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Properties

Label 2-1008-252.223-c1-0-46
Degree $2$
Conductor $1008$
Sign $-0.728 + 0.684i$
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  + (1.44 − 0.960i)3-s + (−0.675 + 0.390i)5-s + (0.645 − 2.56i)7-s + (1.15 − 2.76i)9-s + (−4.95 − 2.85i)11-s + (−3.53 + 2.03i)13-s + (−0.599 + 1.21i)15-s − 2.73i·17-s − 7.06·19-s + (−1.53 − 4.31i)21-s + (6.14 − 3.54i)23-s + (−2.19 + 3.80i)25-s + (−0.992 − 5.10i)27-s + (0.910 − 1.57i)29-s + (2.25 + 3.91i)31-s + ⋯
L(s)  = 1  + (0.832 − 0.554i)3-s + (−0.302 + 0.174i)5-s + (0.244 − 0.969i)7-s + (0.385 − 0.922i)9-s + (−1.49 − 0.861i)11-s + (−0.979 + 0.565i)13-s + (−0.154 + 0.312i)15-s − 0.663i·17-s − 1.62·19-s + (−0.334 − 0.942i)21-s + (1.28 − 0.739i)23-s + (−0.439 + 0.760i)25-s + (−0.191 − 0.981i)27-s + (0.169 − 0.292i)29-s + (0.405 + 0.702i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.728 + 0.684i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.728 + 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-0.728 + 0.684i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (223, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ -0.728 + 0.684i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.320404559\)
\(L(\frac12)\) \(\approx\) \(1.320404559\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.44 + 0.960i)T \)
7 \( 1 + (-0.645 + 2.56i)T \)
good5 \( 1 + (0.675 - 0.390i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (4.95 + 2.85i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (3.53 - 2.03i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 2.73iT - 17T^{2} \)
19 \( 1 + 7.06T + 19T^{2} \)
23 \( 1 + (-6.14 + 3.54i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.910 + 1.57i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2.25 - 3.91i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 1.69T + 37T^{2} \)
41 \( 1 + (2.48 - 1.43i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-7.58 - 4.37i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-6.32 + 10.9i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 9.58T + 53T^{2} \)
59 \( 1 + (1.86 + 3.23i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (7.56 + 4.37i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-11.5 + 6.67i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 5.36iT - 71T^{2} \)
73 \( 1 + 4.91iT - 73T^{2} \)
79 \( 1 + (-8.32 - 4.80i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (6.81 - 11.8i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 6.93iT - 89T^{2} \)
97 \( 1 + (1.45 + 0.841i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.588083446439497830383898314222, −8.606019852773665127374264066601, −7.977478202008926473079079221518, −7.20163143785112825172952046904, −6.63090548488765336686786244652, −5.16872303678522557445365038517, −4.21920817484915748661871886202, −3.09492284344121371543983876389, −2.22286703324872014220317234041, −0.50321823773237637559995716660, 2.22586945975404283003353220790, 2.71946820116712591729328400429, 4.19566395348465901115699083192, 4.93510515245590807869545964686, 5.74557545670734211908175254774, 7.26774538441513592345754230066, 7.929075972855222356878626755968, 8.597753979748251829203406425810, 9.367864941490841814897119252871, 10.33837496887848828518756151055

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