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Properties

Label 2-504-168.5-c1-0-5
Degree $2$
Conductor $504$
Sign $0.999 + 0.0285i$
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  + (−1.22 − 0.707i)2-s + (0.999 + 1.73i)4-s + (−2.87 − 1.65i)5-s + (2.5 + 0.866i)7-s − 2.82i·8-s + (2.34 + 4.06i)10-s + (2.87 + 4.97i)11-s − 4.69·13-s + (−2.44 − 2.82i)14-s + (−2.00 + 3.46i)16-s + (1.22 + 2.12i)17-s + (2.34 − 4.06i)19-s − 6.63i·20-s − 8.12i·22-s + (1.22 + 0.707i)23-s + ⋯
L(s)  = 1  + (−0.866 − 0.499i)2-s + (0.499 + 0.866i)4-s + (−1.28 − 0.741i)5-s + (0.944 + 0.327i)7-s − 0.999i·8-s + (0.741 + 1.28i)10-s + (0.866 + 1.50i)11-s − 1.30·13-s + (−0.654 − 0.755i)14-s + (−0.500 + 0.866i)16-s + (0.297 + 0.514i)17-s + (0.538 − 0.931i)19-s − 1.48i·20-s − 1.73i·22-s + (0.255 + 0.147i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0285i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0285i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.999 + 0.0285i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (341, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ 0.999 + 0.0285i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.822876 - 0.0117291i\)
\(L(\frac12)\) \(\approx\) \(0.822876 - 0.0117291i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.22 + 0.707i)T \)
3 \( 1 \)
7 \( 1 + (-2.5 - 0.866i)T \)
good5 \( 1 + (2.87 + 1.65i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.87 - 4.97i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 4.69T + 13T^{2} \)
17 \( 1 + (-1.22 - 2.12i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.34 + 4.06i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.22 - 0.707i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 5.74T + 29T^{2} \)
31 \( 1 + (-4.5 + 2.59i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-7.03 - 4.06i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 9.79T + 41T^{2} \)
43 \( 1 - 8.12iT - 43T^{2} \)
47 \( 1 + (3.67 - 6.36i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.87 + 4.97i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.87 + 1.65i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.34 + 4.06i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 1.41iT - 71T^{2} \)
73 \( 1 + (3 - 1.73i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.5 + 6.06i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 3.31iT - 83T^{2} \)
89 \( 1 + (4.89 - 8.48i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 1.73iT - 97T^{2} \)
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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

−11.12904325984422283425212641841, −9.782390367338062443730789481077, −9.264987279654103152555301292183, −8.081018069702570658364956901354, −7.73993356308687919649677580398, −6.75728123597318013600533391308, −4.73341736071739413864982616358, −4.31682890311997059742172817592, −2.62782326270394987387772824794, −1.15987594951749495044191543573, 0.836239326669811585426799495341, 2.83576822472231661314703626669, 4.18738986204643616292712871544, 5.45846220430211546809866322046, 6.67205706674191253499150518809, 7.52348295859090884703434695187, 8.037138237342509920427442830652, 8.922076360355870096559829613115, 10.10283060153389360272528899839, 10.90477669685708528667316231044

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