Properties

Label 2-1008-9.4-c1-0-12
Degree $2$
Conductor $1008$
Sign $0.766 - 0.642i$
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  + (−1.5 + 0.866i)3-s + (−1 + 1.73i)5-s + (−0.5 − 0.866i)7-s + (1.5 − 2.59i)9-s + (0.5 + 0.866i)11-s + (3 − 5.19i)13-s − 3.46i·15-s − 5·17-s + 7·19-s + (1.5 + 0.866i)21-s + (2 − 3.46i)23-s + (0.500 + 0.866i)25-s + 5.19i·27-s + (2 + 3.46i)29-s + (−3 + 5.19i)31-s + ⋯
L(s)  = 1  + (−0.866 + 0.499i)3-s + (−0.447 + 0.774i)5-s + (−0.188 − 0.327i)7-s + (0.5 − 0.866i)9-s + (0.150 + 0.261i)11-s + (0.832 − 1.44i)13-s − 0.894i·15-s − 1.21·17-s + 1.60·19-s + (0.327 + 0.188i)21-s + (0.417 − 0.722i)23-s + (0.100 + 0.173i)25-s + 0.999i·27-s + (0.371 + 0.643i)29-s + (−0.538 + 0.933i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.766 - 0.642i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ 0.766 - 0.642i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.066941349\)
\(L(\frac12)\) \(\approx\) \(1.066941349\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.5 - 0.866i)T \)
7 \( 1 + (0.5 + 0.866i)T \)
good5 \( 1 + (1 - 1.73i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-3 + 5.19i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 5T + 17T^{2} \)
19 \( 1 - 7T + 19T^{2} \)
23 \( 1 + (-2 + 3.46i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2 - 3.46i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3 - 5.19i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 + (1.5 - 2.59i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 12T + 53T^{2} \)
59 \( 1 + (3.5 - 6.06i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6 - 10.3i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.5 + 11.2i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 8T + 71T^{2} \)
73 \( 1 - T + 73T^{2} \)
79 \( 1 + (3 + 5.19i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-8 - 13.8i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 + (-2.5 - 4.33i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.36888140826630746176488953348, −9.390380442065762142082684866181, −8.437375492676719228509422488476, −7.23215565870312661102432643279, −6.77451638415183494774830382539, −5.72133422824476750017118463218, −4.88781362742784228309991891096, −3.74294154192593674966189724795, −3.02128274132811354066849985770, −0.906756118738555286380305438060, 0.812235298555252510886035275990, 2.07452330142743886164965910853, 3.80716737012656742753774995804, 4.69645324057902840504579966857, 5.59414197443045563636132795290, 6.47773240772197719564003062171, 7.21114181375660063017265460879, 8.232515500873521451287137570030, 9.024756521375990362504658290706, 9.750602134832573164493971686400

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