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Properties

Label 2-504-504.461-c1-0-8
Degree $2$
Conductor $504$
Sign $-0.617 + 0.786i$
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 + (−0.420 + 1.68i)3-s + (0.999 + 1.73i)4-s + (−3.87 + 2.23i)5-s + (1.70 − 1.76i)6-s + (−1.32 + 2.29i)7-s − 2.82i·8-s + (−2.64 − 1.41i)9-s + 6.32·10-s + (−3.33 + 0.951i)12-s + (2.79 + 4.84i)13-s + (3.24 − 1.87i)14-s + (−2.12 − 7.44i)15-s + (−2.00 + 3.46i)16-s + (2.24 + 3.60i)18-s − 3.48·19-s + ⋯
L(s)  = 1  + (−0.866 − 0.499i)2-s + (−0.242 + 0.970i)3-s + (0.499 + 0.866i)4-s + (−1.73 + 0.999i)5-s + (0.695 − 0.718i)6-s + (−0.499 + 0.866i)7-s − 0.999i·8-s + (−0.881 − 0.471i)9-s + 1.99·10-s + (−0.961 + 0.274i)12-s + (0.775 + 1.34i)13-s + (0.866 − 0.499i)14-s + (−0.549 − 1.92i)15-s + (−0.500 + 0.866i)16-s + (0.528 + 0.849i)18-s − 0.799·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.617 + 0.786i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.617 + 0.786i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.617 + 0.786i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (461, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ -0.617 + 0.786i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0820056 - 0.168604i\)
\(L(\frac12)\) \(\approx\) \(0.0820056 - 0.168604i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.420 - 1.68i)T \)
7 \( 1 + (1.32 - 2.29i)T \)
good5 \( 1 + (3.87 - 2.23i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.79 - 4.84i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + 3.48T + 19T^{2} \)
23 \( 1 + (-1.25 + 0.727i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + (12.4 - 7.21i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.62 + 11.4i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 16.5iT - 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 + (6.02 - 10.4i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-1.21 - 0.700i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + (48.5 + 84.0i)T^{2} \)
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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

−11.27638712527229625336509068250, −10.80855487058536437403952499477, −9.786731338750599823628426398610, −8.802372881566656091159080636629, −8.294477749925115796498336977245, −6.98717572289906048803461253869, −6.29265735492533646752606880935, −4.33779552161069299712867745464, −3.63364982334904129979536069265, −2.67781183079577348752917847655, 0.18260844250930219157382698140, 1.11611866935123965278639067781, 3.34042494261648275229226160237, 4.75574649511472559113520479320, 5.94959289258848546622747537539, 7.03024932515937233609701094891, 7.76228865021694401480312894364, 8.245873230117232997875531598108, 9.053751320080799696934935625144, 10.56577895656229179743346481059

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