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Properties

Label 2-1008-63.58-c1-0-19
Degree $2$
Conductor $1008$
Sign $0.650 + 0.759i$
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  + (−1.39 + 1.02i)3-s + (−0.667 − 1.15i)5-s + (−1.90 + 1.83i)7-s + (0.880 − 2.86i)9-s + (0.756 − 1.31i)11-s + (−2.58 + 4.48i)13-s + (2.11 + 0.923i)15-s + (0.774 + 1.34i)17-s + (1.25 − 2.16i)19-s + (0.757 − 4.51i)21-s + (−3.68 − 6.37i)23-s + (1.60 − 2.78i)25-s + (1.72 + 4.90i)27-s + (−0.0309 − 0.0536i)29-s + 3.84·31-s + ⋯
L(s)  = 1  + (−0.804 + 0.594i)3-s + (−0.298 − 0.516i)5-s + (−0.719 + 0.694i)7-s + (0.293 − 0.955i)9-s + (0.228 − 0.395i)11-s + (−0.717 + 1.24i)13-s + (0.547 + 0.238i)15-s + (0.187 + 0.325i)17-s + (0.287 − 0.497i)19-s + (0.165 − 0.986i)21-s + (−0.767 − 1.32i)23-s + (0.321 − 0.557i)25-s + (0.332 + 0.943i)27-s + (−0.00575 − 0.00996i)29-s + 0.691·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.650 + 0.759i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.650 + 0.759i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.650 + 0.759i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (625, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ 0.650 + 0.759i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7415745423\)
\(L(\frac12)\) \(\approx\) \(0.7415745423\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.39 - 1.02i)T \)
7 \( 1 + (1.90 - 1.83i)T \)
good5 \( 1 + (0.667 + 1.15i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.756 + 1.31i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.58 - 4.48i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.774 - 1.34i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.25 + 2.16i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.68 + 6.37i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.0309 + 0.0536i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 3.84T + 31T^{2} \)
37 \( 1 + (0.281 - 0.487i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.51 + 7.81i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.09 + 8.83i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 9.51T + 47T^{2} \)
53 \( 1 + (-0.755 - 1.30i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 8.44T + 59T^{2} \)
61 \( 1 - 3.23T + 61T^{2} \)
67 \( 1 + 6.93T + 67T^{2} \)
71 \( 1 - 12.3T + 71T^{2} \)
73 \( 1 + (1.37 + 2.38i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 - 5.91T + 79T^{2} \)
83 \( 1 + (2.80 + 4.85i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-0.703 + 1.21i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (6.09 + 10.5i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.876511343646890748359664693281, −9.036068661404697476325697611582, −8.539077366409366680976931607028, −7.05081413703242354236112658475, −6.37649666554423730990808501038, −5.51238002511966042802035405553, −4.56824732092182528335606035773, −3.82165879370733583904353931482, −2.39756355375532837165302699089, −0.45606050691765940457999696647, 1.06580315227596789057935993124, 2.72010998386897578700393255373, 3.78019158931629649391936300260, 5.01268855573577131890349187019, 5.89383725930807359518902255640, 6.77744076736908785540620990310, 7.51421402853889710223759949767, 7.955930377964472220580014867216, 9.649161092784018867712028482452, 10.06854484904462968083941047326

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