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Properties

Label 2-504-63.47-c1-0-5
Degree $2$
Conductor $504$
Sign $0.931 - 0.364i$
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  + (−0.859 − 1.50i)3-s − 4.19·5-s + (1.67 + 2.05i)7-s + (−1.52 + 2.58i)9-s + 1.66i·11-s + (5.64 − 3.25i)13-s + (3.60 + 6.31i)15-s + (1.45 + 2.52i)17-s + (−2.39 − 1.38i)19-s + (1.64 − 4.27i)21-s + 2.21i·23-s + 12.6·25-s + (5.19 + 0.0719i)27-s + (5.69 + 3.28i)29-s + (0.414 + 0.239i)31-s + ⋯
L(s)  = 1  + (−0.495 − 0.868i)3-s − 1.87·5-s + (0.631 + 0.775i)7-s + (−0.507 + 0.861i)9-s + 0.503i·11-s + (1.56 − 0.903i)13-s + (0.931 + 1.63i)15-s + (0.353 + 0.611i)17-s + (−0.549 − 0.317i)19-s + (0.359 − 0.933i)21-s + 0.462i·23-s + 2.52·25-s + (0.999 + 0.0138i)27-s + (1.05 + 0.610i)29-s + (0.0743 + 0.0429i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.931 - 0.364i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.931 - 0.364i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.931 - 0.364i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (425, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ 0.931 - 0.364i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.837471 + 0.158162i\)
\(L(\frac12)\) \(\approx\) \(0.837471 + 0.158162i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.859 + 1.50i)T \)
7 \( 1 + (-1.67 - 2.05i)T \)
good5 \( 1 + 4.19T + 5T^{2} \)
11 \( 1 - 1.66iT - 11T^{2} \)
13 \( 1 + (-5.64 + 3.25i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.45 - 2.52i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.39 + 1.38i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 2.21iT - 23T^{2} \)
29 \( 1 + (-5.69 - 3.28i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.414 - 0.239i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.378 - 0.655i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.769 + 1.33i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.79 - 8.31i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-4.05 - 7.02i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-7.11 + 4.10i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.426 + 0.739i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.89 + 2.25i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (7.69 - 13.3i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 1.89iT - 71T^{2} \)
73 \( 1 + (6.22 - 3.59i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-5.52 - 9.57i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.162 + 0.280i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (2.86 - 4.95i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-4.22 - 2.44i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.20690982199672551177881461348, −10.54612505454489316474614176705, −8.563262469579732299363903957733, −8.304999096505390650807952217411, −7.49010299519081059764215143253, −6.46751069699311217991907281169, −5.37408760288449801748286176120, −4.27344300802913923163287278708, −3.01205929788154675453083638447, −1.18804078480875938786445309643, 0.68632417077846591462601516281, 3.48428363869807347602979036832, 4.06396362558065203691606147531, 4.81759369736839114333841652772, 6.28172376127179635696480982836, 7.26968727536009266980891685259, 8.381390373490816822098459588459, 8.791131655319864220936979857665, 10.40950617989071682392797004416, 10.87429955818018037062424413044

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