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Properties

Label 2-1008-63.25-c1-0-14
Degree $2$
Conductor $1008$
Sign $0.994 - 0.106i$
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  + (−1.34 − 1.09i)3-s + (0.918 − 1.59i)5-s + (0.361 + 2.62i)7-s + (0.616 + 2.93i)9-s + (−1.54 − 2.68i)11-s + (2.40 + 4.16i)13-s + (−2.97 + 1.13i)15-s + (1.87 − 3.24i)17-s + (2.71 + 4.70i)19-s + (2.37 − 3.91i)21-s + (−3.97 + 6.89i)23-s + (0.813 + 1.40i)25-s + (2.37 − 4.62i)27-s + (−0.325 + 0.563i)29-s − 1.03·31-s + ⋯
L(s)  = 1  + (−0.776 − 0.630i)3-s + (0.410 − 0.711i)5-s + (0.136 + 0.990i)7-s + (0.205 + 0.978i)9-s + (−0.466 − 0.808i)11-s + (0.666 + 1.15i)13-s + (−0.767 + 0.293i)15-s + (0.453 − 0.786i)17-s + (0.622 + 1.07i)19-s + (0.518 − 0.855i)21-s + (−0.829 + 1.43i)23-s + (0.162 + 0.281i)25-s + (0.457 − 0.889i)27-s + (−0.0604 + 0.104i)29-s − 0.186·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.106i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 - 0.106i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.994 - 0.106i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (529, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ 0.994 - 0.106i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.302811782\)
\(L(\frac12)\) \(\approx\) \(1.302811782\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.34 + 1.09i)T \)
7 \( 1 + (-0.361 - 2.62i)T \)
good5 \( 1 + (-0.918 + 1.59i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.54 + 2.68i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.40 - 4.16i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.87 + 3.24i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.71 - 4.70i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.97 - 6.89i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.325 - 0.563i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 1.03T + 31T^{2} \)
37 \( 1 + (-0.873 - 1.51i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.52 - 4.36i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-6.09 + 10.5i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 4.61T + 47T^{2} \)
53 \( 1 + (-4.55 + 7.88i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 5.79T + 59T^{2} \)
61 \( 1 + 4.81T + 61T^{2} \)
67 \( 1 - 14.4T + 67T^{2} \)
71 \( 1 - 5.00T + 71T^{2} \)
73 \( 1 + (1.81 - 3.14i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 14.3T + 79T^{2} \)
83 \( 1 + (3.83 - 6.63i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (5.76 + 9.99i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.04 - 1.80i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.871251987052790870854893735510, −9.143148485460267107088040113512, −8.296362025902338316877674352743, −7.50397473664860953913034336409, −6.37205338629959182513139213687, −5.47296686563181207174348168687, −5.31906918914816951731904026277, −3.74107393336866490440225212347, −2.19723469902636483628904796248, −1.18309792096383640850483934545, 0.794226503688600075116307622955, 2.64314725546995447022801975207, 3.82873240346488773935370698142, 4.66551087593974280014840920828, 5.69855904156668551766244252742, 6.45457569164454047945808325665, 7.30957458680039668879865731502, 8.196163333337269413578767591524, 9.461518301454607671792227047622, 10.26606022888929013085154915286

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