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Properties

Label 2-1008-63.58-c1-0-15
Degree $2$
Conductor $1008$
Sign $0.902 - 0.431i$
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  + (−1.73 − 0.0184i)3-s + (0.790 + 1.36i)5-s + (−2.57 + 0.601i)7-s + (2.99 + 0.0640i)9-s + (2.58 − 4.47i)11-s + (−0.681 + 1.18i)13-s + (−1.34 − 2.38i)15-s + (−2.30 − 3.99i)17-s + (−0.0321 + 0.0557i)19-s + (4.47 − 0.994i)21-s + (3.37 + 5.84i)23-s + (1.24 − 2.16i)25-s + (−5.19 − 0.166i)27-s + (4.70 + 8.15i)29-s + 2.66·31-s + ⋯
L(s)  = 1  + (−0.999 − 0.0106i)3-s + (0.353 + 0.612i)5-s + (−0.973 + 0.227i)7-s + (0.999 + 0.0213i)9-s + (0.779 − 1.35i)11-s + (−0.189 + 0.327i)13-s + (−0.347 − 0.616i)15-s + (−0.559 − 0.969i)17-s + (−0.00738 + 0.0127i)19-s + (0.976 − 0.216i)21-s + (0.703 + 1.21i)23-s + (0.249 − 0.432i)25-s + (−0.999 − 0.0320i)27-s + (0.874 + 1.51i)29-s + 0.478·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.902 - 0.431i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.902 - 0.431i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.902 - 0.431i$
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: \((2,\ 1008,\ (\ :1/2),\ 0.902 - 0.431i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.088412129\)
\(L(\frac12)\) \(\approx\) \(1.088412129\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.73 + 0.0184i)T \)
7 \( 1 + (2.57 - 0.601i)T \)
good5 \( 1 + (-0.790 - 1.36i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.58 + 4.47i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.681 - 1.18i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.30 + 3.99i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.0321 - 0.0557i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.37 - 5.84i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.70 - 8.15i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 2.66T + 31T^{2} \)
37 \( 1 + (-0.880 + 1.52i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.858 - 1.48i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.12 - 8.86i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 5.20T + 47T^{2} \)
53 \( 1 + (0.479 + 0.831i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 9.33T + 59T^{2} \)
61 \( 1 - 14.3T + 61T^{2} \)
67 \( 1 - 12.4T + 67T^{2} \)
71 \( 1 - 4.49T + 71T^{2} \)
73 \( 1 + (0.941 + 1.63i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + 6.53T + 79T^{2} \)
83 \( 1 + (-5.08 - 8.81i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (4.12 - 7.14i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (7.26 + 12.5i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.960862643310531676635929750921, −9.447311441026437789235168750223, −8.500558419834844306407211515887, −6.95898102527743685785105458607, −6.70049516348071958728458025582, −5.87266391119547836949464398066, −4.99317958180120288338840615843, −3.68893396162874186594086168063, −2.73048240598811405082632134286, −0.943881466082071393086063855300, 0.794758937268402733944326751915, 2.21781919861293584544449991931, 3.96479453547305414648603663885, 4.63503644843900516965213782972, 5.63799641966201589040989315941, 6.63651327247862392936929314184, 6.93635700792709662094482723227, 8.312604269256964289706021756568, 9.334225332392388090437961821072, 9.977189117870323698851141178951

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