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Properties

Label 2-1008-63.4-c1-0-5
Degree $2$
Conductor $1008$
Sign $0.543 - 0.839i$
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  + (−0.266 − 1.71i)3-s − 2.92·5-s + (−2.35 − 1.20i)7-s + (−2.85 + 0.913i)9-s + 1.35·11-s + (−0.733 + 1.26i)13-s + (0.779 + 4.99i)15-s + (1.65 − 2.86i)17-s + (1.10 + 1.91i)19-s + (−1.42 + 4.35i)21-s − 2.62·23-s + 3.53·25-s + (2.32 + 4.64i)27-s + (0.521 + 0.903i)29-s + (1.63 + 2.83i)31-s + ⋯
L(s)  = 1  + (−0.154 − 0.988i)3-s − 1.30·5-s + (−0.891 − 0.453i)7-s + (−0.952 + 0.304i)9-s + 0.408·11-s + (−0.203 + 0.352i)13-s + (0.201 + 1.29i)15-s + (0.401 − 0.695i)17-s + (0.253 + 0.438i)19-s + (−0.311 + 0.950i)21-s − 0.548·23-s + 0.706·25-s + (0.447 + 0.894i)27-s + (0.0968 + 0.167i)29-s + (0.294 + 0.509i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.543 - 0.839i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.543 - 0.839i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.543 - 0.839i$
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: \((2,\ 1008,\ (\ :1/2),\ 0.543 - 0.839i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4594363821\)
\(L(\frac12)\) \(\approx\) \(0.4594363821\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.266 + 1.71i)T \)
7 \( 1 + (2.35 + 1.20i)T \)
good5 \( 1 + 2.92T + 5T^{2} \)
11 \( 1 - 1.35T + 11T^{2} \)
13 \( 1 + (0.733 - 1.26i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.65 + 2.86i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.10 - 1.91i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 2.62T + 23T^{2} \)
29 \( 1 + (-0.521 - 0.903i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.63 - 2.83i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-5.43 - 9.41i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.904 - 1.56i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.17 - 3.76i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.98 + 3.44i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (3.22 - 5.59i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (6.10 + 10.5i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.279 - 0.484i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.40 - 11.0i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 12.9T + 71T^{2} \)
73 \( 1 + (-5.22 + 9.05i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.383 + 0.664i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.983 - 1.70i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-3.20 - 5.54i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (4.14 + 7.17i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.10066888547787985300672579146, −9.192040076529574593087285169074, −8.127857846587875668100974310641, −7.58801143233291941317517886027, −6.81804951721022386589893741340, −6.13013386625863887640897936874, −4.79613670256991008716572145874, −3.71325018101802825308605631195, −2.83682426474199315652829290591, −1.09652929812555940970877271989, 0.25480006666668320152506448325, 2.74651274723580558015714290190, 3.71940278558044528476653008216, 4.26091876177923270740505809976, 5.50134937039504212443931075717, 6.27288140422384825116365127390, 7.44033557824494892895388294158, 8.264095211924055211985493023901, 9.113970489359614487652829987728, 9.786478281589578092240149450216

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