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Properties

Label 2-504-63.16-c1-0-9
Degree $2$
Conductor $504$
Sign $0.643 - 0.765i$
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  + (−1.40 + 1.01i)3-s + 3.84·5-s + (0.676 + 2.55i)7-s + (0.953 − 2.84i)9-s + 1.80·11-s + (−0.692 − 1.19i)13-s + (−5.40 + 3.88i)15-s + (−0.833 − 1.44i)17-s + (−0.0802 + 0.138i)19-s + (−3.53 − 2.91i)21-s + 3.20·23-s + 9.75·25-s + (1.53 + 4.96i)27-s + (−3.78 + 6.54i)29-s + (−1.61 + 2.78i)31-s + ⋯
L(s)  = 1  + (−0.811 + 0.584i)3-s + 1.71·5-s + (0.255 + 0.966i)7-s + (0.317 − 0.948i)9-s + 0.544·11-s + (−0.192 − 0.332i)13-s + (−1.39 + 1.00i)15-s + (−0.202 − 0.350i)17-s + (−0.0184 + 0.0318i)19-s + (−0.772 − 0.635i)21-s + 0.667·23-s + 1.95·25-s + (0.295 + 0.955i)27-s + (−0.701 + 1.21i)29-s + (−0.289 + 0.500i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.643 - 0.765i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.643 - 0.765i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.643 - 0.765i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (457, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ 0.643 - 0.765i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.39037 + 0.647046i\)
\(L(\frac12)\) \(\approx\) \(1.39037 + 0.647046i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.40 - 1.01i)T \)
7 \( 1 + (-0.676 - 2.55i)T \)
good5 \( 1 - 3.84T + 5T^{2} \)
11 \( 1 - 1.80T + 11T^{2} \)
13 \( 1 + (0.692 + 1.19i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (0.833 + 1.44i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.0802 - 0.138i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 3.20T + 23T^{2} \)
29 \( 1 + (3.78 - 6.54i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.61 - 2.78i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.58 + 2.74i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-6.00 - 10.3i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.45 + 5.98i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (5.71 + 9.90i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.37 - 2.38i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (7.53 - 13.0i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.60 - 7.96i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.16 + 10.6i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 6.93T + 71T^{2} \)
73 \( 1 + (6.22 + 10.7i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (8.03 + 13.9i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.45 - 2.51i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-5.04 + 8.73i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-4.18 + 7.25i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.90135954527524634914616329232, −10.17546933122065857243629394986, −9.213314656216958133487151506135, −8.952066453788774060402109403781, −7.09221973557845614970021182640, −6.08837384213536255725632658061, −5.52936020995091292345988918560, −4.72793504412673048047904097407, −2.98659860882191519621250600529, −1.58962396303074821694752084904, 1.20717093618910807321623518352, 2.26476034277108504524132064076, 4.26345854055238837512642820556, 5.35462617561607626592055624742, 6.22663492472924622371332876568, 6.87050840427494049130761487756, 7.904087570184520968953435812321, 9.305356414489593547075518775236, 9.943685348718405237321151291999, 10.87275246214999305600427201191

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