 2.29i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.966906773841039186739769244985, −9.781216336559307510728708503224, −8.672438647532192734097655910089, −7.18859240852518343853374777663, −6.52378147714060286698372096070, −6.21000483027686354846892809458, −4.73683940826282664958941161743, −3.99885483814621628298115503792, −2.96694203162364902761124657043, −1.20050729023528433023757137432, 0.848511445286727442299929280562, 1.94697843532544127241787176994, 3.49220565230631065436089580976, 4.84218550220877663672895855130, 5.57841408197253657654310685748, 6.42614998713148186809127463632, 6.87431165807809164812742856002, 8.508859433918521300308814220043, 8.820685529807891002029154546904, 9.693993827148785771996629063038

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