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Properties

Label 2-504-63.20-c1-0-7
Degree $2$
Conductor $504$
Sign $0.985 + 0.171i$
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  + (−1.01 + 1.40i)3-s + (−1.60 − 2.77i)5-s + (−1.96 + 1.77i)7-s + (−0.927 − 2.85i)9-s + (4.13 + 2.38i)11-s + (0.861 − 0.497i)13-s + (5.52 + 0.580i)15-s + 6.51·17-s − 5.38i·19-s + (−0.487 − 4.55i)21-s + (6.56 − 3.79i)23-s + (−2.63 + 4.57i)25-s + (4.94 + 1.60i)27-s + (−2.93 − 1.69i)29-s + (−3.51 + 2.02i)31-s + ⋯
L(s)  = 1  + (−0.587 + 0.809i)3-s + (−0.716 − 1.24i)5-s + (−0.741 + 0.670i)7-s + (−0.309 − 0.951i)9-s + (1.24 + 0.720i)11-s + (0.239 − 0.137i)13-s + (1.42 + 0.149i)15-s + 1.57·17-s − 1.23i·19-s + (−0.106 − 0.994i)21-s + (1.36 − 0.790i)23-s + (−0.527 + 0.914i)25-s + (0.951 + 0.309i)27-s + (−0.545 − 0.314i)29-s + (−0.630 + 0.364i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.171i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.985 + 0.171i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.985 + 0.171i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (209, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ 0.985 + 0.171i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.999459 - 0.0864686i\)
\(L(\frac12)\) \(\approx\) \(0.999459 - 0.0864686i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.01 - 1.40i)T \)
7 \( 1 + (1.96 - 1.77i)T \)
good5 \( 1 + (1.60 + 2.77i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-4.13 - 2.38i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.861 + 0.497i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 6.51T + 17T^{2} \)
19 \( 1 + 5.38iT - 19T^{2} \)
23 \( 1 + (-6.56 + 3.79i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.93 + 1.69i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.51 - 2.02i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 6.29T + 37T^{2} \)
41 \( 1 + (-3.48 - 6.04i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.81 + 6.60i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (3.78 - 6.54i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 7.77iT - 53T^{2} \)
59 \( 1 + (-2.25 - 3.90i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.26 + 3.03i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.493 - 0.854i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 3.20iT - 71T^{2} \)
73 \( 1 + 2.63iT - 73T^{2} \)
79 \( 1 + (7.96 - 13.7i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-4.49 + 7.77i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 4.42T + 89T^{2} \)
97 \( 1 + (-3.13 - 1.81i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.06336298872652245752783296792, −9.715679421477171264518796978049, −9.254737074792388127059883319070, −8.554023802201810211086615660581, −7.16823988714412639571338641757, −6.08043160854568911192708094478, −5.08029413676313296078377418810, −4.30525684709904151275802815602, −3.23214446094075576794176014079, −0.853833465691729268031606296604, 1.14316079001768499421946103283, 3.17639407696988173192552603369, 3.83273717996095260701197312620, 5.72635129766343610244364802045, 6.42371874679691591860651387500, 7.31657781777011850926602870245, 7.76870414170918864247206111202, 9.232297004104568586762311359298, 10.34136027307821440394167556458, 11.09123717060981098396953312239

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