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Properties

Label 2-1008-63.4-c1-0-39
Degree $2$
Conductor $1008$
Sign $0.647 + 0.762i$
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  + (1.07 − 1.35i)3-s + 3.40·5-s + (2.05 + 1.66i)7-s + (−0.668 − 2.92i)9-s − 5.39·11-s + (1.89 − 3.28i)13-s + (3.67 − 4.61i)15-s + (0.411 − 0.713i)17-s + (−0.233 − 0.404i)19-s + (4.47 − 0.993i)21-s + 5.49·23-s + 6.61·25-s + (−4.68 − 2.25i)27-s + (0.400 + 0.693i)29-s + (−4.95 − 8.57i)31-s + ⋯
L(s)  = 1  + (0.623 − 0.781i)3-s + 1.52·5-s + (0.778 + 0.628i)7-s + (−0.222 − 0.974i)9-s − 1.62·11-s + (0.525 − 0.910i)13-s + (0.949 − 1.19i)15-s + (0.0999 − 0.173i)17-s + (−0.0535 − 0.0928i)19-s + (0.976 − 0.216i)21-s + 1.14·23-s + 1.32·25-s + (−0.901 − 0.433i)27-s + (0.0743 + 0.128i)29-s + (−0.889 − 1.54i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.647 + 0.762i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.647 + 0.762i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.647 + 0.762i$
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: \((2,\ 1008,\ (\ :1/2),\ 0.647 + 0.762i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.613211600\)
\(L(\frac12)\) \(\approx\) \(2.613211600\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.07 + 1.35i)T \)
7 \( 1 + (-2.05 - 1.66i)T \)
good5 \( 1 - 3.40T + 5T^{2} \)
11 \( 1 + 5.39T + 11T^{2} \)
13 \( 1 + (-1.89 + 3.28i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.411 + 0.713i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.233 + 0.404i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 5.49T + 23T^{2} \)
29 \( 1 + (-0.400 - 0.693i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.95 + 8.57i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4.34 - 7.52i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.84 + 3.19i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.36 - 7.55i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (5.24 - 9.09i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.71 - 8.17i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.830 + 1.43i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.474 - 0.821i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.269 - 0.466i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 3.86T + 71T^{2} \)
73 \( 1 + (-2.58 + 4.48i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.91 + 6.78i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.79 - 6.57i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (3.73 + 6.46i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (3.22 + 5.58i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.641063590994462914723501431350, −9.059027805196488401363743223550, −8.047158342004833256104431969866, −7.63123865152854624092439949005, −6.24874465155756185422568099221, −5.69419146368877466081263791183, −4.88067383396429276877831303765, −2.92869907479886315125159543992, −2.42347739978705435834845834432, −1.26853425228062516009738368619, 1.70694695562936896727513000655, 2.57825555417249995743526973802, 3.83154025591059649601791729239, 5.08893475891802652453881442312, 5.35988019390766492310768587333, 6.72845229209797222047666556697, 7.71700710064666152938873522084, 8.614202602787147816076450183788, 9.277755125343755175286651640004, 10.13293995408588120696586582294

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