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Properties

Label 2-1008-63.4-c1-0-42
Degree $2$
Conductor $1008$
Sign $0.190 + 0.981i$
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  + (0.349 − 1.69i)3-s + 3.69·5-s + (1.40 − 2.24i)7-s + (−2.75 − 1.18i)9-s + 1.47·11-s + (−1.34 + 2.33i)13-s + (1.29 − 6.27i)15-s + (3.28 − 5.69i)17-s + (0.444 + 0.769i)19-s + (−3.31 − 3.16i)21-s − 6.28·23-s + 8.68·25-s + (−2.97 + 4.25i)27-s + (1.25 + 2.17i)29-s + (3.40 + 5.89i)31-s + ⋯
L(s)  = 1  + (0.201 − 0.979i)3-s + 1.65·5-s + (0.531 − 0.847i)7-s + (−0.918 − 0.395i)9-s + 0.445·11-s + (−0.374 + 0.648i)13-s + (0.334 − 1.62i)15-s + (0.797 − 1.38i)17-s + (0.101 + 0.176i)19-s + (−0.722 − 0.691i)21-s − 1.31·23-s + 1.73·25-s + (−0.572 + 0.819i)27-s + (0.233 + 0.403i)29-s + (0.611 + 1.05i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.190 + 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.190 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.190 + 0.981i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ 0.190 + 0.981i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.365170434\)
\(L(\frac12)\) \(\approx\) \(2.365170434\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.349 + 1.69i)T \)
7 \( 1 + (-1.40 + 2.24i)T \)
good5 \( 1 - 3.69T + 5T^{2} \)
11 \( 1 - 1.47T + 11T^{2} \)
13 \( 1 + (1.34 - 2.33i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-3.28 + 5.69i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.444 - 0.769i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 6.28T + 23T^{2} \)
29 \( 1 + (-1.25 - 2.17i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.40 - 5.89i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.38 + 2.40i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.05 - 3.56i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.00618 + 0.0107i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (3.49 - 6.05i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.60 - 2.78i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.45 - 5.98i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.86 + 4.96i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.73 + 8.19i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 5.46T + 71T^{2} \)
73 \( 1 + (6.03 - 10.4i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-5.72 + 9.91i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (2.23 + 3.87i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (4.43 + 7.68i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (6.58 + 11.4i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.733864437816586783745216670303, −9.048279954618423477633324843195, −8.018653958606438185261821160049, −7.13536003472815780391527582223, −6.50371408377511550642947600548, −5.62131860316359037425551068739, −4.68701181304165230751247305788, −3.12480129779635462651186558067, −1.98945166766857644454963637492, −1.17688736407831848937463953416, 1.80679661295125280428201686411, 2.66538709856972608246273600499, 3.94543305387240282319054500557, 5.16061646378169559277793887487, 5.71278249256307805134153042209, 6.34462652926123119907121213276, 8.043495636901880489024006085301, 8.568576385477626517475421844961, 9.597820959061696129369322458518, 9.956018844603940940271257664526

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