Properties

Label 2-504-63.41-c1-0-21
Degree $2$
Conductor $504$
Sign $-0.979 - 0.200i$
Analytic cond. $4.02446$
Root an. cond. $2.00610$
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 − 0.0591i)3-s + (1.11 − 1.92i)5-s + (−2.58 + 0.566i)7-s + (2.99 + 0.204i)9-s + (−1.51 + 0.876i)11-s + (−3.37 − 1.94i)13-s + (−2.04 + 3.27i)15-s − 1.78·17-s + 2.73i·19-s + (4.50 − 0.828i)21-s + (1.64 + 0.947i)23-s + (0.0196 + 0.0339i)25-s + (−5.16 − 0.531i)27-s + (−7.48 + 4.32i)29-s + (−7.18 − 4.14i)31-s + ⋯
L(s)  = 1  + (−0.999 − 0.0341i)3-s + (0.498 − 0.862i)5-s + (−0.976 + 0.214i)7-s + (0.997 + 0.0682i)9-s + (−0.457 + 0.264i)11-s + (−0.935 − 0.540i)13-s + (−0.527 + 0.845i)15-s − 0.433·17-s + 0.628i·19-s + (0.983 − 0.180i)21-s + (0.342 + 0.197i)23-s + (0.00392 + 0.00678i)25-s + (−0.994 − 0.102i)27-s + (−1.39 + 0.802i)29-s + (−1.29 − 0.745i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.979 - 0.200i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.979 - 0.200i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.979 - 0.200i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ -0.979 - 0.200i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00683115 + 0.0676058i\)
\(L(\frac12)\) \(\approx\) \(0.00683115 + 0.0676058i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.73 + 0.0591i)T \)
7 \( 1 + (2.58 - 0.566i)T \)
good5 \( 1 + (-1.11 + 1.92i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.51 - 0.876i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (3.37 + 1.94i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 1.78T + 17T^{2} \)
19 \( 1 - 2.73iT - 19T^{2} \)
23 \( 1 + (-1.64 - 0.947i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (7.48 - 4.32i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (7.18 + 4.14i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 9.91T + 37T^{2} \)
41 \( 1 + (3.42 - 5.93i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.22 + 7.32i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.47 - 2.54i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 5.45iT - 53T^{2} \)
59 \( 1 + (0.449 - 0.778i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-10.3 + 5.98i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.38 + 9.32i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 14.2iT - 71T^{2} \)
73 \( 1 - 4.71iT - 73T^{2} \)
79 \( 1 + (4.70 + 8.15i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.326 + 0.565i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 9.70T + 89T^{2} \)
97 \( 1 + (0.0294 - 0.0169i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.31721767460504724566344401948, −9.712560767232351843593011725681, −8.895841548964818649950282706868, −7.51115238067146187956445080133, −6.69229845083032892771460914743, −5.43021723147484938594993424666, −5.20542745266517869274525049570, −3.65458328032038771452539649094, −1.89739107145504849838108657369, −0.04250814503588927511363965369, 2.25155718178267877435070442881, 3.60521721180190406209106435338, 4.96463456641562405061988358012, 5.92117578599734678989054895298, 6.88446695863059881374108555596, 7.22694348312173015868581467998, 9.005621700648743069925250832220, 9.877245396431039813927590759267, 10.52230623138084847268466230954, 11.21237489291331348551417700602

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