↑

Properties

Label 2-1008-9.7-c1-0-8
Degree $2$
Conductor $1008$
Sign $-0.5 - 0.866i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.11 + 1.32i)3-s + (1.43 + 2.49i)5-s + (0.5 − 0.866i)7-s + (−0.520 − 2.95i)9-s + (−0.592 + 1.02i)11-s + (2.37 + 4.12i)13-s + (−4.91 − 0.866i)15-s + 5.41·17-s − 1.10·19-s + (0.592 + 1.62i)21-s + (2.95 + 5.11i)23-s + (−1.64 + 2.84i)25-s + (4.5 + 2.59i)27-s + (−2.49 + 4.31i)29-s + (−2.78 − 4.82i)31-s + ⋯
L(s)  = 1  + (−0.642 + 0.766i)3-s + (0.643 + 1.11i)5-s + (0.188 − 0.327i)7-s + (−0.173 − 0.984i)9-s + (−0.178 + 0.309i)11-s + (0.659 + 1.14i)13-s + (−1.26 − 0.223i)15-s + 1.31·17-s − 0.253·19-s + (0.129 + 0.355i)21-s + (0.615 + 1.06i)23-s + (−0.329 + 0.569i)25-s + (0.866 + 0.499i)27-s + (−0.462 + 0.801i)29-s + (−0.500 − 0.866i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.5 - 0.866i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.5 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-0.5 - 0.866i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (673, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ -0.5 - 0.866i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.391162891\)
\(L(\frac12)\) \(\approx\) \(1.391162891\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.11 - 1.32i)T \)
7 \( 1 + (-0.5 + 0.866i)T \)
good5 \( 1 + (-1.43 - 2.49i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.592 - 1.02i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.37 - 4.12i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 5.41T + 17T^{2} \)
19 \( 1 + 1.10T + 19T^{2} \)
23 \( 1 + (-2.95 - 5.11i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.49 - 4.31i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.78 + 4.82i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 2.42T + 37T^{2} \)
41 \( 1 + (3.81 + 6.61i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.78 - 6.55i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.141 + 0.245i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 4.22T + 53T^{2} \)
59 \( 1 + (-5.86 - 10.1i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.992 - 1.71i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.76 + 11.7i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 5.11T + 71T^{2} \)
73 \( 1 + 0.327T + 73T^{2} \)
79 \( 1 + (5.24 - 9.08i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.62 + 6.28i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 14.7T + 89T^{2} \)
97 \( 1 + (9.04 - 15.6i)T + (-48.5 - 84.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.24689948165042614650307251492, −9.662063862029974506666932872943, −8.867713751626576405998554998439, −7.48464322353433902756471600247, −6.76398923968704313927543819670, −5.94018449337007602390819763510, −5.15322549965987162379983877547, −3.96361638528571575286311004204, −3.16158505888743613240350162483, −1.61372240167283562437948452942, 0.73731417234963549790849247879, 1.74668054940701417677491024070, 3.15562331937802466851616446493, 4.81013358089598661199902571428, 5.49680109724432648504277542129, 5.99328130805500509673854016873, 7.13137647813886560739409045939, 8.303575907121097860854599106018, 8.486577329500815367990121950284, 9.766530501157270606624426460854

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